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2.16k
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86
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401 |
Semiconductors
|
A thermistor made of intrinsic silicon material has a resistance value of $500 \Omega$ at 290 K. Assuming the band gap of silicon $E_{\mathrm{q}}=1.12 \mathrm{eV}$ and does not change with temperature, if we assume the carrier mobility remains unchanged, try to estimate the approximate value of the thermistor at 325 K.
|
[] |
Numeric
|
{"$E_{\\mathrm{q}}$": "band gap of silicon", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$q$": "elementary charge", "$\\mu_{\\mathrm{n}}$": "mobility of electrons", "$\\mu_{\\mathrm{p}}$": "mobility of holes", "$T$": "temperature", "$N_{\\mathrm{c}}$": "effective density of states in conduction band", "$N_{\\mathrm{v}}$": "effective density of states in valence band", "$k_{0}$": "Boltzmann constant", "$R$": "resistance", "$C$": "proportionality constant related to resistance"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
402 |
Semiconductors
|
The resistivity of intrinsic germanium material with temperature $T$ can be tabulated as follows:
\begin{tabular}{|c|c|c|c|c|}
\hline$T(\mathrm{~K})$ & 385 & 458 & 556 & 714 \\
\hline$\rho(\Omega \cdot \mathrm{~cm})$ & 0.028 & 0.0061 & 0.0013 & 0.00027 \\
\hline
\end{tabular}
Assume $E_{\mathrm{g}}$ is independent of temperature $T$, and the mobilities of electrons and holes $\mu_{\mathrm{n}} \mu_{\mathrm{p}}$ both vary as $T^{-\frac{3}{2}}$. Find the band gap $E_{\mathrm{g}}$ of germanium.
|
[] |
Numeric
|
{"$T$": "temperature", "$E_{\\mathrm{g}}$": "band gap of germanium", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$\\sigma_{\\mathrm{i}}$": "intrinsic conductivity", "$\\rho_{\\mathrm{i}}$": "intrinsic resistivity", "$k_{0}$": "Boltzmann constant", "$C$": "constant"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
403 |
Semiconductors
|
Calculate the resistivity of intrinsic silicon at room temperature (unit $\Omega \cdot \mathrm{~cm}$). It is known that the electron mobility of intrinsic silicon at room temperature is $1350 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, the hole mobility is $500 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and the intrinsic carrier concentration is $n_{\mathrm{i}}=1.5 \times 10^{10} / \mathrm{cm}^{3}$, elementary charge $q=1.6 \times 10^{-19} \mathrm{C}$.
|
[] |
Numeric
|
{"$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$q$": "elementary charge", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$\\sigma_{\\mathrm{i}}$": "intrinsic conductivity", "$N_{\\mathrm{D}}$": "concentration of the dopant", "$n_{0}$": "majority carrier concentration", "$N_{\\mathrm{i}}$": "impurity concentration", "$\\sigma$": "conductivity"}
|
Conductivity of semiconductors
|
Conductivity of semiconductors
|
|
404 |
Semiconductors
|
For a certain n-type semiconductor silicon with a doping concentration $N_{\mathrm{D}}=10^{15} \mathrm{~cm}^{-3}$, and a minority carrier lifetime $\tau_{\mathrm{p}}=5 \mu \mathrm{~s}$, if due to external influences all minority carriers are removed (such as near a reverse-biased pn junction), what is the electron-hole generation rate at this time (let $n_{\mathrm{i}}=1.5 \times 10^{10} \mathrm{~cm}^{-3}$)?
|
[] |
Numeric
|
{"$N_{\\mathrm{D}}$": "doping concentration", "$\\tau_{\\mathrm{p}}$": "minority carrier lifetime", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$p$": "minority carrier concentration", "$p_{0}$": "equilibrium minority carrier concentration", "$\\Delta_{p}$": "change in minority carrier concentration", "$R$": "recombination rate"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
405 |
Semiconductors
|
The concentration of copper in a copper-doped germanium sample is $10^{15} \mathrm{~cm}^{-3}$, and the concentration of antimony is $10^{17} \mathrm{~cm}^{-3}$. Its minority carrier lifetime measured under small injection conditions is $10^{-7} \mathrm{~s}^{-1}$. Given that $N_{\mathrm{c}}=1.04 \times 10^{19} \mathrm{~cm}^{-1}$. If the effective mass of holes in germanium $m_{\mathrm{p}}^{*}=0.30 m_{0}(m_{0}$ is the free electron mass), find the hole capture cross-section?
|
[] |
Numeric
|
{"$m_{\\mathrm{p}}^{*}$": "effective mass of holes in germanium", "$m_{0}$": "free electron mass", "$\\gamma_{\\mathrm{p}}$": "hole capture coefficient", "$N_{\\mathrm{t}}$": "trap concentration", "$\\tau_{\\mathrm{p}}$": "hole capture time", "$\\sigma_{\\mathrm{p}}$": "hole capture cross-section", "$v_{\\mathrm{t}}$": "thermal velocity of holes", "$k_{0}$": "Boltzmann constant", "$T$": "temperature"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
406 |
Semiconductors
|
In a piece of p-type semiconductor, there exists a recombination-generation center. When slightly doped, the electrons captured by these centers are reemitted to the conduction band with the same probability as their recombination with holes. Try to find the energy level position of this recombination-generation center. You should return your answer as an equation.
|
[] |
Equation
|
{"$n_{1}$": "number of conduction band electrons from recombination centers", "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", "$E_{\\mathrm{c}}$": "energy level of the conduction band", "$E_{\\mathrm{t}}$": "energy level position of the recombination-generation center", "$k_{0}$": "Boltzmann constant", "$T$": "absolute temperature", "$s_{-}$": "electron excitation probability", "$n_{\\mathrm{t}}$": "number of electrons at the recombination-generation center", "$r_{\\mathrm{n}}$": "electron capture probability", "$r_{\\mathrm{p}}$": "hole capture probability", "$p$": "hole concentration", "$p_{0}$": "equilibrium hole concentration", "$\\Delta p$": "excess hole concentration", "$N_{\\mathrm{v}}$": "effective density of states in the valence band", "$E_{\\mathrm{v}}$": "energy level of the valence band", "$E_{\\mathrm{F}}$": "Fermi energy level", "$E_{\\mathrm{i}}$": "intrinsic energy level"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
407 |
Semiconductors
|
Illuminating an n-type silicon sample with a resistivity of $1 \Omega \cdot \mathrm{~cm}$, non-equilibrium carriers are uniformly generated, with the generation rate of electron-hole pairs being $10^{17} \mathrm{~cm}^{-3} \cdot \mathrm{~s}^{-1}$. Assume the lifetime of the sample is $10 \mu \mathrm{~s}$, and the surface recombination velocity is $100 \mathrm{~cm} / \mathrm{s}$. Calculate: The number of holes recombined at the surface per unit time per unit surface area;
|
[] |
Numeric
|
{"$u_{\\mathrm{s}}$": "recombination rate", "$\\nu_{\\mathrm{s}}$": "recombination rate at the surface", "$s_{\\mathrm{p}}$": "surface recombination velocity", "$p(x)$": "hole concentration as a function of x", "$p_{0}$": "equilibrium hole concentration", "$\\tau_{\\mathrm{p}}$": "hole lifetime", "$g_{\\mathrm{p}}$": "generation rate of holes", "$L_{\\mathrm{p}}$": "diffusion length of holes", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$N_{\\mathrm{D}}$": "donor concentration"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
408 |
Semiconductors
|
Illuminate a $1 \Omega \cdot \mathrm{~cm}$ n-type silicon sample, uniformly generating non-equilibrium carriers, with an electron-hole pair generation rate of $10^{17} \mathrm{~cm}^{-3} \cdot \mathrm{~s}^{-1}$. Assume the sample's lifetime is $10 \mu \mathrm{~s}$, and the surface recombination velocity is $100 \mathrm{~cm} / \mathrm{s}$. Calculate: The number of holes recombined within three diffusion lengths from the surface, per unit time and per unit surface area.
|
[] |
Numeric
|
{"$\\Delta p$": "change in hole concentration", "$L_{\\mathrm{p}}$": "diffusion length of holes", "$p(x)$": "hole concentration as a function of position", "$p_{0}$": "equilibrium hole concentration", "$\\tau_{\\mathrm{p}}$": "hole lifetime", "$g_{\\mathrm{p}}$": "hole generation rate", "$s_{\\mathrm{p}}$": "surface recombination velocity of holes"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
409 |
Semiconductors
|
A silicon wafer with a donor concentration of $2 \times 10^{16} \mathrm{~cm}^{-3}$ is saturated with gold at $920^{\circ} \mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \mathrm{~cm}^{-2}$. Calculate the bulk lifetime;
|
[] |
Numeric
|
{"$\\tau$": "bulk lifetime", "$r_{\\mathrm{p}}$": "hole capture rate", "$N_{\\mathrm{t}}$": "trap concentration", "$\\gamma_{\\mathrm{p}}$": "hole capture rate of gold", "$\\mu_{\\mathrm{p}}$": "mobility", "$N_{\\mathrm{i}}$": "total impurity concentration", "$N_{\\mathrm{D}}$": "donor concentration", "$D_{\\mathrm{p}}$": "diffusion coefficient", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$q$": "elementary charge", "$L_{\\mathrm{p}}$": "diffusion length", "$\\tau_{\\mathrm{p}}$": "carrier lifetime", "$s_{\\mathrm{p}}$": "surface recombination velocity", "$N_{\\mathrm{st}}$": "surface trap density"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
410 |
Semiconductors
|
A silicon wafer with a donor concentration of $2 \times 10^{16} \mathrm{~cm}^{-3}$ is saturated with gold at $920^{\circ} \mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \mathrm{~cm}^{-2}$. Calculate the surface recombination velocity;
|
[] |
Numeric
|
{"$\\tau$": "bulk lifetime", "$r_{\\mathrm{p}}$": "hole recombination rate", "$N_{\\mathrm{t}}$": "trap density", "$\\gamma_{\\mathrm{p}}$": "hole capture rate of gold", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$N_{\\mathrm{i}}$": "total impurity concentration", "$N_{\\mathrm{D}}$": "donor concentration", "$D_{\\mathrm{p}}$": "hole diffusion coefficient", "$L_{\\mathrm{p}}$": "hole diffusion length", "$s_{\\mathrm{p}}$": "surface recombination velocity", "$N_{\\mathrm{st}}$": "surface trap density"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
411 |
Semiconductors
|
A silicon wafer with a dopant concentration of $2 \times 10^{16} \mathrm{~cm}^{-3}$ is gold-doped to saturation concentration at $920^{\circ} \mathrm{C}$. After oxidation and other treatments, the surface recombination center of the silicon wafer is $10^{10} \mathrm{~cm}^{-2}$. If the silicon wafer is uniformly illuminated, and the generation rate of electron-hole pairs is $10^{11} \mathrm{~cm}^{-3} \cdot \mathrm{~s}^{-1}$, what is the hole concentration at the surface?
|
[] |
Numeric
|
{"$p(x)$": "hole concentration at position x", "$p_{0}$": "equilibrium hole concentration", "$\\tau_{\\mathrm{p}}$": "hole lifetime", "$g_{\\mathrm{p}}$": "generation rate of holes", "$s_{\\mathrm{p}}$": "surface recombination velocity for holes", "$L_{\\mathrm{p}}$": "diffusion length for holes", "$x$": "distance from the surface", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$n_{0}$": "equilibrium electron concentration", "$N_{\\mathrm{D}}$": "donor concentration", "$N_{\\mathrm{t}}$": "trap concentration", "$J_{\\mathrm{p}}$": "hole current density"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
412 |
Semiconductors
|
A silicon wafer doped with a donor concentration of $2 \times 10^{16} \mathrm{~cm}^{-3}$ is saturated with gold at $920^{\circ} \mathrm{C}$. After oxidation and other treatments, the surface recombination center of this silicon wafer is $10^{10} \mathrm{~cm}^{-2}$. If the silicon wafer is illuminated and uniformly absorbed by the sample, the generation rate of electron-hole pairs is $10^{11} \mathrm{~cm}^{-3} \cdot \mathrm{~s}^{-1}$. What is the hole current density flowing towards the surface?
|
[] |
Numeric
|
{"$p$": "hole density", "$p_{0}$": "equilibrium hole density", "$\\tau_{\\mathrm{p}}$": "hole lifetime", "$g_{\\mathrm{p}}$": "generation rate of holes", "$s_{\\mathrm{p}}$": "surface recombination velocity", "$L_{\\mathrm{p}}$": "diffusion length of holes", "$n_{0}$": "equilibrium electron density", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$N_{\\mathrm{D}}$": "donor concentration", "$N_{\\mathrm{t}}$": "trap concentration", "$J_{\\mathrm{p}}$": "hole current density"}
|
Non-equilibrium carriers
|
Non-equilibrium carriers
|
|
413 |
Semiconductors
|
A pn junction composed of p-type germanium with a resistivity of $1 \Omega \cdot \mathrm{~cm}$ and n-type germanium with a resistivity of $0.1 \Omega \cdot \mathrm{~cm}$, calculate the built-in potential difference $V_{\mathrm{D}}$ at room temperature (300 K). Given that at the above resistivities, the hole mobility in the p-type region $\mu_{\mathrm{p}}=1650 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, electron mobility in the n-type region $\mu_{\mathrm{n}}=3000 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and the intrinsic carrier concentration of germanium $n_{\mathrm{i}}=2.5 \times 10^{13} \mathrm{~cm}^{-3}$.
|
[] |
Numeric
|
{"$V_{\\mathrm{D}}$": "built-in potential difference", "$\\mu_{\\mathrm{p}}$": "hole mobility in the p-type region", "$\\mu_{\\mathrm{n}}$": "electron mobility in the n-type region", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration of germanium", "$N_{\\mathrm{A}}$": "acceptor concentration in p-type region", "$N_{\\mathrm{D}}$": "donor concentration in n-type region"}
|
pn junction
|
pn junction
|
|
414 |
Semiconductors
|
A pn junction composed of p-type germanium with resistivity $1 \Omega \cdot \mathrm{~cm}$ and n-type germanium with resistivity $0.1 \Omega \cdot \mathrm{~cm}$, calculate the width of the depletion region at room temperature (300 K). Given that at these resistivities, the hole mobility in the p region is $\mu_{\mathrm{p}}=1650 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and the electron mobility in the n region is $\mu_{\mathrm{n}}=3000 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, the intrinsic carrier concentration of germanium $n_{\mathrm{i}}=2.5 \times 10^{13} \mathrm{~cm}^{-3} $.
|
[] |
Numeric
|
{"$\\mu_{\\mathrm{p}}$": "hole mobility in the p region", "$\\mu_{\\mathrm{n}}$": "electron mobility in the n region", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration of germanium", "$p_{\\mathrm{p} 0}$": "hole concentration in the p-type region at equilibrium", "$N_{\\mathrm{A}}$": "acceptor concentration in the p-type region", "$q$": "elementary charge", "$\\rho_{\\mathrm{p}}$": "resistivity of p-type germanium", "$n_{n 0}$": "electron concentration in the n-type region at equilibrium", "$N_{\\mathrm{D}}$": "donor concentration in the n-type region", "$\\rho_{\\mathrm{n}}$": "resistivity of n-type germanium", "$V_{\\mathrm{D}}$": "built-in potential difference", "$k_{0}$": "Boltzmann constant", "$T$": "absolute temperature", "$x_{\\mathrm{p}}$": "depletion width in the p-type region", "$\\varepsilon$": "permittivity", "$\\varepsilon_{0}$": "vacuum permittivity", "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity of germanium", "$x_{\\mathrm{n}}$": "depletion width in the n-type region", "$x$": "total depletion width"}
|
pn junction
|
pn junction
|
|
415 |
Semiconductors
|
Given a silicon abrupt junction, with resistivities on both sides being $\rho_{\mathrm{n}}=10 \Omega \cdot \mathrm{~cm}$ for $\mathrm{n}-\mathrm{Si}$ and $\rho_{\mathrm{p}}=0.01 \Omega \cdot \mathrm{~cm}$ for $\mathrm{p}-\mathrm{Si}$, and mobilities $\mu_{\mathrm{n}}=100 \mathrm{~cm}^{2}/(\mathrm{V} \cdot \mathrm{s}), \mu_{\mathrm{p}}=300 \mathrm{~cm}^{2}/(\mathrm{V} \cdot \mathrm{s})$, find the barrier width at room temperature.
|
[] |
Numeric
|
{"$\\rho_{\\mathrm{n}}$": "resistivity of n-type silicon", "$\\rho_{\\mathrm{p}}$": "resistivity of p-type silicon", "$\\mu_{\\mathrm{n}}$": "mobility of electrons in n-type silicon", "$\\mu_{\\mathrm{p}}$": "mobility of holes in p-type silicon", "$q$": "charge of an electron", "$N_{\\mathrm{D}}$": "donor concentration", "$N_{\\mathrm{A}}$": "acceptor concentration", "$n_{\\mathrm{n} 0}$": "minority electron concentration in p-region", "$p_{\\mathrm{p} 0}$": "minority hole concentration in n-region", "$V_{\\mathrm{D}}$": "built-in potential", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$x_{\\mathrm{n}}$": "depletion width in n-region", "$x_{\\mathrm{D}}$": "total barrier width", "$\\varepsilon_{0}$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of silicon"}
|
pn junction
|
pn junction
|
|
416 |
Semiconductors
|
Assume a silicon abrupt junction with the impurity concentrations on both sides as $N_{\mathrm{A}}=10^{17} / \mathrm{cm}^{3}, N_{\mathrm{D}}=4.5 \times 10^{15} / \mathrm{cm}^{3}$. The intrinsic carrier concentration of silicon at room temperature is known as $n_{\mathrm{i}} = 1.5 \times 10^{10} / \mathrm{cm}^{3}$, vacuum permittivity $\varepsilon_{0} = 8.85 \times 10^{-14} \mathrm{F/cm}$, the relative permittivity of silicon $\varepsilon_{\mathrm{rs}} = 11.9$, elementary charge $q = 1.6 \times 10^{-19} \mathrm{C}$, thermal voltage $kT/q \approx 0.026 \mathrm{V}$. Determine the value of total depletion width $x_{\mathrm{D}}$ at zero applied voltage.
|
[] |
Numeric
|
{"$N_{\\mathrm{A}}$": "acceptor impurity concentration", "$N_{\\mathrm{D}}$": "donor impurity concentration", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration of silicon", "$\\varepsilon_{0}$": "vacuum permittivity", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of silicon", "$q$": "elementary charge", "$kT/q$": "thermal voltage", "$x_{\\mathrm{D}}$": "total depletion width", "$V_{\\mathrm{D}}$": "built-in potential", "$x_{\\mathrm{n}}$": "depletion width in the n-region", "$x_{\\mathrm{p}}$": "depletion width in the p-region", "$V_{\\mathrm{R}}$": "reverse bias voltage", "$V$": "applied voltage across the junction"}
|
pn junction
|
pn junction
|
|
417 |
Semiconductors
|
Suppose a silicon abrupt junction, with impurity concentrations on either side as $N_{\mathrm{A}}=10^{17} / \mathrm{cm}^{3}, N_{\mathrm{D}}=4.5 \times 10^{15} / \mathrm{cm}^{3}$, when a reverse bias voltage of 10 V is applied, find the value of $x_{\mathrm{D}}$.
|
[] |
Numeric
|
{"$N_{\\mathrm{A}}$": "acceptor impurity concentration", "$N_{\\mathrm{D}}$": "donor impurity concentration", "$x_{\\mathrm{D}}$": "depletion layer width", "$x_{\\mathrm{n}}$": "width of the depletion region on the n-side", "$x_{\\mathrm{p}}$": "width of the depletion region on the p-side", "$V_{\\mathrm{D}}$": "built-in potential of the diode", "$V_{\\mathrm{R}}$": "reverse bias voltage", "$\\varepsilon_{0}$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", "$q$": "elementary charge", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration"}
|
pn junction
|
pn junction
|
|
418 |
Semiconductors
|
Given that the impurity concentration gradient $\alpha_{\mathrm{j}}$ of the linear graded junction of silicon is $10^{22} / \mathrm{cm}^{4}, V_{\mathrm{D}}=0.68 \mathrm{~V}$, Find the value of the barrier width $x_{\mathrm{D}}$.
|
[] |
Numeric
|
{"$x_{\\mathrm{D}}$": "barrier width", "$\\varepsilon_{0}$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity", "$V_{\\mathrm{D}}$": "built-in potential", "$\\alpha_{\\mathrm{j}}$": "doping gradient", "$E_{\\max}$": "maximum electric field", "$q$": "elementary charge", "$V$": "external voltage"}
|
pn junction
|
pn junction
|
|
419 |
Semiconductors
|
Assume the impurity concentration gradient $\alpha_{\mathrm{j}}$ of silicon's linearly graded junction is $10^{22} / \mathrm{cm}^{4}, V_{\mathrm{D}}=0.68 \mathrm{~V}$, Find its maximum electric field,
|
[] |
Numeric
|
{"$x_{\\mathrm{D}}$": "barrier width of the junction", "$\\varepsilon_{0}$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", "$V_{\\mathrm{D}}$": "built-in voltage of the diode", "$q$": "elementary charge", "$\\alpha_{\\mathrm{j}}$": "gradient of the doping concentration", "$E_{\\max}$": "maximum electric field", "$V$": "applied voltage across the junction"}
|
pn junction
|
pn junction
|
|
420 |
Semiconductors
|
Assume the impurity concentration gradient $\alpha_{\mathrm{j}}$ of a silicon linear gradient junction is $10^{22} / \mathrm{cm}^{4}, V_{\mathrm{D}}=0.68 \mathrm{~V}$, Find the value of $x_{\mathrm{D}}$ when an external reverse bias of 10 V is applied.
|
[] |
Numeric
|
{"$x_{\\mathrm{D}}$": "barrier width of the linear gradient junction", "$\\varepsilon_{0}$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", "$V_{\\mathrm{D}}$": "built-in potential across the p-n junction", "$q$": "elementary charge", "$\\alpha_{\\mathrm{j}}$": "doping gradient coefficient", "$E_{\\max }$": "maximum electric field", "$V$": "external voltage"}
|
pn junction
|
pn junction
|
|
421 |
Semiconductors
|
For a silicon $n^+p$ junction, given $x_p = 0.2\ \mu\text{m}$, $L_n = 200\ \mu\text{m}$, $N_A = 10^{15}\ \text{cm}^{-3}$, $n_i = 1.5 \times 10^{10}\ \text{cm}^{-3}$, at room temperature $T = 300\ \text{K}$, find the value of voltage V corresponding to the condition where the barrier recombination current equals the diffusion current.
|
[] |
Numeric
|
{"$x_p$": "width of the p-type region", "$L_n$": "electron diffusion length", "$N_A$": "acceptor concentration", "$n_i$": "intrinsic carrier concentration", "$T$": "temperature", "$V$": "voltage", "$I_{\\mathrm{r}}$": "barrier recombination current", "$I_{\\mathrm{FD}}$": "diffusion current", "$q$": "elementary charge", "$k_{0}$": "Boltzmann constant"}
|
pn junction
|
pn junction
|
|
422 |
Semiconductors
|
The relationship between the barrier capacitance $C_{\mathrm{T}}$ of the $\mathrm{p}^{+} \mathrm{n}$ junction made from GaP material and the reverse voltage $V_{\mathrm{R}}$ is measured as follows
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline$V_{\mathrm{R}}(\mathrm{V})$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline$C_{\mathrm{T}}(\mathrm{pF})$ & 20 & 17.3 & 15.6 & 14.3 & 13.3 & 12.4 & 11.6 \\
\hline
\end{tabular}
The $pn$ junction area $A=4 \times 10^{-4} \mathrm{~cm}^{2}$, try to find the built-in potential $V_{\mathrm{D}}$ of the $\mathrm{p}^{+} \mathrm{n}$ junction.
|
[] |
Numeric
|
{"$C_{\\mathrm{T}}$": "barrier capacitance", "$V_{\\mathrm{R}}$": "reverse voltage", "$A$": "$pn$ junction area", "$V_{\\mathrm{D}}$": "built-in potential", "$N_{\\mathrm{A}}$": "acceptor concentration", "$N_{\\mathrm{D}}$": "donor concentration", "$B$": "constant related to material properties", "$q$": "elementary charge", "$\\varepsilon_{0}$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of GaP material"}
|
pn junction
|
pn junction
|
|
423 |
Semiconductors
|
The relationship between the barrier capacitance $C_{\mathrm{T}}$ and reverse voltage $V_{\mathrm{R}}$ of a $\mathrm{p}^{+} \mathrm{n}$ junction made of GaP material is measured as follows
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline$V_{\mathrm{R}}(\mathrm{V})$ & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline$C_{\mathrm{T}}(\mathrm{pF})$ & 20 & 17.3 & 15.6 & 14.3 & 13.3 & 12.4 & 11.6 \\
\hline
\end{tabular}
The pn junction area $A=4 \times 10^{-4} \mathrm{~cm}^{2}$, try to find the built-in field $N_{\mathrm{D}}$ of this $\mathrm{p}^{+} \mathrm{n}$ junction.
|
[] |
Numeric
|
{"$C_{\\mathrm{T}}$": "barrier capacitance", "$V_{\\mathrm{R}}$": "reverse voltage", "$A$": "pn junction area", "$N_{\\mathrm{D}}$": "built-in field of the junction", "$N_{\\mathrm{A}}$": "acceptor concentration", "$q$": "elementary charge", "$\\varepsilon_{0}$": "vacuum permittivity", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity", "$V_{\\mathrm{D}}$": "diffusion potential", "$B$": "constant related to capacitance and voltage"}
|
pn junction
|
pn junction
|
|
424 |
Semiconductors
|
A pn junction diode has the following parameters: $N_{\mathrm{D}}=10^{16} \mathrm{~cm}^{-3}, ~ N_{\mathrm{A}}=5 \times 10^{18} \mathrm{~cm}^{-3}, ~ \tau_{\mathrm{n}}=\tau_{p}=$ $1 \mu \mathrm{~s}, ~ A=0.01 \mathrm{~cm}^{2}$. Assume that the widths on both sides of the junction are much larger than the diffusion lengths of minority carriers. Find the applied voltage at room temperature (300 K) when the forward current is 1 mA. Assume the electron mobility in the p-type region $\mu_{\mathrm{n}}=500 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and the hole mobility in the n-type region $\mu_{\mathrm{p}}=180 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$.
|
[] |
Numeric
|
{"$N_{\\mathrm{D}}$": "donor concentration", "$N_{\\mathrm{A}}$": "acceptor concentration", "$\\tau_{\\mathrm{n}}$": "electron lifetime", "$\\tau_{p}$": "hole lifetime", "$A$": "area of the diode", "$\\mu_{\\mathrm{n}}$": "electron mobility in the p-type region", "$\\mu_{\\mathrm{p}}$": "hole mobility in the n-type region", "$D_{\\mathrm{p}}$": "hole diffusion coefficient", "$D_{\\mathrm{n}}$": "electron diffusion coefficient", "$L_{\\mathrm{p}}$": "hole diffusion length", "$L_{\\mathrm{n}}$": "electron diffusion length", "$p_{\\mathrm{n} 0}$": "minority hole concentration in n-type region", "$n_{\\mathrm{p} 0}$": "minority electron concentration in p-type region", "$I_{0}$": "saturation current", "$V$": "applied voltage", "$V_{\\mathrm{T}}$": "thermal voltage", "$I$": "forward current"}
|
pn junction
|
pn junction
|
|
425 |
Semiconductors
|
An n-type single crystal silicon wafer with [100] crystal orientation forms a Schottky diode with a certain metal contact. The parameters are $W_{\mathrm{m}}=4.7 \mathrm{eV}, \chi_{\mathrm{s}}=4.0 \mathrm{eV}, N_{\mathrm{c}}=10^{19} \mathrm{~cm}^{-3}, N_{\mathrm{D}}=10^{15} \mathrm{~cm}^{-3}$, and the relative permittivity of semiconductor silicon is $\varepsilon_{\mathrm{r}}=$ 12. Ignoring the effect of surface states, calculate at room temperature: Zero-bias depletion width;
|
[] |
Numeric
|
{"$N_{\\mathrm{D}}$": "donor concentration", "$n_{0}$": "initial electron concentration", "$N_{\\mathrm{c}}$": "effective density of states in the conduction band", "$E_{\\mathrm{c}}$": "conduction band energy", "$E_{\\mathrm{F}}$": "Fermi level energy", "$k_{0}$": "Boltzmann constant", "$T$": "absolute temperature", "$W_{\\mathrm{s}}$": "barrier height with respect to semiconductor", "$\\chi_{\\mathrm{s}}$": "electron affinity of the semiconductor", "$q$": "elementary charge", "$V_{\\mathrm{D}}$": "built-in potential difference", "$W_{\\mathrm{m}}$": "work function of the material", "$\\varepsilon_{\\mathrm{r}}$": "relative permittivity", "$\\varepsilon_{0}$": "permittivity of free space", "$d$": "depletion width"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
426 |
Semiconductors
|
An n-type monocrystalline silicon wafer with [100] crystal orientation forms a Schottky diode upon contact with a certain metal. Its parameters are $W_{\mathrm{m}}=4.7 \mathrm{eV}, \chi_{\mathrm{s}}=4.0 \mathrm{eV}, N_{\mathrm{c}}=10^{19} \mathrm{~cm}^{-3}, N_{\mathrm{D}}=10^{15} \mathrm{~cm}^{-3}$, and the relative dielectric constant of semiconductor silicon is $\varepsilon_{\mathrm{r}}=$ 12. Ignoring the effect of surface states, calculate at room temperature: The thermionic emission current when forward biased at 0.2 V. Assume $\frac{A^{*}}{A}=2.1, A=120 \mathrm{~A} / \mathrm{cm}^{2}$.
|
[] |
Numeric
|
{"$A$": "Richardson constant for a material", "$A^{*}$": "modified Richardson constant", "$q$": "elementary charge", "$\\varphi_{\\mathrm{ns}}$": "surface potential difference", "$V_{\\mathrm{D}}$": "diffusion voltage", "$E_{\\mathrm{n}}$": "energy level difference", "$J$": "current density", "$T$": "temperature", "$\\varphi_{\\mathrm{n}}$": "potential barrier height", "$k_{0}$": "Boltzmann constant", "$V$": "applied voltage"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
427 |
Semiconductors
|
Consider a metal forming a Schottky diode with (111) crystal plane $\mathrm{n}-\mathrm{Si}$. It is known that the barrier height on the semiconductor side after contact is $0.50 \mathrm{eV}, N_{\mathrm{D}}=10^{15} \mathrm{~cm}^{-3}, N_{\mathrm{c}}=2.8 \times 10^{19} \mathrm{~cm}^{-3}$, electron affinity $X=4.05 \mathrm{eV}, I_{\mathrm{p}}=10 \mu \mathrm{~m}$, $D_{\mathrm{p}}=15 \mathrm{~cm}^{2} / \mathrm{s}, n_{\mathrm{i}}=1.5 \times 10^{10} \mathrm{~cm}^{-3}, A^{*}=252 \mathrm{~A} / \mathrm{cm}^{2} \mathrm{~K}^{2}$ (Richardson constant), find: Calculate the minority carrier injection ratio.
|
[] |
Numeric
|
{"$\\gamma$": "minority carrier injection ratio", "$J_{\\mathrm{p}}$": "hole current density", "$J_{\\mathrm{n}}$": "electron current density"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
428 |
Semiconductors
|
A metal contacts a uniformly doped $n-Si$ material, forming a Schottky barrier diode. The barrier height on the semiconductor side is known as $qV_{\mathrm{D}}=0.6 \mathrm{eV}, N_{\mathrm{D}}=5 \times 10^{16} \mathrm{~cm}^{-3}$. Calculate the maximum electric field in the semiconductor at the interface under a reverse bias of 5 V.
|
[] |
Numeric
|
{"$n-Si$": "n-type silicon", "$q$": "elementary charge", "$V_{\\mathrm{D}}$": "Schottky barrier height on the semiconductor side", "$N_{\\mathrm{D}}$": "doping concentration in silicon", "$E_{\\mathrm{M}}$": "maximum electric field at the semiconductor interface", "$\\varepsilon_{0}$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of silicon", "$d$": "depletion width", "$C$": "unit-area barrier capacitance"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
429 |
Semiconductors
|
A metal contacts a uniformly doped $n-Si$ material, forming a Schottky barrier diode. Given the barrier height on the semiconductor side $q V_{\mathrm{D}}=0.6 \mathrm{eV}, N_{\mathrm{D}}=5 \times 10^{16} \mathrm{~cm}^{-3}$, determine the barrier capacitance per unit area under a reverse bias voltage of 5 V.
|
[] |
Numeric
|
{"$n$": "indicates n-type semiconductor, such as silicon (Si)", "$q$": "elementary charge", "$V_{\\mathrm{D}}$": "barrier height on the semiconductor side", "$N_{\\mathrm{D}}$": "doping concentration of the n-type semiconductor", "$d$": "space charge region width", "$E_{\\mathrm{M}}$": "maximum electric field in the semiconductor at the interface", "$N$": "doping concentration, replaced with $N_{\\mathrm{D}}$ for calculation", "$\\varepsilon_{0}$": "permittivity of free space", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", "$V$": "applied reverse bias voltage", "$C$": "barrier capacitance per unit area"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
430 |
Semiconductors
|
A metal plate and n-type silicon are separated by $0.4 \mu \mathrm{~m}$, forming a parallel plate capacitor. The relative permittivity of the dry air between them is $\varepsilon_{\mathrm{ra}}=1$. When a negative voltage is applied to the metal side, the semiconductor is in a depletion state. Ignoring the work function difference between the metal and the semiconductor, what is the voltage $V_{\mathrm{G}}$ on the metal plate when the depletion layer width just reaches its maximum? ($N_{\mathrm{D}}=10^{16} \mathrm{~cm}^{-3}$)
|
[] |
Numeric
|
{"$V_{\\mathrm{G}}$": "gate voltage", "$N_{\\mathrm{D}}$": "donor concentration", "$X_{\\mathrm{dm}}$": "maximum depletion layer width", "$Q_{\\mathrm{s}}$": "surface charge density on the semiconductor", "$V_{0}$": "voltage across the air gap", "$C_{0}$": "air capacitance", "$\\varepsilon_{\\mathrm{r} 0}$": "relative permittivity of air", "$\\varepsilon_{0}$": "vacuum permittivity", "$\\mathrm{d}_{0}$": "air gap distance"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
431 |
Semiconductors
|
The MOS structure capacitor formed by metal-$\mathrm{SiO}_{2}-\mathrm{Si}$ (p-type), with hole concentration $N_{\mathrm{A}}=1.5 \times 10^{15}$ $\mathrm{cm}^{-3}, ~ \mathrm{SiO}_{2}$ layer thickness $d_{0}=0.2 \mu \mathrm{~m}$, its relative permittivity $\varepsilon_{\mathrm{r}_{0}}=3.9$, the relative permittivity of silicon $\varepsilon_{\mathrm{rs}}=12, \varepsilon_{0}=$ $8.85 \times 10^{-14} \mathrm{~F} / \mathrm{cm}$, at room temperature $n_{\mathrm{i}}=1.5 \times 10^{10} \mathrm{~cm}^{-3}$. If there is a fixed positive charge at the SiO$_2$-silicon interface, and the measured $V_{\mathrm{T}}=2.6 \mathrm{~V}$, find the amount of fixed positive charge per unit area (neglecting the influence of the work function difference);
|
[] |
Numeric
|
{"$V_{\\mathrm{T}}$": "threshold voltage", "$Q_{\\mathrm{fc}}$": "amount of fixed positive charge", "$C_{0}$": "capacitance per unit area", "$\\Delta V_{\\mathrm{T}}$": "change in threshold voltage", "$N_{\\mathrm{fc}}$": "number of fixed positive charges per unit area", "$q$": "elementary charge"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
432 |
Semiconductors
|
The MOS structure capacitor composed of metal-$\mathrm{SiO}_{2}-\mathrm{Si}$ (p-type), with hole concentration $N_{\mathrm{A}}=1.5 \times 10^{15}$ $\mathrm{cm}^{-3}, ~ \mathrm{SiO}_{2}$ layer thickness $d_{0}=0.2 \mu \mathrm{~m}$, relative permittivity of $\varepsilon_{\mathrm{r}_{0}}=3.9$, silicon's relative permittivity $\varepsilon_{\mathrm{rs}}=12, \varepsilon_{0}=$ $8.85 \times 10^{-14} \mathrm{~F} / \mathrm{cm}$, intrinsic carrier concentration at room temperature $n_{\mathrm{i}}=1.5 \times 10^{10} \mathrm{~cm}^{-3}$. If the above positive charges are uniformly distributed in $\mathrm{SiO}_{2}$, what is the measured $V_{\mathrm{T}}$? (neglecting the effect of work function difference);
|
[] |
Numeric
|
{"$V_{\\mathrm{T}}$": "threshold voltage", "$V_{\\mathrm{FB}}$": "flatband voltage", "$C_{0}$": "capacitance", "$d_{0}$": "thickness of SiO2 layer", "$\\rho_{0}$": "charge density", "$x$": "position variable in integration"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
433 |
Semiconductors
|
The metallurgical junction area of a gate-controlled $\mathrm{p}^{+} \mathrm{n}$ diode is $10^{-3} \mathrm{~cm}^{2}$, and the overlap area of the gate with the n region is $10^{-3} \mathrm{cm}^{2}$. The substrate impurity concentration is $10^{16} \mathrm{~cm}^{-3}$, the junction depth is $5 \mu \mathrm{~m}$, the oxide layer thickness is $0.2 \mu \mathrm{~m}$, the lifetime $\tau=1 \mu \mathrm{~s}$, the surface recombination velocity $s_{0}=5 \mathrm{~cm} / \mathrm{s}$, the flat-band voltage is -2 V. Calculate the gate voltage when the substrate surface is intrinsic (when the junction voltage is zero at room temperature).
|
[] |
Numeric
|
{"$V_{\\mathrm{G}}$": "gate voltage", "$V_{\\mathrm{s}}$": "substrate voltage", "$V_{0}$": "voltage drop across the SiO2 layer", "$V_{\\mathrm{FB}}$": "flat-band voltage", "$V_{\\mathrm{B}}$": "voltage related to the band bending", "$E_{\\mathrm{i}}$": "intrinsic energy level", "$E_{\\mathrm{F}}$": "Fermi energy level", "$q$": "elementary charge", "$n_{0}$": "carrier concentration at equilibrium", "$N_{\\mathrm{D}}$": "donor concentration", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$Q_{\\mathrm{s}}$": "surface charge", "$C_{0}$": "oxide capacitance", "$x_{\\mathrm{d}}$": "depletion layer thickness", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the substrate", "$\\varepsilon_{0}$": "permittivity of free space", "$d_{0}$": "oxide thickness"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
434 |
Semiconductors
|
The metallurgical junction area of a gate-controlled $\mathrm{p}^{+} \mathrm{n}$ diode is $10^{-3} \mathrm{~cm}^{2}$, the overlap area between the gate and the n region is $10^{-3} \mathrm{~cm}^{2}$, the substrate impurity concentration is $10^{16} \mathrm{~cm}^{-3}$, the junction depth is $5 \mu \mathrm{~m}$, the oxide thickness is $0.2 \mu \mathrm{~m}$, the lifetime $\tau=1 \mu \mathrm{~s}$, the surface recombination velocity $s_{0}=5 \mathrm{~cm} / \mathrm{s}$, the flat-band voltage is -2 V. For the gate-controlled $\mathrm{p}^{+} \mathrm{n}$ diode described in the problem (metallurgical junction area $10^{-3} \mathrm{~cm}^{2}$, overlap area between the gate and n region $10^{-3} \mathrm{~cm}^{2}$, substrate impurity concentration $10^{16} \mathrm{~cm}^{-3}$, oxide thickness $0.2 \mu \mathrm{~m}$, minority carrier lifetime $\tau=1 \mu \mathrm{~s}$, flat-band voltage $V_{FB} = -2 \mathrm{~V}$), when the diode is subjected to a reverse bias voltage of $V_{\mathrm{R}}=1 \mathrm{~V}$ at room temperature, calculate the change of the forward current $\Delta I_p = I_p(V_G=-20V) - I_p(V_G=0V)$ induced by varying the gate voltage $V_G$ from $0 \mathrm{~V}$ to $-20 \mathrm{~V}$.
|
[] |
Numeric
|
{"$\\tau$": "minority carrier lifetime", "$V_{FB}$": "flat-band voltage", "$V_{\\mathrm{R}}$": "reverse bias voltage", "$V_G$": "gate voltage", "$V_{\\mathrm{s}}$": "surface potential", "$V_{\\mathrm{B}}$": "built-in potential", "$V_{\\mathrm{F}}$": "forward voltage", "$x_{\\mathrm{dm}}$": "maximum width of the depletion layer", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of the semiconductor", "$N_{\\mathrm{D}}$": "dopant concentration", "$I_p$": "forward current", "$I_{\\mathrm{rF}}$": "recombination current in the barrier region", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$S_{0}$": "interface state density", "$A_{\\mathrm{s}}$": "interfacial surface area", "$I_{\\mathrm{rs}}$": "interface state contribution to the forward recombination current", "$I_{\\mathrm{D}}$": "drift current", "$I_{\\mathrm{rm}}$": "metallurgical junction recombination current"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
435 |
Semiconductors
|
The metallurgical junction area of a gate-controlled $\mathrm{p}^{+} \mathrm{n}$ diode is $10^{-3} \mathrm{~cm}^{2}$, and the overlap area between the gate and n-region is $10^{-3} \mathrm{~cm}^{2}$. The substrate impurity concentration is $10^{16} \mathrm{~cm}^{-3}$, the junction depth is $5 \mu \mathrm{~m}$, the oxide layer thickness is $0.2 \mu \mathrm{~m}$, lifetime $\tau=1 \mu \mathrm{~s}$, surface recombination velocity $s_{0}=5 \mathrm{~cm} / \mathrm{s}$, and the flat-band voltage is -2 V. Calculate: For the gate-controlled $\mathrm{p}^{+} \mathrm{n}$ diode (with relevant parameters: gate to n-region overlap area $A_s = 10^{-3} \mathrm{~cm}^{2}$, used to calculate the diffusion current; n-region substrate impurity concentration $N_D = 10^{16} \mathrm{~cm}^{-3}$; minority carrier hole lifetime $\tau_p = 1 \mu \mathrm{s}$; hole diffusion coefficient $D_p = 13 \mathrm{~cm}^{2} / \mathrm{s}$; intrinsic carrier concentration $n_i = 1.5 \times 10^{10} \mathrm{~cm}^{-3}$; thermal voltage $k_0T/q = 0.026 \mathrm{~V}$), when the diode is under forward bias voltage $V_F = 0.4 \mathrm{~V}$ at room temperature, calculate the diffusion current component $I_D$.
|
[] |
Numeric
|
{"$A_s$": "gate to n-region overlap area", "$N_D$": "n-region substrate impurity concentration", "$\\tau_p$": "minority carrier hole lifetime", "$D_p$": "hole diffusion coefficient", "$n_i$": "intrinsic carrier concentration", "$k_0T/q$": "thermal voltage", "$V_F$": "forward bias voltage", "$I_D$": "diffusion current component", "$I_p$": "hole current", "$q$": "elementary charge", "$p_{n_0}$": "equilibrium hole concentration in the n-region", "$L_{p}$": "hole diffusion length", "$I_{\\mathrm{rm}}$": "barrier region recombination current", "$x_{\\mathrm{D}}$": "width of the depletion region", "$\\varepsilon_{\\mathrm{rs}}$": "relative permittivity of silicon", "$\\varepsilon_{0}$": "permittivity of free space", "$V$": "voltage across the diode"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
436 |
Semiconductors
|
Estimate the injection ratio between GaAs and $\mathrm{Al}_{0.3} \mathrm{Ga}_{0.7} \mathrm{As}$ at 300 K.
|
[] |
Numeric
|
{"$x$": "composition variable", "$E_{\\mathrm{g} x}^{\\mathrm{Al}_{x} \\mathrm{Ga}_{1-x} x_{\\mathrm{s}}}$": "band gap energy of AlGaAs as a function of composition", "$E_{\\mathrm{g}}^{\\mathrm{Al}_{0.3} \\mathrm{Ga}_{0.7} \\mathrm{As}^{\\mathrm{As}}}$": "band gap energy of Al0.3Ga0.7As", "$E_{\\mathrm{g}}^{\\mathrm{GaAs}}$": "band gap energy of GaAs", "$\\Delta E_{\\mathrm{g}}$": "difference in band gap energies", "$j_{\\mathrm{n} 1}$": "electron injection current density", "$j_{\\mathrm{p} 2}$": "hole injection current density", "$k_{0}$": "Boltzmann constant", "$T$": "temperature"}
|
Metal-Semiconductor Contact
|
Metal-Semiconductor Contact
|
|
437 |
Semiconductors
|
Try to derive the relationship between the absorption coefficient $\alpha$ and the extinction coefficient $\bar{k}$.
|
[] |
Expression
|
{"$\\alpha$": "absorption coefficient", "$\\bar{k}$": "extinction coefficient", "$\\omega$": "angular frequency", "$c$": "speed of light in vacuum", "$I$": "intensity of light", "$E_{x}$": "electric field component in the x-direction", "$E_{0}$": "initial electric field amplitude", "$t$": "time", "$n$": "refractive index", "$z$": "propagation distance", "$\\lambda$": "wavelength in vacuum"}
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
|
438 |
Semiconductors
|
There is an n-type CdS cubic chip, with an edge length of 1 mm and a thickness of 0.1 mm, with a wavelength absorption limit of $5100 \AA$. Now, a violet light of intensity $1 \mathrm{~mW} / \mathrm{cm}^{2}$ $(\lambda=4096 \AA)$ is used to illuminate the square surface, with a quantum yield of $\beta=1$. Assuming all photo-generated holes are trapped and the lifetime of photo-generated electrons is $\tau_{\mathrm{n}}=10^{-5} \mathrm{~s}$, the electron mobility is $\mu_{\mathrm{n}}=100 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$; and assuming the illuminative energy is completely absorbed by the chip. Calculate the number of electron-hole pairs generated per second in the sample;
|
[] |
Numeric
|
{"$\\beta$": "quantum efficiency", "$Q$": "number of electron-hole pairs generated per second", "$I$": "light intensity represented by the number of photons", "$E_{0}$": "energy of each photon", "$h$": "Planck's constant", "$\\nu$": "frequency of the photon", "$c$": "speed of light", "$\\lambda$": "wavelength of the photon", "$S$": "sample area"}
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
|
439 |
Semiconductors
|
There is an n-type CdS cubic wafer with a side length of 1 mm and a thickness of 0.1 mm, with an absorption edge wavelength of $5100 \AA$. Now, the square surface is irradiated with purple light $(\lambda=4096 \AA)$ at an intensity of $1 \mathrm{~mW} / \mathrm{cm}^{2}$, with a quantum yield of $\beta=1$. Assume all photogenerated holes are trapped, the lifetime of photogenerated electrons is $\tau_{\mathrm{n}}=10^{-5} \mathrm{~s}$, and the electron mobility is $\mu_{\mathrm{n}}=100 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$; further assume all illumination energy is absorbed by the wafer. calculate the increase in the number of electrons in the sample;
|
[] |
Numeric
|
{"$\\Delta n^{\\prime}$": "increase in the number of electrons in the sample", "$Q$": "total charge", "$\\tau_{\\mathrm{n}}$": "lifetime of charge carriers", "$\\Delta n$": "increase in the number of electrons per unit volume", "$V$": "volume"}
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
|
440 |
Semiconductors
|
There is an n-type CdS square crystal with a side length of 1 mm and a thickness of 0.1 mm. Its wavelength absorption limit is $5100 \AA$. Now, violet light with an intensity of $1 \mathrm{~mW} / \mathrm{cm}^{2}$ $(\lambda=4096 \AA)$ illuminates the square surface, and the quantum yield is $\beta=1$. Assume all photogenerated holes are trapped, the lifetime of the photogenerated electrons is $\tau_{\mathrm{n}}=10^{-5} \mathrm{~s}$, and the electron mobility is $\mu_{\mathrm{n}}=100 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$; also assume that the illumination energy is completely absorbed by the crystal. When the photocurrent when a 50 V voltage is applied to the sample, calculate photoconductive gain factor.
|
[] |
Numeric
|
{"$G$": "photoconductive gain factor", "$\\tau_{\\mathrm{n}}$": "carrier lifetime", "$\\tau_{\\mathrm{t}}$": "transit time", "$\\mu_{\\mathrm{n}}$": "mobility", "$V$": "voltage", "$l$": "characteristic length"}
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
|
441 |
Semiconductors
|
Given a piece of n-type semiconductor material with a room temperature dark conductivity of $100 \mathrm{~S} / \mathrm{cm}$, when illuminated with light at an intensity of $I=$ $10^{-6} \mathrm{~W} / \mathrm{cm}^{2}$, its absorption coefficient $\alpha=10^{2} / \mathrm{cm}$, the measured ratio of steady-state photoconductivity to dark conductivity $\gamma$ $=10$, and the lifetime $\tau$ is $10^{-4} \mathrm{~s}, b=\mu_{\mathrm{n}} / \mu_{\mathrm{p}}=10, \mu_{\mathrm{n}}=10000 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, determine the corresponding quantum yield.
|
[] |
Numeric
|
{"$I$": "light intensity", "$\\alpha$": "absorption coefficient", "$\\gamma$": "ratio of steady-state photoconductivity to dark conductivity", "$b$": "mobility ratio of electrons to holes", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$\\tau$": "lifetime", "$q$": "elementary charge", "$\\sigma_{0}$": "dark conductivity", "$\\sigma$": "photoconductivity", "$\\Delta n$": "change in electron concentration", "$\\beta$": "quantum yield", "$\\tau_{\\mathrm{n}}$": "electron lifetime", "$n_{0}$": "initial electron concentration", "$p_{0}$": "initial hole concentration"}
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
Optical properties of semiconductors, photoelectric and luminescence phenomena
|
|
442 |
Semiconductors
|
In a p-type silicon with a hole concentration of $10^{16} \mathrm{~cm}^{-3}$, a cold end temperature of $0^{\circ} \mathrm{C}$, and a hot end temperature of $50^{\circ} \mathrm{C}$, assuming long wavelength acoustic wave scattering, calculate the thermoelectric power.
|
[] |
Numeric
|
{"$\\alpha_{\\mathrm{p}}$": "thermopower of a p-type semiconductor", "$k_{0}$": "Boltzmann constant", "$q$": "elementary charge", "$\\gamma$": "scattering parameter for acoustic wave scattering", "$\\xi_{\\mathrm{p}}$": "reduced Fermi energy fraction for p-type semiconductor", "$E_{\\mathrm{F}}$": "Fermi energy", "$E_{\\mathrm{v}}$": "valence band edge energy", "$p$": "hole concentration", "$N_{\\mathrm{v}}$": "effective density of states in the valence band", "$T$": "absolute temperature", "$\\Delta T$": "temperature difference between hot and cold ends", "$N_{\\mathrm{v}^{\\prime}}$": "adjusted effective density of states in the valence band at cold end"}
|
Thermoelectric properties of semiconductors
|
Thermoelectric properties of semiconductors
|
|
443 |
Semiconductors
|
For n-type PoTe with a conductivity of $2000 \mathrm{~S} / \mathrm{cm}$, an electron mobility of $6000 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and an electron effective mass of $0.2 m_{0}$, determine the thermoelectric power factor at room temperature assuming long-wavelength acoustic phonon scattering.
|
[] |
Numeric
|
{"$n$": "charge carrier concentration", "$g$": "degeneracy factor", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$q$": "elementary charge", "$\\gamma$": "gamma parameter for scattering", "$\\xi_{\\mathrm{n}}$": "reduced Fermi energy", "$\\sigma$": "electrical conductivity", "$m_{\\mathrm{n}}^{*}$": "electron effective mass", "$m_{0}$": "electron rest mass", "$N_{\\mathrm{c}}$": "effective density of states for conduction band", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$h$": "Planck constant", "$\\alpha_{\\mathrm{n}}$": "Seebeck coefficient for n-type semiconductor", "$\\pi_{\\mathrm{ab}}$": "Peltier coefficient"}
|
Thermoelectric properties of semiconductors
|
Thermoelectric properties of semiconductors
|
|
444 |
Semiconductors
|
For an n-type PoTe with a conductivity of $2000 \mathrm{~S} / \mathrm{cm}$, electron mobility of $6000 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and an electron effective mass of $0.2 m_{0}$, assuming long-wavelength acoustic scattering, determine the Peltier coefficient at room temperature.
|
[] |
Numeric
|
{"$n$": "electron density", "$\\sigma$": "conductivity", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$m_{n}^{*}$": "electron effective mass", "$q$": "elementary charge", "$N_{\\mathrm{c}}$": "effective density of states", "$k_{0}$": "Boltzmann constant", "$T$": "temperature", "$\\alpha_{\\mathrm{n}}$": "thermoelectric power factor", "$\\gamma$": "scattering parameter", "$\\xi_{\\mathrm{n}}$": "dimensionless factor in thermoelectric power", "$\\pi_{\\mathrm{ab}}$": "Peltier coefficient"}
|
Thermoelectric properties of semiconductors
|
Thermoelectric properties of semiconductors
|
|
445 |
Semiconductors
|
The thermal conductivity of bismuth telluride ($\mathrm{Bi}_{2} \mathrm{Te}_{3}$) is $2.4[\mathrm{~W} /(\mathrm{m} \cdot \mathrm{K})$]. Calculate the percentage contribution of carrier to the thermal conductivity for n-type $\mathrm{Bi}_{2} \mathrm{Te}_{3}$ at $10^{5} \mathrm{~s} / \mathrm{m}$ and $300 \mathrm{~K}$. (Assume long-wavelength acoustic phonon scattering)
|
[] |
Numeric
|
{"$K_{\\mathrm{c}}$": "thermal conductivity contribution of carriers", "$\\sigma$": "electrical conductivity", "$T$": "temperature", "$k_{0}$": "Boltzmann constant", "$q$": "elementary charge", "$K$": "total thermal conductivity"}
|
Thermoelectric properties of semiconductors
|
Thermoelectric properties of semiconductors
|
|
446 |
Semiconductors
|
Try to find the Seebeck coefficient of intrinsic silicon at room temperature.
Assume the effective masses of electrons and holes are equal, the band gap of silicon is 1.12 eV, and the mobilities of electrons and holes are $0.135 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and $0.048 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, respectively.
|
[] |
Numeric
|
{"$k$": "Boltzmann constant", "$q$": "elementary charge", "$p$": "hole concentration", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$n$": "electron concentration", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\xi_{\\mathrm{p}}$": "reduced Fermi level for holes", "$\\xi_{\\mathrm{n}}$": "reduced Fermi level for electrons", "$m_{\\mathrm{n}}^{*}$": "effective mass of electrons", "$m_{\\mathrm{p}}^{*}$": "effective mass of holes", "$E_{\\mathrm{i}}$": "intrinsic Fermi level energy", "$E_{\\mathrm{g}}$": "band gap energy", "$k_{0}$": "Boltzmann constant (alternate symbol)", "$T$": "temperature"}
|
Thermoelectric properties of semiconductors
|
Thermoelectric properties of semiconductors
|
|
447 |
Semiconductors
|
For an indium antimonide sample, the hole concentration at room temperature is 9 times the electron concentration. Calculate the Hall coefficient $R$. Assume at room temperature $b=\mu_{\mathrm{n}} / \mu_{\mathrm{p}}=100, n_{\mathrm{i}}=1.1 \times 10^{16} \mathrm{~cm}^{-3}$.
|
[] |
Numeric
|
{"$R$": "Hall coefficient", "$b$": "mobility ratio", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$n$": "electron concentration", "$p$": "hole concentration", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$q$": "elementary charge"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
448 |
Semiconductors
|
InSb electron mobility is $7.8 \mathrm{~m}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and hole mobility is $780 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, with an intrinsic carrier concentration of $1.6 \times 10^{16} \mathrm{~cm}^{-3}$, at 300 K Hall coefficient of intrinsic material;
|
[] |
Numeric
|
{"$R_{\\mathrm{H}}$": "Hall coefficient", "$p$": "hole concentration", "$n$": "electron concentration", "$b$": "ratio of mobilities of electrons to holes", "$q$": "elementary charge", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
449 |
Semiconductors
|
The electron mobility of InSb is $7.8 \mathrm{~m}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and the hole mobility is $780 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$. The intrinsic carrier concentration is $1.6 \times 10^{16} \mathrm{~cm}^{-3}$, find at 300 K intrinsic resistivity;
|
[] |
Numeric
|
{"$\\rho_{\\mathrm{i}}$": "intrinsic resistivity", "$\\sigma_{\\mathrm{i}}$": "intrinsic conductivity", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
450 |
Semiconductors
|
The electron mobility of InSb is $7.8 \mathrm{~m}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and the hole mobility is $780 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$. The intrinsic carrier concentration is $1.6 \times 10^{16} \mathrm{~cm}^{-3}$ at 300 K. When $B_{z}=0.1 \mathrm{~Wb} / \mathrm{m}^{2}$, calculate the resistivity of the material considering the scattering of long acoustic waves.
|
[] |
Numeric
|
{"$B_{z}$": "magnetic field component in the z direction", "$\\theta$": "angle related to the magnetic field and material parameters", "$\\mu_{\\mathrm{H}}$": "Hall mobility", "$\\sigma$": "electrical conductivity", "$R_{\\mathrm{H}}$": "Hall coefficient", "$\\rho$": "resistivity", "$\\rho_{0}$": "initial resistivity", "$\\xi$": "transverse magnetoresistance coefficient", "$n$": "electron concentration", "$p$": "hole concentration", "$b$": "parameter related to the carrier concentration ratios", "$n_{\\mathrm{i}}$": "intrinsic carrier concentration", "$\\mu_{\\mathrm{n}}$": "mobility of electrons", "$\\mu_{\\mathrm{p}}$": "mobility of holes"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
451 |
Semiconductors
|
For n-type GaAs with a thickness of 0.08 cm, a current of 50 mA is passed along the $x$ direction, and a magnetic field of 0.5 T is applied along the $z$ direction, resulting in a Hall voltage of -0.4 mV. Hall coefficient;
|
[] |
Numeric
|
{"$R_{\\mathrm{H}}$": "Hall coefficient", "$V_{\\mathrm{H}}$": "Hall voltage", "$d$": "thickness of the material", "$I_{x}$": "current in the material", "$B_{z}$": "magnetic field in the z direction"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
452 |
Semiconductors
|
A silicon sample with a conductivity of $0.001 /(\Omega \cdot \mathrm{cm})$ has zero Hall voltage under a weak magnetic field. Assuming the electron mobility $\mu_{\mathrm{n}}=1300 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$ and the hole mobility $\mu_{\mathrm{p}}=300 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$, and the Hall factors for electrons and holes are the same. Try to determine the carrier density of electrons $n$ in the sample.
|
[] |
Numeric
|
{"$n$": "carrier density of electrons", "$b$": "ratio of hole to electron mobility", "$\\sigma$": "conductivity", "$e$": "elementary charge", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$p$": "carrier density of holes", "$R$": "Hall coefficient"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
453 |
Semiconductors
|
Given the band gap of InSb $E_{\mathrm{g}}=0.15 \mathrm{eV}$, the effective mass of electrons $m_{\mathrm{e}}=0.014 m_{0}$, and the effective mass of holes $m_{\mathrm{h}}=0.18 m_{0}$ (with $m_{0}$ as the inertial mass of the electron). If only electrons are the effective carriers, calculate the Hall coefficient of intrinsic InSb at 300 K.
|
[] |
Numeric
|
{"$E_{\\mathrm{g}}$": "band gap energy", "$m_{\\mathrm{e}}$": "effective mass of electrons", "$m_{\\mathrm{h}}$": "effective mass of holes", "$m_{0}$": "inertial mass of the electron", "$R_{\\mathrm{H}}$": "Hall coefficient", "$n$": "electron concentration", "$k_{B}$": "Boltzmann constant", "$T$": "temperature", "$h$": "Planck's constant", "$b$": "width of the sample", "$d$": "thickness of the sample", "$V_{\\mathrm{H}}$": "Hall voltage", "$B_{z}$": "magnetic field component in the z-direction", "$J$": "current density", "$I$": "current intensity"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
454 |
Semiconductors
|
Try to demonstrate in the Hall effect under conditions of two types of carriers and a weak magnetic field, the Hall angle $\theta$ and Hall coefficient $R$ can be expressed as
\begin{align}
& \theta=\arctan \frac{p \mu_{\mathrm{p}}^{2}-n \mu_{\mathrm{n}}^{2}}{p \mu_{\mathrm{p}}+n \mu_{\mathrm{n}}} B_{z} \\
& R=\frac{1}{q} \frac{p \mu_{\mathrm{p}}^{2}-n \mu_{\mathrm{n}}^{2}}{(p \mu_{\mathrm{p}}+n \mu_{\mathrm{n}})^{2}}
\end{align} If a given germanium sample is placed in a magnetic field of $B=0.1$ T, what is $\tan \theta$ when its conductivity is at a minimum? Let $\mu_{\mathrm{n}}=3900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s}), \mu_{\mathrm{p}}=1$ $900 \mathrm{~cm}^{2} /(\mathrm{V} \cdot \mathrm{s})$.
|
[] |
Numeric
|
{"$B$": "magnetic field", "$\\mu_{\\mathrm{n}}$": "electron mobility", "$\\mu_{\\mathrm{p}}$": "hole mobility", "$\\theta$": "angle related to conductivity"}
|
Semiconductor magneto- and piezoresistive effects
|
Semiconductor magneto- and piezoresistive effects
|
|
455 |
Others
|
Briefly answer the following questions: Classified by symmetry type, how many types of point groups are there for Bravais lattices? How many types of space groups are there? How many types of point groups are there for crystal structures? How many types of space groups are there? You should return your answer as a tuple format.
|
[] |
Tuple
|
{}
|
Crystal structure
|
Crystal structure
|
|
456 |
Others
|
A one-dimensional atomic chain consisting of N identical atoms with mass m and spacing a. Each atom has only one valence electron. Using the tight-binding approximation, only nearest-neighbor interactions are considered, Derive the expression for the density of states of the s-band electrons.
|
[] |
Expression
|
{"$G(E)$": "density of states", "$L$": "length of the system", "$a$": "lattice constant", "$J_{1}$": "hopping integral", "$k$": "wave vector", "$N$": "number of states", "$E_{0}$": "edge of the energy band", "$E$": "energy"}
|
Band theory
|
Band theory
|
|
457 |
Semiconductors
|
The valence band of a semiconductor material is almost filled with electrons (nearly full band), and the expression for the energy of valence band electrons is $E(k)=-1.016 \times 10^{-34} k^{2}(J)$, where the energy zero point is taken at the top of the valence band. At this time, if the electron at $k=1 \times 10^{6} \mathrm{~cm}^{-1}$ is excited to a higher energy band (conduction band), a hole is generated at this location. Try to find energy of this hole.
|
[] |
Numeric
|
{"$E$": "energy of valence band electrons", "$k$": "wave vector", "$m_{h}^{*}$": "effective mass of the hole", "$m_{e}^{*}$": "effective mass of the electron", "$m_{0}$": "rest mass of electron", "$k_{h}$": "wave vector of the hole", "$k_{e}$": "wave vector of the electron", "$p_{h}$": "quasi-momentum of the hole", "$v_{h}$": "velocity of the hole", "$E_{h}$": "energy of the hole", "$E_{e}$": "energy of the electron"}
|
Movement of electrons in a crystal in electric and magnetic fields
|
Movement of electrons in a crystal in electric and magnetic fields
|
|
458 |
Others
|
Calculate the mean free path of electrons at room temperature (T=295K). (The density of silver is $10.5 \mathrm{~g} / \mathrm{cm}^{3}$, atomic weight is 107.87, and its resistivity at T=295K is $1.61 \times 10^{-6} \Omega \cdot \mathrm{~cm}$)
|
[] |
Numeric
|
{"$T$": "temperature", "$\\lambda$": "mean free path", "$\\tau$": "relaxation time", "$v_{F}$": "Fermi velocity", "$\\hbar$": "reduced Planck's constant", "$k_{F}$": "Fermi wavevector", "$m$": "mass of electron", "$\\sigma$": "electrical conductivity", "$n$": "electron density", "$e$": "elementary charge", "$\\rho$": "resistivity"}
|
Metal Electron Theory
|
Metal Electron Theory
|
|
459 |
Superconductivity
|
Phase transition in BCS superconducting systems
Consider the Hamiltonian with parameter $\lambda$, $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$, and the corresponding Gibbs free energy is $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\beta \bar{H}_{\mathrm{int}}(\lambda)]$. When $T>0K$, Feynman's theorem gives:
\begin{equation} \label{eq:6.5.44}
\frac{\partial G (\lambda)}{\partial \lambda} = \frac{< \bar{H}_{\text{int}} (\lambda) >_T}{\lambda}
\end{equation}
For a BCS superconducting system, the Hamiltonian satisfies:
\begin{equation}
\bar{H}_{\text{int}} = -\Delta \sum_k (C_k^\dagger C_{-k}^\dagger + C_{-k} C_k) + (\Delta^2/V), \Delta = V \sum_k < C_{-k} C_k >_T
\end{equation}
where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.
Please:
Use Feynman's theorem to derive the difference in Gibbs free energy between the normal state and the superconducting phase
Hint: If the answer is an integral with respect to $\lambda$, you can just output the integrand.
|
[] |
Expression
|
{"$\\lambda$": "coupling parameter", "$\\bar{H}$": "Hamiltonian", "$\\bar{H}_0$": "free Hamiltonian component", "$\\bar{H}_{\\mathrm{int}}$": "interaction Hamiltonian component", "$\\Gamma(\\lambda)$": "Gibbs free energy", "$k_B$": "Boltzmann constant", "$T$": "temperature", "$G(\\lambda)$": "Gibbs free energy as a function of $\\lambda$", "$\\Delta$": "superconducting gap parameter", "$V$": "interaction constant", "$g(0)$": "density of states at the Fermi surface", "$C_k$": "annihilation operator", "$C_k^\\dagger$": "creation operator", "$G_S$": "Gibbs free energy in superconducting state", "$G_N$": "Gibbs free energy in normal state", "$\\epsilon$": "energy variable", "$\\xi(\\lambda)$": "quasiparticle energy"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
460 |
Superconductivity
|
Phase transition in BCS superconducting systems
Consider a Hamiltonian with a parameter $\lambda$ given by $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$, with the corresponding Gibbs free energy $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\beta \bar{H}_{\mathrm{int}}(\lambda)]$, when $T>0K$, Feynman's theorem gives:
\begin{equation} \label{eq:6.5.44}
\frac{\partial G (\lambda)}{\partial \lambda} = \frac{< \bar{H}_{\text{int}} (\lambda) >_T}{\lambda}
\end{equation}
For a BCS superconducting system, the Hamiltonian satisfies:
\begin{equation}
\bar{H}_{\text{int}} = -\Delta \sum_k (C_k^\dagger C_{-k}^\dagger + C_{-k} C_k) + (\Delta^2/V), \Delta = V \sum_k < C_{-k} C_k >_T
\end{equation}
where $V$ is the interaction constant, assuming the density of states on the Fermi surface is $g(0)$.
Please:
Derive the approximate expression for the difference in free energy near the superconducting critical temperature
|
[] |
Expression
|
{"$\\lambda$": "interaction parameter in the Hamiltonian", "$\\bar{H}$": "total Hamiltonian", "$\\bar{H}_0$": "non-interacting part of the Hamiltonian", "$\\bar{H}_{\\mathrm{int}}$": "interaction part of the Hamiltonian", "$\\Gamma$": "Gibbs free energy", "$k_B$": "Boltzmann constant", "$T$": "temperature", "$\\beta$": "inverse temperature", "$\\Delta$": "superconducting energy gap", "$V$": "interaction constant", "$g(0)$": "density of states on Fermi surface", "$C_k$": "annihilation operator for state k", "$\\omega_n$": "Matsubara frequency", "$\\epsilon$": "energy variable", "$T_c$": "critical temperature for superconductivity", "$\\zeta$": "Riemann zeta function"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
461 |
Superconductivity
|
Phase transition in BCS superconducting systems
Consider a Hamiltonian with parameter $\lambda$ given by $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$, and the corresponding Gibbs free energy is $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\beta \bar{H}_{\mathrm{int}}(\lambda)]$. When $T>0K$, the Feynman theorem gives:
\begin{equation} \label{eq:6.5.44}
\frac{\partial G (\lambda)}{\partial \lambda} = \frac{< \bar{H}_{\text{int}} (\lambda) >_T}{\lambda}
\end{equation}
For BCS superconducting systems, the Hamiltonian satisfies:
\begin{equation}
\bar{H}_{\text{int}} = -\Delta \sum_k (C_k^\dagger C_{-k}^\dagger + C_{-k} C_k) + (\Delta^2/V), \Delta = V \sum_k < C_{-k} C_k >_T
\end{equation}
where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.
Please:
Calculate the change in entropy $\Delta S = S_\mathrm{S}-S_\mathrm{N}$, and analyze the physical significance of the results
|
[] |
Expression
|
{"$\\lambda$": "parameter of the Hamiltonian", "$\\bar{H}$": "Hamiltonian", "$\\bar{H}_0$": "unperturbed Hamiltonian", "$\\bar{H}_{\\mathrm{int}}$": "interaction Hamiltonian", "$\\Gamma$": "Gibbs free energy", "$k_B$": "Boltzmann constant", "$T$": "temperature", "$G$": "Gibbs energy", "$\\Delta$": "superconducting gap parameter", "$V$": "interaction constant", "$C_k$": "creation operator for a particle with momentum $k$", "$g(0)$": "density of states at the Fermi surface", "$\\Delta S$": "change in entropy between superconducting and normal states", "$S_\\mathrm{S}$": "entropy of the superconducting state", "$S_\\mathrm{N}$": "entropy of the normal state", "$T_c$": "critical temperature"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
462 |
Superconductivity
|
Phase transition in BCS superconducting system
Consider a Hamiltonian with parameter $\lambda$ given by $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$, the corresponding Gibbs free energy is $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\beta \bar{H}_{\mathrm{int}}(\lambda)]$, when $T>0K$, the Feynman theorem gives:
\begin{equation} \label{eq:6.5.44}
\frac{\partial G (\lambda)}{\partial \lambda} = \frac{< \bar{H}_{\text{int}} (\lambda) >_T}{\lambda}
\end{equation}
For a BCS superconducting system, the Hamiltonian satisfies:
\begin{equation}
\bar{H}_{\text{int}} = -\Delta \sum_k (C_k^\dagger C_{-k}^\dagger + C_{-k} C_k) + (\Delta^2/V), \Delta = V \sum_k < C_{-k} C_k >_T
\end{equation}
where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.
Please:
Calculate the discontinuity in the electronic specific heat.
|
[] |
Expression
|
{"$\\lambda$": "parameter for interaction", "$\\bar{H}$": "Hamiltonian", "$\\bar{H}_0$": "non-interacting Hamiltonian component", "$\\bar{H}_{\\mathrm{int}}$": "interaction Hamiltonian", "$G$": "Gibbs free energy", "$T$": "temperature", "$k_B$": "Boltzmann constant", "$\\Delta$": "energy gap parameter", "$V$": "interaction constant", "$C_k$": "annihilation operator for momentum k", "$C_{-k}$": "annihilation operator for momentum -k", "$g(0)$": "density of states at the Fermi surface", "$T_c$": "critical temperature", "$\\Delta S$": "change in entropy"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
463 |
Superconductivity
|
Phase transition in BCS superconducting system
Consider a Hamiltonian $\bar{H} = \bar{H}_0 + \bar{H}_{\mathrm{int}}(\lambda), \bar{H}_{\mathrm{int}}(\lambda) = \lambda \bar{H}_{\mathrm{int}}$ with parameter $\lambda$, the corresponding Gibbs free energy is $\Gamma(\lambda) = -k_B T \mathrm{Tr}\exp[-\beta \bar{H}_{\mathrm{int}}(\lambda)]$, when $T>0K$, Feynman's theorem gives:
\begin{equation} \label{eq:6.5.44}
\frac{\partial G (\lambda)}{\partial \lambda} = \frac{< \bar{H}_{\text{int}} (\lambda) >_T}{\lambda}
\end{equation}
For BCS superconducting systems, the Hamiltonian satisfies:
\begin{equation}
\bar{H}_{\text{int}} = -\Delta \sum_k (C_k^\dagger C_{-k}^\dagger + C_{-k} C_k) + (\Delta^2/V), \Delta = V \sum_k < C_{-k} C_k >_T
\end{equation}
where $V$ is the interaction constant, assuming the density of states at the Fermi surface is $g(0)$.
Please:
Calculate the critical magnetic field $H_c(T)$ with Euler constant
|
[] |
Expression
|
{"$\\bar{H}$": "Hamiltonian", "$\\bar{H}_0$": "unperturbed Hamiltonian", "$\\bar{H}_{\\mathrm{int}}$": "interaction Hamiltonian", "$\\lambda$": "coupling parameter", "$\\Gamma(\\lambda)$": "Gibbs free energy as a function of lambda", "$k_B$": "Boltzmann constant", "$T$": "temperature", "$\\beta$": "inverse thermal energy (1/k_B T)", "$G(\\lambda)$": "Gibbs free energy with respect to lambda", "$< \\bar{H}_{\\text{int}} (\\lambda) >_T$": "thermal expectation value of interaction Hamiltonian", "$\\Delta$": "superconducting gap parameter", "$V$": "interaction constant", "$C_k$": "annihilation operator for state k", "$C_k^\\dagger$": "creation operator for state k", "$g(0)$": "density of states at the Fermi surface", "$H_c(T)$": "critical magnetic field as a function of temperature", "$H_c(0)$": "critical magnetic field at zero temperature", "$T_c$": "critical temperature", "$e$": "Euler's number", "$\\gamma$": "Euler-Mascheroni constant"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
464 |
Superconductivity
|
London Theory
Superconductors have two properties:
\begin{itemize}
\item[(i)] The DC resistance disappears when $T < T_c$, and a resistance-free supercurrent exists, which is the ideal conductivity of the superconductor.
\item[(ii)] The Meissner effect, a weak magnetic field cannot penetrate the interior of a bulk superconducting sample, exhibiting complete diamagnetism.
\end{itemize}
Now consider the following current
\begin{equation}\label{eq:6.7.1}
\mathbf{j}_s = - \frac{c}{\Lambda}\mathbf{A} \quad \Lambda = \frac{m}{n_se^2},
\end{equation}
where at $T=0K$ $n_s=n$ is the density of conduction electrons, choosing the transverse field condition $\nabla \cdot \mathbf{A}=0$, please explain using Maxwell's equations how the above expression encompasses the two main properties of the superconductor.
1. Ideal Conductivity You should return your answer as an equation.
|
[] |
Equation
|
{"$T$": "temperature", "$T_c$": "critical temperature", "$\\mathbf{j}_s$": "supercurrent density", "$c$": "speed of light", "$\\Lambda$": "London parameter", "$m$": "electron mass", "$n_s$": "density of superconducting electrons", "$e$": "elementary charge", "$\\mathbf{A}$": "vector potential", "$\\boldsymbol{v}_s$": "velocity of superconducting electrons", "$\\boldsymbol{E}$": "electric field", "$n$": "density of conduction electrons", "$\\nabla$": "nabla operator (grad)"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
465 |
Superconductivity
|
London theory
Superconductors have two properties:
\begin{itemize}
\item[(i)] When $T < T_c$, the DC resistance disappears, and there exists a resistance-free supercurrent, which is the ideal conductivity of a superconductor.
\item[(ii)] Meissner effect, a weak magnetic field cannot penetrate inside a bulk superconducting sample, exhibiting perfect diamagnetism.
\end{itemize}
Now consider the following current
\begin{equation}\label{eq:6.7.1}
\mathbf{j}_s = - \frac{c}{\Lambda}\mathbf{A} \quad \Lambda = \frac{m}{n_se^2},
\end{equation}
where when $T=0K$, $n_s=n$ is the density of conduction electrons. Choose the transversal gauge condition $\nabla \cdot \mathbf{A}=0$, and using Maxwell's equations, explain how the above expression encompasses the two main properties of a superconductor.
Consider a semi-infinite superconducting sample with $z > 0$, and explain the perfect diamagnetism when the external magnetic field is parallel to the $z=0$ plane.
|
[] |
Expression
|
{"$T$": "temperature", "$T_c$": "critical temperature", "$c$": "speed of light in vacuum", "$\\Lambda$": "London parameter", "$m$": "mass of an electron", "$n_s$": "density of superconducting electrons", "$e$": "elementary charge", "$n$": "density of conduction electrons", "$\\mathbf{j}_s$": "supercurrent density", "$\\mathbf{A}$": "magnetic vector potential", "$\\boldsymbol{B}$": "magnetic induction", "$\\lambda_L$": "London penetration depth"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
466 |
Superconductivity
|
Pippard Theory
In superconductors, within the coherence length $\xi_0 = \frac{\hbar v_F}{\pi \Delta(0)}$, there exists a correlation of electron motion, hence a perturbing potential acting at one point will inevitably affect the velocity of superconducting electrons and current density within the spatial scale of $\xi_0$. Conversely, the supercurrent density at a certain point in space must also be influenced by the perturbing potentials within its neighboring scale of $\xi_0$. Thus, $j_s (\boldsymbol{r})$ depends not only on $\boldsymbol{A}(\boldsymbol{r})$ at the same point but should also include contributions from the vector potential $\boldsymbol{A}(\boldsymbol{r}')$ at all points $\boldsymbol{r}'$ within the range $|\boldsymbol{r} - \boldsymbol{r}'| < \xi_0$. For this reason, one can consider the nonlocal relationship between $j_s$ and $\boldsymbol{A}$ as (Pippard Theory):
\begin{equation*}
j_s (\boldsymbol{r}) = - \frac{3}{4\pi \xi_0 \lambda_c} \int \frac{\boldsymbol{R} [\boldsymbol{R} \cdot \boldsymbol{A}(\boldsymbol{r}')]}{R^4} e^{-R/\xi_0} d\boldsymbol{r}' \quad (\boldsymbol{R} = \boldsymbol{r} - \boldsymbol{r}')
\end{equation*}
Complete the following question:
Analyze the approximate value of $j_s$ under the condition $q\xi_0 \ll 1$
|
[] |
Expression
|
{"$\\xi_0$": "coherence length", "$\\hbar$": "reduced Planck's constant", "$v_F$": "Fermi velocity", "$\\Delta(0)$": "energy gap at zero temperature", "$j_s$": "supercurrent density", "$\\boldsymbol{r}$": "position vector", "$\\boldsymbol{A}$": "vector potential", "$\\lambda_c$": "characteristic length scale", "$\\boldsymbol{R}$": "relative position vector", "$q$": "wave vector magnitude", "$\\lambda_L(0)$": "London penetration depth at zero temperature"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
467 |
Superconductivity
|
Pippard Theory
In superconductors, there is a correlation of electron motion within the coherence length $\xi_0 = \frac{\hbar v_F}{\pi \Delta(0)}$, thus a perturbative potential acting at one point will inevitably affect the velocity and current density of superconducting electrons within the spatial scale of $\xi_0$. Conversely, the supercurrent density at a certain point in space will inevitably be influenced by the perturbative potential within the scale of $\xi_0$ around it. Therefore, $j_s (\boldsymbol{r})$ is not only determined by $\boldsymbol{A}(\boldsymbol{r})$ at the same point, but should also include the contribution of vector potentials $\boldsymbol{A}(\boldsymbol{r}')$ at various points within the range $|\boldsymbol{r} - \boldsymbol{r}'| < \xi_0$. To this end, one can consider the nonlocal relationship between $j_s$ and $\boldsymbol{A}$ as follows (Pippard theory):
\begin{equation*}
j_s (\boldsymbol{r}) = - \frac{3}{4\pi \xi_0 \lambda_c} \int \frac{\boldsymbol{R} [\boldsymbol{R} \cdot \boldsymbol{A}(\boldsymbol{r}')]}{R^4} e^{-R/\xi_0} d\boldsymbol{r}' \quad (\boldsymbol{R} = \boldsymbol{r} - \boldsymbol{r}')
\end{equation*}
Please complete the following question:
Analyze the approximate value of $j_s$ in the case of $q\xi_0 \gg 1$
|
[] |
Expression
|
{"$\\xi_0$": "coherence length", "$\\hbar$": "reduced Planck's constant", "$v_F$": "Fermi velocity", "$\\Delta(0)$": "energy gap at zero temperature", "$j_s$": "supercurrent density", "$\\boldsymbol{r}$": "position vector", "$\\boldsymbol{A}$": "vector potential", "$\\lambda_c$": "penetration depth", "$\\boldsymbol{r}'$": "position vector (integration variable)", "$R$": "distance between points in space", "$q$": "wave vector magnitude", "$\\lambda_L(0)$": "London penetration depth at zero temperature", "$K(q)$": "kernel function", "$j_1(x)$": "spherical Bessel function of the first kind"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
468 |
Superconductivity
|
The Current in Superconductors
According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation it is given by:
\begin{align}
\hat{\mathbf{j}}(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\nabla \Psi) - (\nabla \Psi^{\dagger}) \Psi] - \frac{e^2}{mc} \Psi^{\dagger} \mathbf{A} \Psi \\
&\equiv \hat{\mathbf{j}}_1(\mathbf{r}) + \hat{\mathbf{j}}_2(\mathbf{r}) \label{eq:6.8.9}
\end{align}
Where
\begin{align}
\hat{\mathbf{j}}_1(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\nabla \Psi) - (\nabla \Psi^{\dagger}) \Psi] \\
&= \frac{e\hbar}{2m} \sum_{\mathbf{k},\mathbf{q}} (2\mathbf{k} + \mathbf{q}) e^{-i\mathbf{q}\cdot\mathbf{r}} (C_{\mathbf{k}+\mathbf{q}}^{\dagger} C_{\mathbf{k}} - C_{-\mathbf{k}\downarrow}^{\dagger} C_{-\mathbf{k}\downarrow}) \label{eq:6.8.10}
\end{align}
\begin{align}
\hat{\mathbf{j}}_2(\mathbf{r}) &= -\frac{e^2}{mc} \Psi^{\dagger} \mathbf{A} \Psi \\
&= -\frac{e^2}{mc} \mathbf{A}(\mathbf{r}) \sum_{\mathbf{k},\mathbf{q}} e^{-i\mathbf{q}\cdot\mathbf{r}} (C_{\mathbf{k}+\mathbf{q}}^{\dagger} C_{\mathbf{k}} + C_{-\mathbf{k}-\mathbf{q}}^{\dagger} C_{-\mathbf{k}-\mathbf{q}}) \label{eq:6.8.11}.
\end{align}
In terms of notation, we set $(\mathbf{k},\uparrow) = k,(-\mathbf{k},\downarrow)=-k$.
The Hamiltonian of the system interaction $H_1$ can be expressed in terms of quasiparticle operators $\alpha,\alpha^+$ as:
\begin{align}
H_1 = &-\frac{e\hbar}{mc} \sum_{\mathbf{k},\mathbf{q}} [\mathbf{k} \cdot \mathbf{A}(\mathbf{q})] [(u_{\mathbf{k}+\mathbf{q}}v_{\mathbf{k}} - u_{\mathbf{k}}v_{\mathbf{k}+\mathbf{q}}) (\alpha_{\mathbf{k}+\mathbf{q}\uparrow}^{\dagger}\alpha_{-\mathbf{k}\downarrow}^{\dagger} + \alpha_{-\mathbf{k}\downarrow}\alpha_{\mathbf{k}+\mathbf{q}\uparrow}) + \nonumber \\
& (u_{\mathbf{k}+\mathbf{q}}u_{\mathbf{k}} + v_{\mathbf{k}+\mathbf{q}}v_{\mathbf{k}}) (\alpha_{\mathbf{k}+\mathbf{q}\uparrow}^{\dagger}\alpha_{\mathbf{k}\uparrow} - \alpha_{-\mathbf{k}\downarrow}^{\dagger}\alpha_{-\mathbf{k}\downarrow})] \label{eq:6:8.8},
\end{align}
Where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:
\begin{equation}
u_k^2 = \frac{1}{2} \left(1 + \frac{\epsilon_k}{\xi_k}\right), \quad v_k^2 = \frac{1}{2} \left(1 - \frac{\epsilon_k}{\xi_k}\right),
\end{equation}
Where $\xi_k = \sqrt{\epsilon_k^2 + \Delta^2}$.
Based on the above information, please complete the following calculation:
1. Under the influence of a weak magnetic field, calculate the current $j_2(r)$ at zero temperature, and express the result using the London penetration depth $\lambda_L = \left( \frac{mc^2}{4\pi ne^2} \right)^{1/2}$;
|
[] |
Expression
|
{"$\\mathbf{j}$": "current density operator", "$e$": "elementary charge", "$\\hbar$": "reduced Planck's constant", "$m_1$": "effective mass of particle 1", "$m$": "effective mass of electron", "$\\Psi$": "field operator", "$\\mathbf{A}$": "vector potential", "$\\mathbf{k}$": "momentum vector", "$\\mathbf{q}$": "momentum transfer", "$C_{\\mathbf{k}}$": "annihilation operator for the state with momentum $\\mathbf{k}$", "$C_{\\mathbf{k}}^{\\dagger}$": "creation operator for the state with momentum $\\mathbf{k}$", "$H_1$": "interaction Hamiltonian", "$\\alpha$": "quasiparticle annihilation operator", "$\\alpha^+$": "quasiparticle creation operator", "$u_k$": "Bogoliubov transformation coefficient (u)", "$v_k$": "Bogoliubov transformation coefficient (v)", "$\\epsilon_k$": "single particle energy", "$\\xi_k$": "quasiparticle energy", "$\\Delta$": "superconducting energy gap", "$\\lambda_L$": "London penetration depth", "$n$": "electron density"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
469 |
Superconductivity
|
Current in Superconductors
According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation it is:
\begin{align}
\hat{\mathbf{j}}(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\nabla \Psi) - (\nabla \Psi^{\dagger}) \Psi] - \frac{e^2}{mc} \Psi^{\dagger} \mathbf{A} \Psi \\
&\equiv \hat{\mathbf{j}}_1(\mathbf{r}) + \hat{\mathbf{j}}_2(\mathbf{r}) \label{eq:6.8.9}
\end{align}
Where
\begin{align}
\hat{\mathbf{j}}_1(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\nabla \Psi) - (\nabla \Psi^{\dagger}) \Psi] \\
&= \frac{e\hbar}{2m} \sum_{\mathbf{k},\mathbf{q}} (2\mathbf{k} + \mathbf{q}) e^{-i\mathbf{q}\cdot\mathbf{r}} (C_{\mathbf{k}+\mathbf{q}}^{\dagger} C_{\mathbf{k}} - C_{-\mathbf{k}\downarrow}^{\dagger} C_{-\mathbf{k}\downarrow}) \label{eq:6.8.10}
\end{align}
\begin{align}
\hat{\mathbf{j}}_2(\mathbf{r}) &= -\frac{e^2}{mc} \Psi^{\dagger} \mathbf{A} \Psi \\
&= -\frac{e^2}{mc} \mathbf{A}(\mathbf{r}) \sum_{\mathbf{k},\mathbf{q}} e^{-i\mathbf{q}\cdot\mathbf{r}} (C_{\mathbf{k}+\mathbf{q}}^{\dagger} C_{\mathbf{k}} + C_{-\mathbf{k}-\mathbf{q}}^{\dagger} C_{-\mathbf{k}-\mathbf{q}}) \label{eq:6.8.11}.
\end{align}
Symbolically, we note $(\mathbf{k},\uparrow) = k,(-\mathbf{k},\downarrow)=-k$.
The Hamiltonian of the system interaction $H_1$ can be expressed using the quasiparticle operators $\alpha,\alpha^+$ as:
\begin{align}
H_1 = &-\frac{e\hbar}{mc} \sum_{\mathbf{k},\mathbf{q}} [\mathbf{k} \cdot \mathbf{A}(\mathbf{q})] [(u_{\mathbf{k}+\mathbf{q}}v_{\mathbf{k}} - u_{\mathbf{k}}v_{\mathbf{k}+\mathbf{q}}) (\alpha_{\mathbf{k}+\mathbf{q}\uparrow}^{\dagger}\alpha_{-\mathbf{k}\downarrow}^{\dagger} + \alpha_{-\mathbf{k}\downarrow}\alpha_{\mathbf{k}+\mathbf{q}\uparrow}) + \nonumber \\
& (u_{\mathbf{k}+\mathbf{q}}u_{\mathbf{k}} + v_{\mathbf{k}+\mathbf{q}}v_{\mathbf{k}}) (\alpha_{\mathbf{k}+\mathbf{q}\uparrow}^{\dagger}\alpha_{\mathbf{k}\uparrow} - \alpha_{-\mathbf{k}\downarrow}^{\dagger}\alpha_{-\mathbf{k}\downarrow})] \label{eq:6:8.8},
\end{align}
where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:
\begin{equation}
u_k^2 = \frac{1}{2} \left(1 + \frac{\epsilon_k}{\xi_k}\right), \quad v_k^2 = \frac{1}{2} \left(1 - \frac{\epsilon_k}{\xi_k}\right),
\end{equation}
where $\xi_k = \sqrt{\epsilon_k^2 + \Delta^2}$.
Based on the above information, please complete the following calculation:
Under a weak magnetic field, the system is in the superconducting ground state. Considering the first-order approximation, calculate the current $j_1$. Hint: if the answer exists in an integral, then find the integrand.
|
[] |
Expression
|
{"$e$": "elementary charge", "$\\hbar$": "reduced Planck's constant", "$m_1$": "mass associated with the first type of particle", "$m$": "mass", "$\\Psi$": "wave function", "$\\Psi^{\\dagger}$": "complex conjugate of the wave function", "$\\mathbf{A}$": "vector potential", "$C_{\\mathbf{k}}$": "quasiparticle operator for momentum state $\\mathbf{k}$", "$C_{\\mathbf{k}}^{\\dagger}$": "creation operator for quasiparticle in momentum state $\\mathbf{k}$", "$H_1$": "Hamiltonian of the system interaction", "$\\alpha$": "quasiparticle operator", "$\\alpha^+$": "creation operator for quasiparticles", "$u_k$": "Bogoliubov transformation coefficient", "$v_k$": "Bogoliubov transformation coefficient", "$\\epsilon_k$": "kinetic energy of quasiparticle with wavevector $k$", "$\\xi_k$": "energy spectrum related to superconductivity", "$\\Delta$": "superconducting gap", "$k_F$": "Fermi wavevector", "$\\lambda_L(0)$": "London penetration depth at zero temperature"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
470 |
Superconductivity
|
Current in Superconductors
According to quantum mechanics, the current density operator in an electromagnetic field can be derived from the continuity equation, and in the second quantization representation, it is expressed as:
\begin{align}
\hat{\mathbf{j}}(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\nabla \Psi) - (\nabla \Psi^{\dagger}) \Psi] - \frac{e^2}{mc} \Psi^{\dagger} \mathbf{A} \Psi \\
&\equiv \hat{\mathbf{j}}_1(\mathbf{r}) + \hat{\mathbf{j}}_2(\mathbf{r}) \label{eq:6.8.9}
\end{align}
Where
\begin{align}
\hat{\mathbf{j}}_1(\mathbf{r}) &= \frac{e\hbar}{2m_1} [\Psi^{\dagger} (\nabla \Psi) - (\nabla \Psi^{\dagger}) \Psi] \\
&= \frac{e\hbar}{2m} \sum_{\mathbf{k},\mathbf{q}} (2\mathbf{k} + \mathbf{q}) e^{-i\mathbf{q}\cdot\mathbf{r}} (C_{\mathbf{k}+\mathbf{q}}^{\dagger} C_{\mathbf{k}} - C_{-\mathbf{k}\downarrow}^{\dagger} C_{-\mathbf{k}\downarrow}) \label{eq:6.8.10}
\end{align}
\begin{align}
\hat{\mathbf{j}}_2(\mathbf{r}) &= -\frac{e^2}{mc} \Psi^{\dagger} \mathbf{A} \Psi \\
&= -\frac{e^2}{mc} \mathbf{A}(\mathbf{r}) \sum_{\mathbf{k},\mathbf{q}} e^{-i\mathbf{q}\cdot\mathbf{r}} (C_{\mathbf{k}+\mathbf{q}}^{\dagger} C_{\mathbf{k}} + C_{-\mathbf{k}-\mathbf{q}}^{\dagger} C_{-\mathbf{k}-\mathbf{q}}) \label{eq:6.8.11}.
\end{align}
In terms of notation, we consider $(\mathbf{k},\uparrow) = k,(-\mathbf{k},\downarrow)=-k$.
The Hamiltonian $H_1$ of the system interaction can be expressed using the quasiparticle operators $\alpha,\alpha^+$ as:
\begin{align}
H_1 = &-\frac{e\hbar}{mc} \sum_{\mathbf{k},\mathbf{q}} [\mathbf{k} \cdot \mathbf{A}(\mathbf{q})] [(u_{\mathbf{k}+\mathbf{q}}v_{\mathbf{k}} - u_{\mathbf{k}}v_{\mathbf{k}+\mathbf{q}}) (\alpha_{\mathbf{k}+\mathbf{q}\uparrow}^{\dagger}\alpha_{-\mathbf{k}\downarrow}^{\dagger} + \alpha_{-\mathbf{k}\downarrow}\alpha_{\mathbf{k}+\mathbf{q}\uparrow}) + \nonumber \\
& (u_{\mathbf{k}+\mathbf{q}}u_{\mathbf{k}} + v_{\mathbf{k}+\mathbf{q}}v_{\mathbf{k}}) (\alpha_{\mathbf{k}+\mathbf{q}\uparrow}^{\dagger}\alpha_{\mathbf{k}\uparrow} - \alpha_{-\mathbf{k}\downarrow}^{\dagger}\alpha_{-\mathbf{k}\downarrow})] \label{eq:6:8.8},
\end{align}
where $u_k,v_k$ are the coefficients of the Bogoliubov transformation:
\begin{equation}
u_k^2 = \frac{1}{2} \left(1 + \frac{\epsilon_k}{\xi_k}\right), \quad v_k^2 = \frac{1}{2} \left(1 - \frac{\epsilon_k}{\xi_k}\right),
\end{equation}
where $\xi_k = \sqrt{\epsilon_k^2 + \Delta^2}$.
Based on the above information, please complete the following calculation:
The total current can be written as $\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})$, find the expression for $K(q)$.
If the answer exists in an integral, then find the integrand.
|
[] |
Expression
|
{"$e$": "elementary charge", "$\\hbar$": "reduced Planck's constant", "$m_1$": "effective mass (component 1)", "$m$": "electron mass", "$c$": "speed of light", "$\\Psi$": "wave function", "$\\Psi^{\\dagger}$": "conjugate of the wave function", "$\\mathbf{A}$": "vector potential", "$H_1$": "interaction Hamiltonian", "$\\alpha$": "quasiparticle operator", "$\\alpha^+$": "conjugate quasiparticle operator", "$\\mathbf{k}$": "wave vector", "$\\mathbf{q}$": "momentum transfer vector", "$u_k$": "Bogoliubov transformation coefficient (u)", "$v_k$": "Bogoliubov transformation coefficient (v)", "$\\xi_k$": "quasiparticle energy", "$\\epsilon_k$": "kinetic energy", "$\\Delta$": "superconducting gap", "$n$": "conduction electron density", "$\\lambda_L(0)$": "London penetration depth at T=0"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
471 |
Superconductivity
|
Meissner Effect in Superconductors
Assume the current in a superconductor follows $\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})$. If $K(0)=0$, it indicates the absence of the Meissner effect in the superconductor; otherwise, it exists.
At the microscopic level, the current in a BCS superconductor can be written as
\begin{equation*}
\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})
\end{equation*}
\begin{equation}
K(q) = \frac{1}{\lambda_L^2(0)} \left\{1 - \frac{3}{4} \int_{-1}^{+1} (1-Z^2) dZ \times \int_{-\infty}^{\infty} d\epsilon \frac{1}{2} \frac{\xi_+ \xi_- - \epsilon_+ \epsilon_- - \Delta^2}{\xi_+ \xi_- (\xi_+ + \xi_-)} \right\} \label{eq:6.8.24}
\end{equation}
Here, $K(q)$ is isotropic, and
\begin{align}
\epsilon_{\pm} &= \epsilon_{\mathbf{k}\pm\mathbf{q}/2}, \quad \xi_{\pm} = \xi_{\mathbf{k}\pm\mathbf{q}/2} = \sqrt{\epsilon_{\pm}^2 + \Delta^2} \\
u_{\pm} &= u_{\mathbf{k}\pm\mathbf{q}/2}, \quad v_{\pm} = v_{\mathbf{k}\pm\mathbf{q}/2}, \quad Z = \cos\theta
\end{align}
The coherence length of the superconductor is $\xi_0 = \frac{\hbar v_F}{\pi \Delta(0)}$
According to the information, perform the calculation under the assumption $q\ll k_F$:
1. For a normal conductor, calculate $K(q)$
|
[] |
Expression
|
{"$\\mathbf{j}(\\mathbf{q})$": "current density as a function of momentum", "$c$": "speed of light", "$K(q)$": "kernel function as a function of momentum", "$\\mathbf{A}(\\mathbf{q})$": "vector potential as a function of momentum", "$q$": "momentum", "$\\lambda_L$": "London penetration depth", "$Z$": "variable related to angle", "$\\epsilon$": "energy variable", "$\\xi_+$": "quasiparticle energy for positive momentum shift", "$\\xi_-$": "quasiparticle energy for negative momentum shift", "$\\Delta$": "superconducting energy gap", "$\\epsilon_+$": "energy for positive momentum shift", "$\\epsilon_-$": "energy for negative momentum shift", "$u_+$": "coherence factor for positive momentum shift", "$u_-$": "coherence factor for negative momentum shift", "$v_+$": "coherence factor for positive momentum shift", "$v_-$": "coherence factor for negative momentum shift", "$\\theta$": "angle variable", "$\\xi_0$": "coherence length of the superconductor", "$\\hbar$": "reduced Planck's constant", "$v_F$": "Fermi velocity", "$k_F$": "Fermi wave number", "$K_n(q)$": "kernel function for a normal conductor"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
472 |
Superconductivity
|
Meissner Effect in Superconductors
Assume the current in a superconductor follows $\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})$. If $K(0)=0$, it can be shown that there is no Meissner effect in the superconductor, otherwise it exists.
Microscopically, the BCS superconducting current can be written as
\begin{equation*}
\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})
\end{equation*}
\begin{equation}
K(q) = \frac{1}{\lambda_L^2(0)} \left\{1 - \frac{3}{4} \int_{-1}^{+1} (1-Z^2) dZ \times \int_{-\infty}^{\infty} d\epsilon \frac{1}{2} \frac{\xi_+ \xi_- - \epsilon_+ \epsilon_- - \Delta^2}{\xi_+ \xi_- (\xi_+ + \xi_-)} \right\} \label{eq:6.8.24}
\end{equation}
where $K(q)$ is orientation-independent, and
\begin{align}
\epsilon_{\pm} &= \epsilon_{\mathbf{k}\pm\mathbf{q}/2}, \quad \xi_{\pm} = \xi_{\mathbf{k}\pm\mathbf{q}/2} = \sqrt{\epsilon_{\pm}^2 + \Delta^2} \\
u_{\pm} &= u_{\mathbf{k}\pm\mathbf{q}/2}, \quad v_{\pm} = v_{\mathbf{k}\pm\mathbf{q}/2}, \quad Z = \cos\theta
\end{align}
The coherence length of the superconductor is $\xi_0 =\frac{\hbar v_F}{\pi \Delta(0)}$
Based on the information, please complete the calculation under the assumption $q\ll k_F$:
If $q\xi_0 \ll 1$, calculate $K(q)$
|
[] |
Expression
|
{"$\\mathbf{j}$": "current", "$\\mathbf{q}$": "momentum vector", "$c$": "speed of light", "$K(q)$": "function representing the kernel at momentum q", "$\\mathbf{A}$": "vector potential", "$\\lambda_L$": "London penetration depth", "$Z$": "cosine of the angle theta", "$\\xi_+$": "energy term for positive momentum shift", "$\\xi_-$": "energy term for negative momentum shift", "$\\epsilon_+$": "kinetic energy term for positive momentum shift", "$\\epsilon_-$": "kinetic energy term for negative momentum shift", "$\\Delta$": "superconducting energy gap", "$\\epsilon$": "energy", "$\\xi$": "energy related to the superconducting gap", "$u_+$": "coefficient for particle-like quasiparticles with positive momentum shift", "$u_-$": "coefficient for particle-like quasiparticles with negative momentum shift", "$v_+$": "coefficient for hole-like quasiparticles with positive momentum shift", "$v_-$": "coefficient for hole-like quasiparticles with negative momentum shift", "$\\xi_0$": "coherence length of the superconductor", "$\\hbar$": "reduced Planck's constant", "$v_F$": "Fermi velocity", "$k_F$": "Fermi momentum"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
473 |
Superconductivity
|
Meissner effect in superconductors
Assuming the current in a superconductor follows $\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})$, if $K(0)=0$, it can be stated that the superconductor does not exhibit the Meissner effect, otherwise it does.
Microscopically, BCS superconductor current can be expressed as
\begin{equation*}
\mathbf{j}(\mathbf{q}) = -\frac{c}{4\pi} K(q) \mathbf{A}(\mathbf{q})
\end{equation*}
\begin{equation}
K(q) = \frac{1}{\lambda_L^2(0)} \left\{1 - \frac{3}{4} \int_{-1}^{+1} (1-Z^2) dZ \times \int_{-\infty}^{\infty} d\epsilon \frac{1}{2} \frac{\xi_+ \xi_- - \epsilon_+ \epsilon_- - \Delta^2}{\xi_+ \xi_- (\xi_+ + \xi_-)} \right\} \label{eq:6.8.24}
\end{equation}
Where $K(q)$ is orientation-independent, and
\begin{align}
\epsilon_{\pm} &= \epsilon_{\mathbf{k}\pm\mathbf{q}/2}, \quad \xi_{\pm} = \xi_{\mathbf{k}\pm\mathbf{q}/2} = \sqrt{\epsilon_{\pm}^2 + \Delta^2} \\
u_{\pm} &= u_{\mathbf{k}\pm\mathbf{q}/2}, \quad v_{\pm} = v_{\mathbf{k}\pm\mathbf{q}/2}, \quad Z = \cos\theta
\end{align}
Coherence length of the superconductor $\xi_0 =\frac{\hbar v_F}{\pi \Delta(0)}$
Based on the information, under the assumption $q\ll k_F$, complete the calculations:
If $q\xi_0 \gg 1$, calculate $K(q)$
|
[] |
Expression
|
{"$\\mathbf{j}(\\mathbf{q})$": "current density as a function of wave vector $\\mathbf{q}$", "$c$": "speed of light", "$K(q)$": "kernel function for wave vector $q$", "$\\mathbf{A}(\\mathbf{q})$": "vector potential as a function of wave vector $\\mathbf{q}$", "$\\lambda_L^2(0)$": "squared London penetration depth at zero temperature", "$Z$": "cosine of the angle $\\theta$", "$\\epsilon_{\\pm}$": "energy at $\\mathbf{k} \\pm \\mathbf{q}/2$", "$\\xi_{\\pm}$": "quasiparticle excitation energy at $\\mathbf{k} \\pm \\mathbf{q}/2$", "$\\Delta$": "superconducting gap parameter", "$u_{\\pm}$": "coherence factor $u$ at $\\mathbf{k} \\pm \\mathbf{q}/2$", "$v_{\\pm}$": "coherence factor $v$ at $\\mathbf{k} \\pm \\mathbf{q}/2$", "$\\xi_0$": "coherence length of the superconductor", "$\\hbar$": "reduced Planck constant", "$v_F$": "Fermi velocity", "$k_F$": "Fermi wave vector"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
474 |
Superconductivity
|
London penetration depth at finite temperatures
The London equation is a significant equation describing superconductors, reflecting the perfect diamagnetism of superconductors and can be written as
\begin{equation}
\mathbf{j}_s(\mathbf{r}) = - \frac{c}{4\pi}\frac{1}{\lambda_L^2}\mathbf{A}(\mathbf{r}),
\end{equation}
where $\lambda_L$ is called the London penetration depth, depicting the depth to which a magnetic field penetrates into a superconducting sample, and at zero temperature can be expressed as
\begin{equation}
\lambda_L^2(0) = \left( \frac{mc^2}{4\pi ne^2} \right)^{1/2}.
\end{equation}
Based on the context, complete the calculation:
Calculate the linear response coefficient of the diamagnetic current under first order approximation. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\mathbf{j}_s(\\mathbf{r})$": "supercurrent density at position r", "$\\lambda_L$": "London penetration depth", "$c$": "speed of light", "$m$": "mass", "$n$": "number density of Cooper pairs", "$e$": "elementary charge", "$T$": "temperature", "$\\mathbf{A}(\\mathbf{r})$": "vector potential at position r", "$\\mathbf{r}$": "position vector", "$\\mathbf{k}$": "wave vector", "$\\mathbf{q}$": "wave vector", "$\\Psi$": "superconducting wave function", "$C_{\\mathbf{k}}$": "annihilation operator for wave vector k", "$\\Delta$": "superconducting gap", "$\\epsilon_{\\mathbf{k}}$": "energy dispersion at wave vector k", "$\\xi_{\\mathbf{k}}$": "quasiparticle energy at wave vector k", "$K_2(q, T)$": "linear response coefficient of the diamagnetic current"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
475 |
Superconductivity
|
Ginzberg Landau Theory
Ginzburg and Landau proposed using a complex quantity $\psi(r)$ to describe the 'effective wave function' of superconducting electrons, with charge $e^*$ and mass $m^*$, and the corresponding system free energy density and free energy are:
\begin{equation}
f_s = f_n + \alpha(T) |\psi(\mathbf{r})|^2 + \frac{1}{2} \beta(T) |\psi(\mathbf{r})|^4 + \frac{1}{8\pi} (\nabla \times \mathbf{A}) \cdot (\nabla \times \mathbf{A}) + \frac{1}{2m^*} \left| \left(-i\hbar \nabla - \frac{e^*}{c}\mathbf{A}\right) \psi(\mathbf{r}) \right|^2 \label{eq:6.10.6},
\end{equation}
The system's free energy is
\begin{equation}
F_s = \int f_s dr \label{eq:6.10.7}
\end{equation}
Please calculate:
The condition for thermodynamic equilibrium (consider variational performance with respect to the order parameter and vector potential) to derive the equation satisfied by $\psi$, namely the Ginzburg-Landau equation. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\psi$": "effective wave function", "$r$": "position", "$e^*$": "effective charge of superconducting electrons", "$m^*$": "effective mass of superconducting electrons", "$f_s$": "system free energy density", "$f_n$": "normal state free energy density", "$\\alpha(T)$": "temperature-dependent coefficient for the quadratic term", "$\\beta(T)$": "temperature-dependent coefficient for the quartic term", "$\\mathbf{A}$": "vector potential", "$\\hbar$": "reduced Planck's constant", "$c$": "speed of light", "$F_s$": "system free energy", "$\\delta F_s$": "variation of system free energy", "$\\mathbf{j}$": "current density", "$\\mathbf{B}$": "magnetic field"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
476 |
Superconductivity
|
Ginzburg-Landau Theory
Ginzburg and Landau proposed using a complex variable $\psi(r)$ to describe the "effective wave function" of superconducting electrons, with charge $e^*$ and mass $m^*$. The corresponding system free energy density and free energy are:
\begin{equation}
f_s = f_n + \alpha(T) |\psi(\mathbf{r})|^2 + \frac{1}{2} \beta(T) |\psi(\mathbf{r})|^4 + \frac{1}{8\pi} (\nabla \times \mathbf{A}) \cdot (\nabla \times \mathbf{A}) + \frac{1}{2m^*} \left| \left(-i\hbar \nabla - \frac{e^*}{c}\mathbf{A}\right) \psi(\mathbf{r}) \right|^2 \label{eq:6.10.6},
\end{equation}
The free energy of the system is
\begin{equation}
F_s = \int f_s dr \label{eq:6.10.7}
\end{equation}
Please calculate:
Solve the Ginzburg-Landau equation under $A=0$;
|
[] |
Expression
|
{"$\\psi(r)$": "effective wave function of superconducting electrons", "$e^*$": "charge of superconducting electrons", "$m^*$": "mass of superconducting electrons", "$f_s$": "superconducting free energy density", "$f_n$": "normal state free energy density", "$\\alpha(T)$": "temperature-dependent parameter (alpha)", "$\\beta(T)$": "temperature-dependent parameter (beta)", "$\\mathbf{r}$": "position vector", "$\\mathbf{A}$": "magnetic vector potential", "$\\hbar$": "reduced Planck's constant", "$c$": "speed of light", "$F_s$": "superconducting free energy", "$\\psi_0$": "uniform solution for the effective wave function", "$T$": "temperature", "$T_c$": "critical temperature"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
477 |
Superconductivity
|
Ginzburg Landau Theory
Ginzburg and Landau propose using a complex quantity $\psi(r)$ to describe the 'effective wave function' of superconducting electrons, with charge $e^*$ and mass $m^*$. The corresponding system free energy density and free energy are:
\begin{equation}
f_s = f_n + \alpha(T) |\psi(\mathbf{r})|^2 + \frac{1}{2} \beta(T) |\psi(\mathbf{r})|^4 + \frac{1}{8\pi} (\nabla \times \mathbf{A}) \cdot (\nabla \times \mathbf{A}) + \frac{1}{2m^*} \left| \left(-i\hbar \nabla - \frac{e^*}{c}\mathbf{A}\right) \psi(\mathbf{r}) \right|^2 \label{eq:6.10.6},
\end{equation}
The system's free energy is
\begin{equation}
F_s = \int f_s dr \label{eq:6.10.7}
\end{equation}
Please calculate:
The penetration depth $\lambda(T)$ in a weak magnetic field (expressed in terms of the result from question two);
|
[] |
Expression
|
{"$\\psi(r)$": "effective wave function of superconducting electrons", "$e^*$": "effective charge of superconducting electrons", "$m^*$": "effective mass of superconducting electrons", "$f_s$": "system free energy density", "$f_n$": "normal state free energy density", "$\\alpha(T)$": "temperature-dependent coefficient", "$\\beta(T)$": "temperature-dependent coefficient", "$\\mathbf{A}$": "magnetic vector potential", "$\\hbar$": "reduced Planck's constant", "$c$": "speed of light in vacuum", "$F_s$": "system's free energy", "$\\lambda(T)$": "penetration depth", "$\\psi_0$": "field-free order parameter"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
478 |
Superconductivity
|
Ginzburg-Landau Theory
Ginzburg and Landau proposed using a complex quantity $\psi(r)$ to describe the "effective wave function" of superconducting electrons, with charge $e^*$ and mass $m^*$; the corresponding system free energy density and free energy are:
\begin{equation}
f_s = f_n + \alpha(T) |\psi(\mathbf{r})|^2 + \frac{1}{2} \beta(T) |\psi(\mathbf{r})|^4 + \frac{1}{8\pi} (\nabla \times \mathbf{A}) \cdot (\nabla \times \mathbf{A}) + \frac{1}{2m^*} \left| \left(-i\hbar \nabla - \frac{e^*}{c}\mathbf{A}\right) \psi(\mathbf{r}) \right|^2 \label{eq:6.10.6},
\end{equation}
The free energy of the system is
\begin{equation}
F_s = \int f_s dr \label{eq:6.10.7}
\end{equation}
Please calculate:
Calculate the magnetic flux and thereby demonstrate the quantization of magnetic flux.
|
[] |
Expression
|
{"$\\psi$": "effective wave function of superconducting electrons", "$r$": "position vector", "$e^*$": "effective charge of superconducting electrons", "$m^*$": "effective mass of superconducting electrons", "$f_s$": "system free energy density", "$f_n$": "normal state free energy density", "$\\alpha(T)$": "temperature-dependent Ginzburg-Landau coefficient", "$\\beta(T)$": "temperature-dependent Ginzburg-Landau coefficient", "$\\mathbf{A}$": "vector potential", "$\\hbar$": "reduced Planck constant", "$c$": "speed of light", "$\\mathbf{j}$": "current density", "$\\varphi$": "phase of the wave function", "$C$": "circuit path", "$\\mathbf{B}$": "magnetic flux density", "$S$": "surface enclosed by loop C", "$\\Phi$": "magnetic flux", "$\\Phi_0$": "magnetic flux quantum", "$n$": "integer (quantization index)"}
|
Superconductivity: BCS theory and microscopic mechanism
| ||
479 |
Strongly Correlated Systems
|
Hubbard Model in Narrow Bandwidth
Consider the following single band Hubbard model,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c,c^\dagger$ are the annihilation and creation operators for electrons, $n$ is the particle number operator
Please complete the following calculation:
Calculate the off-diagonal elements of the single-particle Green's function obtained in the Bloch representation You should return your answer as an equation.
|
[] |
Equation
|
{"$H$": "Hamiltonian", "$T_{ij}$": "hopping parameter between sites i and j", "$c_{i\\sigma}$": "annihilation operator for an electron with spin sigma at site i", "$c_{i\\sigma}^{\\dagger}$": "creation operator for an electron with spin sigma at site i", "$U$": "on-site interaction energy", "$n_{i\\sigma}$": "particle number operator for spin sigma at site i", "$n_{i\\bar{\\sigma}}$": "particle number operator for the opposite spin at site i", "$G_{kk'}^\\sigma(\\omega)$": "off-diagonal Green's function in Bloch representation", "$C_{k\\sigma}$": "annihilation operator for an electron with wave vector k and spin sigma", "$C_{k\\sigma}^+$": "creation operator for an electron with wave vector k and spin sigma", "$N$": "number of lattice sites", "$\\mathbf{k}$": "wave vector", "$\\mathbf{R}_i$": "position vector of site i", "$G_k^\\sigma(\\omega)$": "diagonal Green's function in Bloch representation", "$\\delta_{kk'}$": "Kronecker delta, equals 1 if k=k' and 0 otherwise"}
|
Strongly correlated system
| ||
480 |
Strongly Correlated Systems
|
Hubbard Model under Narrow Bandwidth — Green's Function Analysis
Consider the following single-band Hubbard model,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c, c^\dagger$ are electron annihilation and creation operators, $n$ is the particle number operator.
In the case of bandwidth $\Delta \neq 0$, the single-particle green function approximately satisfies
\begin{equation}
G_k^\sigma(\omega) = \frac{\omega - T_0 - U(1 - (n_{\bar{\sigma}}))}{(\omega - E_k)(\omega - T_0 - U) + (n_{\bar{\sigma}}) U(T_0 - E_k)}
\end{equation}
Please solve the following problem:
Calculate the energy spectrum of the system;
|
[] |
Expression
|
{"$H$": "Hamiltonian", "$T_{ij}$": "hopping matrix element", "$c_{i\\sigma}$": "annihilation operator for an electron with spin \\sigma at site i", "$c_{i\\sigma}^{\\dagger}$": "creation operator for an electron with spin \\sigma at site i", "$U$": "onsite Coulomb interaction", "$n_{i\\sigma}$": "number operator for electrons with spin \\sigma at site i", "$n_{i\\bar{\\sigma}}$": "number operator for electrons with opposite spin of \\sigma at site i", "$\\Delta$": "bandwidth", "$G_k^\\sigma(\\omega)$": "single-particle Green's function for a given wave vector k and spin \\sigma", "$\\omega$": "frequency variable", "$T_0$": "average hopping energy", "$E_k$": "energy spectrum related to wave vector k", "$E_{k\\sigma}^{(1)}$": "first pole of the Green's function for wave vector k and spin \\sigma", "$E_{k\\sigma}^{(2)}$": "second pole of the Green's function for wave vector k and spin \\sigma", "$A_{k\\sigma}^{(1)}$": "amplitude or spectral weight related to the first pole of the Green's function", "$A_{k\\sigma}^{(2)}$": "amplitude or spectral weight related to the second pole of the Green's function"}
|
Strongly correlated system
| ||
481 |
Strongly Correlated Systems
|
Hubbard Model in Narrow Band - Green's Function Analysis
Consider the following single-band Hubbard model,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c,c^\dagger$ are the annihilation and creation operators of electrons, and $n$ is the particle number operator.
For non-zero bandwidth $\Delta \neq 0$, the single-particle Green's function approximately satisfies
\begin{equation}
G_k^\sigma(\omega) = \frac{\omega - T_0 - U(1 - < n_{\bar{\sigma}} >)}{(\omega - E_k)(\omega - T_0 - U) + < n_{\bar{\sigma}} > U(T_0 - E_k)}
\end{equation}
Please complete the following problem:
Calculate the Green's function for $U=0$ You should return your answer as an equation.
|
[] |
Equation
|
{"$H$": "Hamiltonian", "$T_{ij}$": "hopping parameter between sites i and j", "$c_{i\\sigma}$": "annihilation operator for an electron at site i with spin sigma", "$c_{i\\sigma}^{\\dagger}$": "creation operator for an electron at site i with spin sigma", "$U$": "on-site interaction energy", "$n_{i\\sigma}$": "particle number operator for electrons at site i with spin sigma", "$n_{i\\bar{\\sigma}}$": "particle number operator for electrons at site i with opposite spin to sigma", "$\\Delta$": "bandwidth", "$G_k^\\sigma(\\omega)$": "single-particle Green's function for momentum k and spin sigma", "$\\omega$": "frequency (or energy variable) in the Green's function", "$T_0$": "on-site energy", "$< n_{\\bar{\\sigma}} >$": "average particle number for opposite spin to sigma", "$E_k$": "energy of the band electron with momentum k"}
|
Strongly correlated system
| ||
482 |
Strongly Correlated Systems
|
Hubbard Model in the Narrow Band Limit—Green's Function Analysis
Consider the single-band Hubbard model as follows,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c,c^\dagger$ are the annihilation and creation operators for electrons, and $n$ is the particle number operator.
In the case where bandwidth $\Delta \neq 0$, the single-particle Green's function approximately satisfies
\begin{equation}
G_k^\sigma(\omega) = \frac{\omega - T_0 - U(1 - < n_{\bar{\sigma}} >)}{(\omega - E_k)(\omega - T_0 - U) + < n_{\bar{\sigma}} > U(T_0 - E_k)}
\end{equation}
Please complete the following problem:
3. Calculate the density of states for the case of $U=0$ You should return your answer as an equation.
|
[] |
Equation
|
{"$H$": "Hamiltonian", "$T_{ij}$": "hopping matrix element between sites i and j", "$c_{i\\sigma}^{\\dagger}$": "creation operator for electrons with spin σ at site i", "$c_{j\\sigma}$": "annihilation operator for electrons with spin σ at site j", "$U$": "Coulomb interaction energy", "$n_{i\\sigma}$": "particle number operator for electrons with spin σ at site i", "$n_{i\\bar{\\sigma}}$": "particle number operator for electrons with the opposite spin at site i", "$\\Delta$": "bandwidth", "$G_k^\\sigma(\\omega)$": "single-particle Green's function for momentum k and spin σ", "$\\omega$": "frequency", "$T_0$": "center of the energy band", "$E_k$": "energy of electrons with momentum k", "$< n_{\\bar{\\sigma}} >$": "expectation value of the particle number operator for electrons with the opposite spin", "$\\rho_\\sigma (\\omega)$": "density of states for spin σ", "$N$": "number of lattice sites", "$D(\\omega)$": "density of states function of a single band"}
|
Strongly correlated system
| ||
483 |
Strongly Correlated Systems
|
Narrow-band Hubbard model—Green's function analysis
Consider the following single-band Hubbard model,
\begin{equation}
H = \sum_{i,j,\sigma} T_{ij} c_{i\sigma}^{\dagger} c_{j\sigma} + \frac{U}{2} \sum_{i,\sigma} n_{i\sigma} n_{i\bar{\sigma}} \label{eq:11.1.13}
\end{equation}
where $c,c^\dagger$ are the annihilation and creation operators for electrons, and $n$ is the particle number operator.
When the band width $\Delta \neq 0$, the single-particle Green's function approximately satisfies
\begin{equation}
G_k^\sigma(\omega) = \frac{\omega - T_0 - U(1 - (n_{\bar{\sigma}}))}{(\omega - E_k)(\omega - T_0 - U) + (n_{\bar{\sigma}}) U(T_0 - E_k)}
\end{equation}
Please complete the following problem:
Calculate the Green's function in the case $U\geq \Delta$ You should return your answer as an equation.
|
[] |
Equation
|
{"$H$": "Hamiltonian", "$T_{ij}$": "hopping matrix element between sites i and j", "$c_{i\\sigma}$": "annihilation operator for an electron with spin sigma at site i", "$c_{i\\sigma}^{\\dagger}$": "creation operator for an electron with spin sigma at site i", "$U$": "on-site Coulomb repulsion", "$n_{i\\sigma}$": "number operator for electrons with spin sigma at site i", "$n_{i\\bar{\\sigma}}$": "number operator for electrons with opposite spin to sigma at site i", "$\\Delta$": "band width", "$G_k^\\sigma$": "Green's function for wave vector k and spin sigma", "$\\omega$": "frequency variable", "$T_0$": "reference energy level", "$E_k$": "energy dispersion relation for wave vector k", "$E_{k\\sigma}^{(1)}$": "first energy eigenvalue for wave vector k and spin sigma", "$E_{k\\sigma}^{(2)}$": "second energy eigenvalue for wave vector k and spin sigma", "$A_{k\\sigma}^{(1)}$": "spectral weight factor for first energy level for wave vector k and spin sigma", "$A_{k\\sigma}^{(2)}$": "spectral weight factor for second energy level for wave vector k and spin sigma"}
|
Strongly correlated system
| ||
484 |
Strongly Correlated Systems
|
Anderson s-d exchange model and Green's function equation of motion
In 1961, Anderson proposed the s-d mixing model, suggesting that to discuss the formation of local magnetic moments by transition metal impurity atoms in a non-magnetic metal matrix, two factors must be considered: First, similar to the formation of intrinsic magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant impact on the formation of local magnetic moments in the crystal; second, since the free atomic d-orbital states $\phi_d(\mathbf{r})$ of impurities in crystals are no longer purely eigenstates, especially due to the tendency for electron delocalization in metal crystals into Bloch states (s orbitals), there exists an electron transfer between $\phi_d$ and $\phi_k$ states, which Anderson called s-d mixing. Therefore, he pointed out that the system's Hamiltonian $H$ should consist of the following four parts:
\begin{equation}
H = \sum_{k,\sigma} E_{k\sigma} n_{k\sigma} + \sum_{\sigma} E_{d\sigma} n_{d\sigma} + \frac{U}{2}\sum_{\sigma} n_{d\sigma} n_{d\bar{\sigma}} + \sum_{k,\sigma} V_{kd} (C_{k\sigma}^\dagger d_\sigma + d_\sigma^\dagger C_{k\sigma})
\label{eq:11.2.5}
\end{equation}
where
\begin{equation}
E_{k\sigma} = E_k + \sigma \mu_B h, \quad E_{d\sigma} = E_d + \sigma \mu_B h
\label{eq:11.2.6}
\end{equation}
Here $\mu_B = \left( \left| \frac{e}{2mc} \right| \hbar \right)$ is the Bohr magneton, and suppose the Lande factor of electrons and impurities $g_0 = g_i = 2$, which is the non-degenerate orbital Anderson s-d mixing model.
In dealing with the $s-d$ exchange model, the Green's function equation of motion method is often used:
Starting from the double-time Green's function
\begin{equation*}
\ll A(t); B(t') \gg = -i \theta(t-t') <[A(t), B(t')]_+>
\end{equation*}
By utilizing a technique, differentiate the function $\ll A(t); B(t') \gg$ with respect to $t$ and $t'$, yielding the following two equations of motion:
\begin{equation}
i \frac{d}{dt} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll [A, H]; B(t') \gg
\label{eq:11.2.7}
\end{equation}
\begin{equation}
-i \frac{d}{dt'} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll A(t); [B, H] \gg
\label{eq:11.2.8}
\end{equation}
Perform a Fourier transform
\begin{equation}
\ll A | B \gg_\omega = \int dte^{i\omega(t-t')} \ll A(t); B(t') \gg
\label{eq:11.2.9}
\end{equation}
Obtaining two forms of the general equation of motion for Green’s function
\begin{align}
\omega \ll A | B \gg_\omega &= <[A, B]_+> + \ll [A, H] | B \gg_\omega
\label{eq:11.2.10} \\
\omega \ll A | B \gg_\omega &= <[A, B]_+> - \ll A | [B, H] \gg_\omega
\label{eq:11.2.11}
\end{align}
Please complete the problem
Calculate the Green's function equation of motion for $\ll C_{k\sigma} | C_{k'\sigma}^+ \gg_\omega$ in the s-d exchange model. Hint: Let $a_{kk'\sigma}$ symbolize $\ll C_{k\sigma} | C_{k'\sigma}^+ \gg_\omega$, and $b_{k'\sigma}$ symbolize $\ll d_\sigma | C_{k'\sigma}^+ \gg_\omega$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\phi_d$": "d-orbital state of impurity atom", "$\\phi_k$": "Bloch state of metal crystal", "$H$": "system Hamiltonian", "$k$": "momentum index", "$\\sigma$": "spin index", "$E_{k\\sigma}$": "energy of state with momentum $k$ and spin $\\sigma$", "$n_{k\\sigma}$": "number operator for state with momentum $k$ and spin $\\sigma$", "$E_{d\\sigma}$": "energy of d-state with spin $\\sigma$", "$n_{d\\sigma}$": "number operator for d-state with spin $\\sigma$", "$U$": "Coulomb interaction parameter", "$\\bar{\\sigma}$": "opposite spin of $\\sigma$", "$V_{kd}$": "s-d mixing term", "$C_{k\\sigma}^\\dagger$": "creation operator for Bloch state with momentum $k$ and spin $\\sigma$", "$d_\\sigma$": "annihilation operator for d-state with spin $\\sigma$", "$\\mu_B$": "Bohr magneton", "$\\omega$": "frequency variable in Fourier transform", "$a_{kk'\\sigma}$": "Green's function for states $k\\sigma$ and $k'\\sigma$", "$b_{k'\\sigma}$": "Green's function for d-state and state $k'\\sigma$", "$\\delta_{k, k'}$": "Kronecker delta for momentum indices"}
|
Strongly correlated system
| ||
485 |
Strongly Correlated Systems
|
Anderson s-d exchange model and Green's function equation of motion
In 1961, Anderson proposed the s-d mixing model, suggesting that when discussing the formation of local magnetic moments by transition metal impurity atoms in a non-magnetic metal matrix, it is essential to consider two factors: Firstly, similar to the inherent magnetic moment formed by a free atom, the Coulomb interaction among d-shell electrons in impurity atoms significantly affects the formation of local magnetic moments in the crystal. Secondly, since the d-orbital state of impurity atoms, $\phi_d(\mathbf{r})$, in a crystal are no longer rigorous eigenstates, especially due to the tendency of electrons becoming delocalized into Bloch states (s orbitals) in the metal crystal, there exists an electron transfer between $\phi_d$ and $\phi_k$ states, a phenomenon that Anderson termed as s-d mixing. Therefore, he stated that the Hamiltonian $H$ of the system should be composed of the following four parts:
\begin{equation}
H = \sum_{k,\sigma} E_{k\sigma} n_{k\sigma} + \sum_{\sigma} E_{d\sigma} n_{d\sigma} + \frac{U}{2}\sum_{\sigma} n_{d\sigma} n_{d\bar{\sigma}} + \sum_{k,\sigma} V_{kd} (C_{k\sigma}^\dagger d_\sigma + d_\sigma^\dagger C_{k\sigma})
\label{eq:11.2.5}
\end{equation}
where
\begin{equation}
E_{k\sigma} = E_k + \sigma \mu_B h, \quad E_{d\sigma} = E_d + \sigma \mu_B h
\label{eq:11.2.6}
\end{equation}
Here, $\mu_B = \left( \left| \frac{e}{2mc} \right| \hbar \right)$ is the Bohr magneton, and assuming the Lande's g-factor of electron and impurity $g_0 = g_i = 2$, this constitutes the non-orbitally degenerate Anderson s-d mixing model.
When dealing with the $s-d$ exchange model, the following Green's function equation of motion is often used:
Starting from the double-time Green's function
\begin{equation*}
\ll A(t); B(t') \gg = -i \theta(t-t') <[A(t), B(t')]_+>
\end{equation*}
a technique is used to take derivatives of the function $\ll A(t); B(t') \gg$ with respect to $t$ and $t'$ separately, yielding the following two equations of motion:
\begin{equation}
i \frac{d}{dt} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll [A, H]; B(t') \gg
\label{eq:11.2.7}
\end{equation}
\begin{equation}
-i \frac{d}{dt'} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll A(t); [B, H] \gg
\label{eq:11.2.8}
\end{equation}
By performing a Fourier transform
\begin{equation}
\ll A | B \gg_\omega = \int dte^{i\omega(t-t')} \ll A(t); B(t') \gg
\label{eq:11.2.9}
\end{equation}
two representations of the general form of the Green's function equation of motion are obtained:
\begin{align}
\omega \ll A | B \gg_\omega &= <[A, B]_+> + \ll [A, H] | B \gg_\omega
\label{eq:11.2.10} \\
\omega \ll A | B \gg_\omega &= <[A, B]_+> - \ll A | [B, H] \gg_\omega
\label{eq:11.2.11}
\end{align}
Please complete the question
Calculate the Green's function equation of motion related to $\ll d_\sigma | d_\sigma^+ \gg_\omega$ in the s-d exchange model. Hint: Let $a_{\sigma}$ syimbolize $\ll d_\sigma | d_\sigma^+ \gg_\omega$, $b_\sigma$ symbolize $\sum_k V_{kd} \ll C_{k\sigma} | d_\sigma^+ \gg_\omega$, and $c_{\sigma}$ symbolize $\ll n_{d\bar{\sigma}} d_\sigma | d_\sigma^+ \gg_\omega$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\phi_d$": "d-orbital state of impurity atoms", "$\\phi_k$": "Bloch state (s orbitals) in the metal crystal", "$H$": "Hamiltonian of the system", "$E_{k\\sigma}$": "energy of the k-th Bloch state with spin \\sigma", "$n_{k\\sigma}$": "number operator for k-th Bloch state with spin \\sigma", "$E_{d\\sigma}$": "energy of the d-orbital state with spin \\sigma", "$n_{d\\sigma}$": "number operator for d-orbital state with spin \\sigma", "$U$": "Coulomb interaction energy", "$n_{d\\bar{\\sigma}}$": "number operator for d-orbital state with opposite spin", "$V_{kd}$": "s-d coupling constant", "$C_{k\\sigma}$": "annihilation operator for k-th Bloch state with spin \\sigma", "$d_\\sigma$": "annihilation operator for d-orbital state with spin \\sigma", "$\\mu_B$": "Bohr magneton", "$h$": "external magnetic field", "$g_0$": "Lande's g-factor for electron", "$g_i$": "Lande's g-factor for impurity", "$\\omega$": "angular frequency in Fourier transform", "$a_{\\sigma}$": "Green's function for d-orbital interaction with self", "$b_{\\sigma}$": "Green's function for coupling between Bloch states and d-orbitals", "$c_{\\sigma}$": "Green's function for interaction involving Coulomb interaction"}
|
Strongly correlated system
| ||
486 |
Strongly Correlated Systems
|
Anderson s-d exchange model and Green's function equations of motion
In 1961, Anderson proposed the s-d mixing model, arguing that to discuss the formation of localized magnetic moments by transition metal impurity atoms in a non-magnetic metallic matrix, two factors must be considered simultaneously: First, similar to the intrinsic magnetic moment formation in free atoms, the Coulomb interaction of d shell electrons in impurity atoms significantly influences the formation of localized magnetic moments in the crystal; Second, because the d orbital states $\phi_d(\mathbf{r})$ of the impurity in the crystal are no longer eigenstates, particularly due to the tendency of electron delocalization into Bloch states in the metallic crystal (s orbital), there is mutual electron transfer between the states $\phi_d$ and $\phi_k$, which Anderson termed the s-d mixing interaction. Therefore, he pointed out that the Hamiltonian $H$ of the system should consist of the following four parts:
\begin{equation}
H = \sum_{k,\sigma} E_{k\sigma} n_{k\sigma} + \sum_{\sigma} E_{d\sigma} n_{d\sigma} + \frac{U}{2}\sum_{\sigma} n_{d\sigma} n_{d\bar{\sigma}} + \sum_{k,\sigma} V_{kd} (C_{k\sigma}^\dagger d_\sigma + d_\sigma^\dagger C_{k\sigma})
\label{eq:11.2.5}
\end{equation}
where
\begin{equation}
E_{k\sigma} = E_k + \sigma \mu_B h, \quad E_{d\sigma} = E_d + \sigma \mu_B h
\label{eq:11.2.6}
\end{equation}
Here $\mu_B = \left( \left| \frac{e}{2mc} \right| \hbar \right)$ is the Bohr magneton, and the Landé factor for the electron and impurity is set as $g_0 = g_i = 2$, which defines the non-degenerate orbital Anderson s-d mixing model.
In dealing with the $s-d$ exchange model, the following Green's function equation of motion method is often employed:
Starting from the two-time Green's function
\begin{equation*}
\ll A(t); B(t') \gg = -i \theta(t-t') <[A(t), B(t')]_+>
\end{equation*}
By employing a technique, the differential of the function $\ll A(t); B(t') \gg$ with respect to $t$ and $t'$ can be taken respectively, resulting in the following two equations of motion:
\begin{equation}
i \frac{d}{dt} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll [A, H]; B(t') \gg
\label{eq:11.2.7}
\end{equation}
\begin{equation}
-i \frac{d}{dt'} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll A(t); [B, H] \gg
\label{eq:11.2.8}
\end{equation}
Performing Fourier transform
\begin{equation}
\ll A | B \gg_\omega = \int dte^{i\omega(t-t')} \ll A(t); B(t') \gg
\label{eq:11.2.9}
\end{equation}
achieves the two representations of the Green's function equation of motion
\begin{align}
\omega \ll A | B \gg_\omega &= <[A, B]_+> + \ll [A, H] | B \gg_\omega
\label{eq:11.2.10} \\
\omega \ll A | B \gg_\omega &= <[A, B]_+> - \ll A | [B, H] \gg_\omega
\label{eq:11.2.11}
\end{align}
Please complete the problem
1. Derive the equation of motion for the s-d exchange model concerning the mixed Green's function $\ll C_{k\sigma} | d_\sigma^+ \gg_\omega$. Hint: Let $a_{k\sigma}$ symbolize $\ll C_{k\sigma} | d_\sigma^+ \gg_\omega$, and $b_{\sigma}$ symbolize $\ll d_\sigma | d_\sigma^+ \gg_\omega$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\phi_d$": "d orbital state of the impurity", "$\\phi_k$": "k orbital state in the host crystal", "$H$": "Hamiltonian of the system", "$E_{k\\sigma}$": "energy of the k state with spin sigma", "$n_{k\\sigma}$": "number operator for k state with spin sigma", "$E_{d\\sigma}$": "energy of the d state with spin sigma", "$n_{d\\sigma}$": "number operator for d state with spin sigma", "$U$": "Coulomb interaction term", "$n_{d\\bar{\\sigma}}$": "number operator for d state with opposite spin sigma", "$V_{kd}$": "s-d mixing interaction", "$C_{k\\sigma}$": "annihilation operator for k state with spin sigma", "$d_\\sigma$": "annihilation operator for d state with spin sigma", "$\\mu_B$": "Bohr magneton", "$\\sigma$": "spin index", "$a_{k\\sigma}$": "Green's function symbol for $\\ll C_{k\\sigma} | d_\\sigma^+ \\gg_\\omega$", "$b_{\\sigma}$": "Green's function symbol for $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$", "$\\omega$": "frequency", "$h$": "magnetic field"}
|
Strongly correlated system
| ||
487 |
Strongly Correlated Systems
|
Anderson s-d exchange model and Green's function equation of motion
In 1961, Anderson proposed the s-d mixing model, suggesting that discussing the formation of local magnetic moments of transition-metal impurity atoms in a non-magnetic metal matrix must simultaneously consider two factors: First, similar to the formation of inherent magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant effect on the formation of local magnetic moments in crystals. Second, since the free atomic d-orbital state $\phi_d(\mathbf{r})$ of impurities in crystals is no longer strictly an eigenstate, especially due to the tendency of electron delocalization into Bloch orbital states (s-orbital) in metal crystals, there is electronic transfer between $\phi_d$ and $\phi_k$ states, which Anderson termed s-d mixing. To this end, he pointed out that the system's Hamiltonian $H$ should be composed of the following four parts:
\begin{equation}
H = \sum_{k,\sigma} E_{k\sigma} n_{k\sigma} + \sum_{\sigma} E_{d\sigma} n_{d\sigma} + \frac{U}{2}\sum_{\sigma} n_{d\sigma} n_{d\bar{\sigma}} + \sum_{k,\sigma} V_{kd} (C_{k\sigma}^\dagger d_\sigma + d_\sigma^\dagger C_{k\sigma})
\label{eq:11.2.5}
\end{equation}
where
\begin{equation}
E_{k\sigma} = E_k + \sigma \mu_B h, \quad E_{d\sigma} = E_d + \sigma \mu_B h
\label{eq:11.2.6}
\end{equation}
here $\mu_B = \left( \left| \frac{e}{2mc} \right| \hbar \right)$ is the Bohr magneton, assuming the Lande factor of electrons and impurities $g_0 = g_i = 2$, this is the non-degenerate orbital Anderson s-d mixing model.
In handling the $s-d$ exchange model, the Green's function equation of motion method is often used:
Starting from the double-time Green's function
\begin{equation*}
\ll A(t); B(t') \gg = -i \theta(t-t') <[A(t), B(t')]_+>
\end{equation*}
Using a technique, derive the derivatives of the function $\ll A(t); B(t') \gg$ with respect to $t$ and $t'$ separately to obtain the following two equations of motion:
\begin{equation}
i \frac{d}{dt} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll [A, H]; B(t') \gg
\label{eq:11.2.7}
\end{equation}
\begin{equation}
-i \frac{d}{dt'} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll A(t); [B, H] \gg
\label{eq:11.2.8}
\end{equation}
Performing a Fourier transform
\begin{equation}
\ll A | B \gg_\omega = \int dte^{i\omega(t-t')} \ll A(t); B(t') \gg
\label{eq:11.2.9}
\end{equation}
leads to two expressions for the Green's function equation of motion
\begin{align}
\omega \ll A | B \gg_\omega &= <[A, B]_+> + \ll [A, H] | B \gg_\omega
\label{eq:11.2.10} \\
\omega \ll A | B \gg_\omega &= <[A, B]_+> - \ll A | [B, H] \gg_\omega
\label{eq:11.2.11}
\end{align}
Please complete the problem
Calculate the equation of motion for the mixed Green's function $\ll d_\sigma | C_{k'\sigma}^+ \gg_\omega$ in the s-d exchange model. Hint: Let $a_{k'\sigma}$ symbolize $\ll d_\sigma | C_{k'\sigma}^+ \gg_\omega$, and $b_\sigma$ symbolize $\ll d_\sigma | d_\sigma^+ \gg_\omega$. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\phi_d$": "d-orbital wave function state of impurity", "$\\phi_k$": "Bloch orbital state in the metal", "$H$": "Hamiltonian of the system", "$E_{k\\sigma}$": "energy of k-state with spin σ", "$n_{k\\sigma}$": "number operator for electrons in k-state with spin σ", "$E_{d\\sigma}$": "energy of d-state with spin σ", "$n_{d\\sigma}$": "number operator for d-electrons with spin σ", "$U$": "Coulomb interaction energy", "$V_{kd}$": "s-d mixing (hybridization) matrix element", "$C_{k\\sigma}$": "annihilation operator for electrons in k-state with spin σ", "$d_\\sigma$": "annihilation operator for electrons in d-state with spin σ", "$\\mu_B$": "Bohr magneton", "$\\omega$": "angular frequency in Fourier domain", "$a_{k'\\sigma}$": "symbol for mixed Green's function $\\ll d_\\sigma | C_{k'\\sigma}^+ \\gg_\\omega$", "$b_\\sigma$": "symbol for Green's function $\\ll d_\\sigma | d_\\sigma^+ \\gg_\\omega$"}
|
Strongly correlated system
| ||
488 |
Strongly Correlated Systems
|
In 1961, Anderson proposed the s-d mixing model, where he considered that the formation of localized magnetic moments by transition metal impurity atoms in non-magnetic metal matrices must take into account two factors: first, similar to the formation of intrinsic magnetic moments in free atoms, the Coulomb interaction of d-shell electrons in impurity atoms has a significant influence on forming localized magnetic moments in crystals; second, due to the fact that the d-orbital state $\phi_d(\mathbf{r})$ of free atoms in the crystal is no longer strictly an eigenstate, especially due to the tendency of electron delocalization into Bloch states (s orbitals) in metal crystals, there exists electron transfer between $\phi_d$ and $\phi_k$ states, which Anderson referred to as s-d hybridization. Consequently, he pointed out that the Hamiltonian $H$ of the system should consist of the following four parts:
\begin{equation}
H = \sum_{k,\sigma} E_{k\sigma} n_{k\sigma} + \sum_{\sigma} E_{d\sigma} n_{d\sigma} + \frac{U}{2}\sum_{\sigma} n_{d\sigma} n_{d\bar{\sigma}} + \sum_{k,\sigma} V_{kd} (C_{k\sigma}^\dagger d_\sigma + d_\sigma^\dagger C_{k\sigma})
\label{eq:11.2.5}
\end{equation}
where
\begin{equation}
E_{k\sigma} = E_k + \sigma \mu_B h, \quad E_{d\sigma} = E_d + \sigma \mu_B h
\label{eq:11.2.6}
\end{equation}
Here, $\mu_B = \left( \left| \frac{e}{2mc} \right| \hbar \right)$ is the Bohr magneton, and the Landé factor for electrons and impurities is set as $g_0 = g_i = 2$, which is the non-degenerate s-d mixing model of Anderson.
In dealing with the $s-d$ exchange model, the following method of the equation of motion for Green's functions is often employed:
Starting from the double-time Green's function
\begin{equation*}
\ll A(t); B(t') \gg = -i \theta(t-t') <[A(t), B(t')]_+>
\end{equation*}
Utilizing a technique, the derivatives of the function $\ll A(t); B(t') \gg$ with respect to $t$ and $t'$ respectively yield the following two equations of motion:
\begin{equation}
i \frac{d}{dt} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll [A, H]; B(t') \gg
\label{eq:11.2.7}
\end{equation}
\begin{equation}
-i \frac{d}{dt'} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll A(t); [B, H] \gg
\label{eq:11.2.8}
\end{equation}
Performing the Fourier transform
\begin{equation}
\ll A | B \gg_\omega = \int dte^{i\omega(t-t')} \ll A(t); B(t') \gg
\label{eq:11.2.9}
\end{equation}
leads to two representations of the general form of the equation of motion for Green's functions
\begin{align}
\omega \ll A | B \gg_\omega &= <[A, B]_+> + \ll [A, H] | B \gg_\omega
\label{eq:11.2.10} \\
\omega \ll A | B \gg_\omega &= <[A, B]_+> - \ll A | [B, H] \gg_\omega
\label{eq:11.2.11}
\end{align}
Please complete the question:
Truncate the higher order terms in the equations to write out the approximate equations.
Hint: Let $a_\sigma$ symbolize $\ll d_\sigma | d_\sigma^+ \gg_\omega$, $b_\sigma$ symbolize $\sum_k V_{kd} \ll C_{k\sigma} | d_\sigma^+ \gg_\omega$. Anderson s-d exchange model and the equation of motion for Green's functions You should return your answer as an equation.
|
[] |
Equation
|
{"$\\phi_d$": "d-orbital state", "$\\phi_k$": "k-orbital state", "$H$": "Hamiltonian of the system", "$E_{k\\sigma}$": "energy of the k-state with spin \\sigma", "$E_{d\\sigma}$": "energy of the d-state with spin \\sigma", "$n_{k\\sigma}$": "number operator for k-state with spin \\sigma", "$n_{d\\sigma}$": "number operator for d-state with spin \\sigma", "$U$": "Coulomb interaction energy", "$\\mu_B$": "Bohr magneton", "$V_{kd}$": "hybridization term between k and d states", "$C_{k\\sigma}$": "annihilation operator for k-state with spin \\sigma", "$d_\\sigma$": "annihilation operator for d-state with spin \\sigma", "$g_0$": "Landé factor for electrons", "$g_i$": "Landé factor for impurities", "$A$": "arbitrary operator A", "$B$": "arbitrary operator B", "$a_\\sigma$": "Green's function element for d-state with spin \\sigma", "$b_\\sigma$": "summed Green's function element for hybrid states involving k and d states with spin \\sigma", "$\\bar{\\sigma}$": "opposite spin to \\sigma", "$\\ll A(t); B(t') \\gg$": "double-time Green's function", "$\\omega$": "frequency variable in Fourier transform"}
|
Strongly correlated system
| ||
489 |
Strongly Correlated Systems
|
Anderson s-d exchange model and Green's function equation of motion
In 1961, Anderson proposed the s-d mixing model. He considered that the discussion of the formation of a localized magnetic moment by transition metal impurity atoms in non-magnetic metallic matrices must account for two factors: Firstly, similar to the inherent magnetic moment formation in free atoms, the Coulomb interaction of d shell electrons in impurity atoms has a significant impact on the formation of localized magnetic moments in crystals. Secondly, due to the tendency of electron delocalization to Bloch orbital states (s orbitals) in metallic crystals, there is an exchange between states $\phi_d(\mathbf{r})$ and $\phi_k$, which Anderson termed s-d mixing. Thus, he pointed out that the system's Hamiltonian $H$ should consist of the following four components:
\begin{equation}
H = \sum_{k,\sigma} E_{k\sigma} n_{k\sigma} + \sum_{\sigma} E_{d\sigma} n_{d\sigma} + \frac{U}{2}\sum_{\sigma} n_{d\sigma} n_{d\bar{\sigma}} + \sum_{k,\sigma} V_{kd} (C_{k\sigma}^\dagger d_\sigma + d_\sigma^\dagger C_{k\sigma})
\label{eq:11.2.5}
\end{equation}
where
\begin{equation}
E_{k\sigma} = E_k + \sigma \mu_B h, \quad E_{d\sigma} = E_d + \sigma \mu_B h
\label{eq:11.2.6}
\end{equation}
Here, $\mu_B = \left( \left| \frac{e}{2mc} \right| \hbar \right)$ is the Bohr magneton, with a Landé factor of $g_0 = g_i = 2$ for both electrons and impurities. This is the non-degenerate orbital Anderson s-d mixing model.
When handling the s-d exchange model, the following Green's function equation of motion is often used:
Starting from the double-time Green's function
\begin{equation*}
\ll A(t); B(t') \gg = -i \theta(t-t') <[A(t), B(t')]_+>
\end{equation*}
by employing a technique to differentiate the function $\ll A(t); B(t') \gg$ with respect to $t$ and $t'$, the following two equations of motion can be obtained:
\begin{equation}
i \frac{d}{dt} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll [A, H]; B(t') \gg
\label{eq:11.2.7}
\end{equation}
\begin{equation}
-i \frac{d}{dt'} \ll A(t); B(t') \gg = \delta(t-t') <[A, B]_+> + \ll A(t); [B, H] \gg
\label{eq:11.2.8}
\end{equation}
Performing a Fourier transform
\begin{equation}
\ll A | B \gg_\omega = \int dte^{i\omega(t-t')} \ll A(t); B(t') \gg
\label{eq:11.2.9}
\end{equation}
yields the two forms of the Green's function equation of motion
\begin{align}
\omega \ll A | B \gg_\omega &= <[A, B]_+> + \ll [A, H] | B \gg_\omega
\label{eq:11.2.10} \\
\omega \ll A | B \gg_\omega &= <[A, B]_+> - \ll A | [B, H] \gg_\omega
\label{eq:11.2.11}
\end{align}
Please complete the question
Solve the Green's function using the truncation approximation
|
[] |
Expression
|
{"$E_{k\\sigma}$": "energy of the k-th state with spin sigma", "$n_{k\\sigma}$": "number operator for the k-th state with spin sigma", "$E_{d\\sigma}$": "energy of the d-state with spin sigma", "$n_{d\\sigma}$": "number operator for the d-state with spin sigma", "$U$": "Coulomb interaction energy", "$V_{kd}$": "s-d exchange coupling strength", "$C_{k\\sigma}$": "annihilation operator for the k-th state with spin sigma", "$d_\\sigma$": "annihilation operator for the d-state with spin sigma", "$\\mu_B$": "Bohr magneton", "$h$": "external magnetic field", "$\\omega$": "frequency in the Green's function", "$\\sigma$": "spin index", "$\\rho^{(0)}(E)$": "density of states at energy E", "$\\rho_F^{(0)}$": "density of states at the Fermi surface", "$\\Gamma$": "broadening due to s-d exchange coupling"}
|
Strongly correlated system
| ||
490 |
Superconductivity
|
The known Hamiltonian for electron-phonon interaction:
\begin{equation}
H_{ep} = -i \sum_{\mathbf{k}, \mathbf{k}'} \sum_{\mathbf{q}, s} \sum_{\mathbf{p}}
\left( \frac{N \hbar}{2 M \omega_{\mathbf{q}s}} \right)^{1/2}
(\mathbf{e}_{\mathbf{q}s} \cdot \mathbf{p})
\left\{ \frac{1}{N} \sum_{\mathbf{l}} e^{i(\mathbf{k} + \mathbf{q} - \mathbf{k}')\cdot \mathbf{l}} \right\} \times
V_{\mathbf{p}} \langle \mathbf{k}' | e^{i\mathbf{p}\cdot\mathbf{r}} | \mathbf{k} \rangle
(a_{\mathbf{q}s} + a_{-\mathbf{q}s}^\dagger)
C_{\mathbf{k}}^\dagger C_{\mathbf{k}'}
\end{equation}
Now use plane wave instead of Bloch wave function. Please analyze the conservation laws during the electron-phonon interaction process. You should return your answer as an equation.
|
[] |
Equation
|
{"$H_{ep}$": "Hamiltonian for electron-phonon interaction", "$\\hbar$": "reduced Planck's constant", "$\\mathbf{k}'$": "final wave vector of electron", "$\\mathbf{k}$": "initial wave vector of electron", "$\\mathbf{q}$": "wave vector of the phonon", "$\\mathbf{K}_n$": "reciprocal lattice vector", "$\\mathbf{l}$": "lattice vector", "$N$": "number of lattice points", "$\\Omega$": "crystal cell volume"}
| |||
491 |
Superconductivity
|
The known Hamiltonian for electron-phonon interaction:
\begin{equation}
H_{ep} = -i \sum_{\mathbf{k}, \mathbf{k}'} \sum_{\mathbf{q}, s} \sum_{\mathbf{p}}
\left( \frac{N \hbar}{2 M \omega_{\mathbf{q}s}} \right)^{1/2}
(\mathbf{e}_{\mathbf{q}s} \cdot \mathbf{p})
\left\{ \frac{1}{N} \sum_{\mathbf{l}} e^{i(\mathbf{k} + \mathbf{q} - \mathbf{k}')\cdot \mathbf{l}} \right\} \times
V_{\mathbf{p}} \langle \mathbf{k}' | e^{i\mathbf{p}\cdot\mathbf{r}} | \mathbf{k} \rangle
(a_{\mathbf{q}s} + a_{-\mathbf{q}s}^\dagger)
C_{\mathbf{k}}^\dagger C_{\mathbf{k}'}
\end{equation}
Now replace Bloch wave functions with plane waves Please provide the expression for the transition probability of the system from the initial to the final state during electron-phonon interaction (considering the long-time limit). You should return your answer as an equation.
|
[] |
Equation
|
{"$W$": "transition probability", "$i$": "initial state", "$f$": "final state", "$\\hbar$": "reduced Planck's constant", "$E_f$": "energy of final state", "$E_i$": "energy of initial state", "$t$": "interaction time", "$H_{ep}$": "electron-phonon interaction Hamiltonian"}
| |||
492 |
Superconductivity
|
The known electron-phonon interaction Hamiltonian:
\begin{equation}
H_{ep} = -i \sum_{\mathbf{k}, \mathbf{k}'} \sum_{\mathbf{q}, s} \sum_{\mathbf{p}}
\left( \frac{N \hbar}{2 M \omega_{\mathbf{q}s}} \right)^{1/2}
(\mathbf{e}_{\mathbf{q}s} \cdot \mathbf{p})
\left\{ \frac{1}{N} \sum_{\mathbf{l}} e^{i(\mathbf{k} + \mathbf{q} - \mathbf{k}')\cdot \mathbf{l}} \right\} \times
V_{\mathbf{p}} \langle \mathbf{k}' | e^{i\mathbf{p}\cdot\mathbf{r}} | \mathbf{k} \rangle
(a_{\mathbf{q}s} + a_{-\mathbf{q}s}^\dagger)
C_{\mathbf{k}}^\dagger C_{\mathbf{k}'}
\end{equation}
Now replace Bloch functions with plane waves Please provide the energy conservation relation in the electron-phonon interaction process and explain the specific relationship between electron energy and phonon energy before and after scattering. We only consider the case that the electron absorbs a phonon. You should return your answer as an equation.
|
[] |
Equation
|
{"$E_f$": "final energy", "$E_i$": "initial energy", "$\\varepsilon_{\\mathbf{k}}$": "initial electron energy", "$\\varepsilon_{\\mathbf{k}'}$": "final electron energy", "$\\hbar$": "reduced Planck's constant", "$\\omega_{\\mathbf{q}}$": "phonon frequency", "$\\mathbf{k}$": "initial electron wave vector", "$\\mathbf{k}'$": "final electron wave vector", "$\\mathbf{q}$": "phonon wave vector", "$H_{ep}$": "electron-phonon interaction Hamiltonian"}
| |||
493 |
Superconductivity
|
The known collective coordinate expression of the electron-phonon Hamiltonian is:
\begin{equation}
H_{ep} = \sum_{\mathbf{q}} M_{\mathbf{q}} Q_{\mathbf{q}} \rho_{-\mathbf{q}}, \quad M_{\mathbf{q}} = i \left( \frac{N}{M} \right)^{1/2} \frac{4\pi e^2}{q^2} (\mathbf{e}_{\mathbf{q}} \cdot \mathbf{q})
\end{equation}
Consider a monovalent metal's simple lattice composed of $N$ ions immersed in a uniform electron gas. The frequency of perturbed LA phonons at this moment can be expressed using the plasma collective oscillation frequency:
\begin{equation}
\Omega_q^2 + \frac{4\pi Ne^2}{M},
\end{equation}
Here the crystal occupies unit volume, $\Omega$ is the primitive cell volume, $N=\Omega^{-1}$, corresponding to the LA phonon's Hamiltonian:
\begin{equation}
P_{-q} = \dot{Q}_q,
\end{equation}
The electron-phonon interaction Hamiltonian is:
\begin{equation}
H_{\textrm{ep}} = \sum_q M_q Q_q \rho_{-q},
\end{equation}
The total Hamiltonian is:
\begin{equation}
H = H_p + H_{ep} + H_e + H_{ee},
\end{equation}
Among them, $H_e$ is the free electron approximation Hamiltonian, and $H_{ee}$ represents the Coulomb interaction. These two terms are independent of the electron normal coordinates $Q$.
Under the long-wavelength approximation, complete the following calculation Find the equation of motion for $Q_q$, and discuss how to derive the phonon frequency correction You should return your answer as an equation.
|
[] |
Equation
|
{"$Q_{\\mathbf{q}}$": "normal coordinate (q-component)", "$P_{-\\mathbf{q}}$": "momentum conjugate to Q_q with negative q-component", "$H$": "Hamiltonian", "$\\hbar$": "reduced Planck's constant", "$\\delta_{\\mathbf{q}\\mathbf{q}'}$": "Kronecker delta for wave vectors q and q'", "$\\Omega_{\\mathbf{q}}$": "frequency for normal mode (q-component)", "$M_{-\\mathbf{q}}$": "coupling coefficient", "$\\rho_{\\mathbf{q}}$": "charge density component (q-component)", "$N$": "number of particles", "$M$": "mass", "$e$": "elementary charge", "$\\mathbf{e}_{\\mathbf{q}}$": "polarization vector for q-component", "$\\mathbf{q}$": "wave vector"}
| |||
494 |
Superconductivity
|
The known collective coordinate expression of the electron-phonon Hamiltonian is:
\begin{equation}
H_{ep} = \sum_{\mathbf{q}} M_{\mathbf{q}} Q_{\mathbf{q}} \rho_{-\mathbf{q}}, \quad M_{\mathbf{q}} = i \left( \frac{N}{M} \right)^{1/2} \frac{4\pi e^2}{q^2} (\mathbf{e}_{\mathbf{q}} \cdot \mathbf{q})
\end{equation}
Considering a monovalent metal, where a simple lattice consisting of $N$ ions is immersed in a uniform electron gas, the perturbated LA phonon frequency can be expressed by the plasma collective oscillation frequency:
\begin{equation}
\Omega_q^2 + \frac{4\pi Ne^2}{M},
\end{equation}
Here, the crystal is taken with unit volume, $\Omega$ is the volume of the primitive cell, $N=\Omega^{-1}$, corresponding to the LA phonon Hamiltonian:
\begin{equation}
P_{-q} = \dot{Q}_q,
\end{equation}
The electron-phonon interaction Hamiltonian is:
\begin{equation}
H_{\textrm{ep}} = \sum_q M_q Q_q \rho_{-q},
\end{equation}
The total Hamiltonian is:
\begin{equation}
H = H_p + H_{ep} + H_e + H_{ee},
\end{equation}
where $H_e$ is the Hamiltonian for free electrons approximation, $H_{ee}$ represents the Coulomb interaction, both of which are independent of the normal coordinate $Q$.
Complete the following calculation under the long-wavelength approximation Calculate the ionic density fluctuations $\rho^i_q$ produced by lattice vibrations
|
[] |
Expression
|
{"$\\rho^i_q$": "ionic density fluctuation for wave vector q", "$N$": "number of unit cells", "$e$": "electric charge", "$\\mathbf{u}(\\mathbf{r})$": "displacement field of the lattice", "$M$": "mass of unit cell", "$\\mathbf{q}$": "wave vector", "$\\mathbf{e}_{\\mathbf{q}}$": "polarization vector for wave vector q", "$Q_{\\mathbf{q}}$": "normal coordinate for wave vector q", "$\\rho_{\\mathbf{q}}^i$": "ionic density Fourier component for wave vector q", "$\\mathbf{r}$": "position vector"}
| |||
495 |
Superconductivity
|
The known collective coordinate expression for the electron-phonon Hamiltonian is:
\begin{equation}
H_{ep} = \sum_{\mathbf{q}} M_{\mathbf{q}} Q_{\mathbf{q}} \rho_{-\mathbf{q}}, \quad M_{\mathbf{q}} = i \left( \frac{N}{M} \right)^{1/2} \frac{4\pi e^2}{q^2} (\mathbf{e}_{\mathbf{q}} \cdot \mathbf{q})
\end{equation}
Consider a monovalent metal where a simple lattice composed of $N$ ions is immersed in a uniform electron gas. In this case, the perturbed LA phonon frequency can be expressed by the plasma collective oscillation frequency:
\begin{equation}
\Omega_q^2 + \frac{4\pi Ne^2}{M},
\end{equation}
Here, the crystal is taken to be of unit volume, $\Omega$ is the unit cell volume of the primitive lattice, and $N=\Omega^{-1}$. The Hamiltonian corresponding to the LA phonons is:
\begin{equation}
P_{-q} = \dot{Q}_q,
\end{equation}
The electron-phonon interaction Hamiltonian is:
\begin{equation}
H_{\textrm{ep}} = \sum_q M_q Q_q \rho_{-q},
\end{equation}
The total Hamiltonian is:
\begin{equation}
H = H_p + H_{ep} + H_e + H_{ee},
\end{equation}
where $H_e$ is the Hamiltonian for free electrons approximately, and $H_{ee}$ represents the Coulomb interaction. These two terms are independent of the electronic normal coordinate $Q$.
Under the long-wavelength approximation, complete the following calculation Using linear response theory, calculate the relationship between $\rho_q$ and $\rho^i_q$
|
[] |
Expression
|
{"$\\rho_{\\mathbf{q}}$": "charge density at wavevector q", "$\\rho_{\\mathbf{q}}^i$": "ionic charge density at wavevector q", "$\\epsilon(\\mathbf{q})$": "static dielectric function at wavevector q"}
| |||
496 |
Superconductivity
|
The collective coordinate expression for the known electron-phonon Hamiltonian is:
\begin{equation}
H_{ep} = \sum_{\mathbf{q}} M_{\mathbf{q}} Q_{\mathbf{q}} \rho_{-\mathbf{q}}, \quad M_{\mathbf{q}} = i \left( \frac{N}{M} \right)^{1/2} \frac{4\pi e^2}{q^2} (\mathbf{e}_{\mathbf{q}} \cdot \mathbf{q})
\end{equation}
Consider a monovalent metal immersed in a uniform electron gas, forming a simple lattice composed of $N$ ions. The perturbed LA phonon frequency can be expressed in terms of the plasma collective oscillation frequency:
\begin{equation}
\Omega_q^2 + \frac{4\pi Ne^2}{M},
\end{equation}
Here the crystal is taken with unit volume, $\Omega$ is the unit cell volume of the Bravais lattice, and $N=\Omega^{-1}$. The Hamiltonian corresponding to the LA phonons is:
\begin{equation}
P_{-q} = \dot{Q}_q,
\end{equation}
The electron-phonon interaction Hamiltonian is:
\begin{equation}
H_{\textrm{ep}} = \sum_q M_q Q_q \rho_{-q},
\end{equation}
The total Hamiltonian is:
\begin{equation}
H = H_p + H_{ep} + H_e + H_{ee},
\end{equation}
where $H_e$ is the Hamiltonian of free electrons in approximation, and $H_{ee}$ represents Coulomb interaction, both of which are independent of the normal coordinate $Q$ of phonons.
Under the long-wavelength approximation, complete the following calculations. Under the long-wavelength approximation, considering the electron screening effect, derive the dispersion relation between the LA phonon angular frequency $\omega_{\mathbf{q}}$ and the wave vector $q$, and specify its form $\omega_{\mathbf{q}}$ with respect to $q$. Hint: You can use the Thomas-Fermi dielectric function. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\omega_{\\mathbf{q}}$": "LA phonon angular frequency for wave vector $\\mathbf{q}$", "$q$": "wave vector magnitude", "$M_{\\mathbf{q}}$": "mass term related to wave vector $\\mathbf{q}$", "$\\Omega_{\\mathbf{q}}$": "unperturbed phonon frequency for wave vector $\\mathbf{q}$", "$\\epsilon(\\mathbf{q})$": "dielectric function at wave vector $\\mathbf{q}$", "$\\lambda^2$": "Thomas-Fermi screening parameter squared", "$N$": "electron density", "$e$": "elementary charge", "$E_F$": "Fermi energy", "$M$": "ionic mass in phonon system", "$m$": "electron mass", "$v_F$": "Fermi velocity"}
| |||
497 |
Superconductivity
|
The known collective coordinate expression for the electron-phonon Hamiltonian is:
\begin{equation}
H_{ep} = \sum_{\mathbf{q}} M_{\mathbf{q}} Q_{\mathbf{q}} \rho_{-\mathbf{q}}, \quad M_{\mathbf{q}} = i \left( \frac{N}{M} \right)^{1/2} \frac{4\pi e^2}{q^2} (\mathbf{e}_{\mathbf{q}} \cdot \mathbf{q})
\end{equation}
Consider a monovalent metal, a simple lattice consisting of $N$ ions submerged in a uniform electron gas. At this time, the perturbed LA phonon frequency can be expressed by the plasma collective oscillation frequency:
\begin{equation}
\Omega_q^2 + \frac{4\pi Ne^2}{M},
\end{equation}
Here the crystal takes unit volume, $\Omega$ is the positive lattice unit cell volume, $N=\Omega^{-1}$, corresponding to the Hamiltonian of the LA phonon:
\begin{equation}
P_{-q} = \dot{Q}_q,
\end{equation}
The electron-phonon interaction Hamiltonian is:
\begin{equation}
H_{\textrm{ep}} = \sum_q M_q Q_q \rho_{-q},
\end{equation}
The total Hamiltonian is:
\begin{equation}
H = H_p + H_{ep} + H_e + H_{ee},
\end{equation}
where $H_e$ is the Hamiltonian for free electrons, and $H_{ee}$ represents the Coulomb interaction, both terms are independent of the electron normal coordinate $Q$.
Under the long wavelength approximation, complete the following calculations With the known dispersion relation of LA phonons in the form $\omega_{\mathbf{q}} = c_L q$, please provide the specific expression for the speed of sound of LA phonons $c_L$ (i.e., the Bohm-Staver speed of sound formula), and explain each physical quantity. You should return your answer as an equation.
|
[] |
Equation
|
{"$\\omega_{\\mathbf{q}}$": "angular frequency of the phonon with wavevector \\mathbf{q}", "$c_L$": "speed of sound of longitudinal acoustic (LA) phonons", "$q$": "magnitude of the phonon wavevector", "$m$": "electron mass", "$M$": "ion mass", "$v_F$": "Fermi velocity", "$k_F$": "Fermi wavevector"}
| |||
498 |
Superconductivity
|
The longitudinal optical (LO) vibration mode in ionic crystals generates a polarization field, which strongly couples with conduction electrons in ionic crystals. This interaction is much stronger than the effect of longitudinal acoustic (LA) phonons (which involves center of mass motion and does not generate a polarization field) on conduction electrons. Therefore, the interaction between LO phonons and conduction electrons affects the carrier properties in ionic crystals. When electrons move within an ionic crystal, they cause relative displacements between positive and negative ions, forming a local polarization field. This polarization, accompanying the electron motion, excites LO phonons, leading to the renormalization of the electron ground state energy and effective mass, forming a quasi-particle coupled with phonons—polaron. Please discuss the classification of polarons. Then use the perturbation method starting from the Hamiltonian to calculate the effective mass of the polaron in the case of slow phonons $(\mathbf{k}\rightarrow 0)$.
|
[] |
Expression
|
{"$a$": "lattice constant", "$\\xi$": "range of polarization", "$\\mathbf{k}$": "wave vector of the electron", "$\\varepsilon_{\\mathbf{k}}$": "energy of the electron in state $|\\mathbf{k}\\rangle$", "$\\hbar$": "reduced Planck's constant", "$m$": "effective mass of the band electron", "$\\omega_L$": "longitudinal optical phonon frequency", "$\\mathbf{q}$": "phonon wave vector", "$H_{ep}$": "electron-phonon interaction Hamiltonian", "$e$": "elementary charge", "$F$": "field strength", "$\\alpha$": "electron-phonon coupling constant", "$\\epsilon_\\infty$": "high-frequency dielectric constant", "$\\epsilon_0$": "static dielectric constant", "$m^*$": "effective mass of the polaron"}
| |||
499 |
Superconductivity
|
The longitudinal optical (LO) phonon mode in an ionic crystal generates a polarization field, which strongly couples with the conduction electrons in the ionic crystal. This coupling is much stronger than the effect of longitudinal acoustic (LA) phonons (which represent center of mass motion and do not generate a polarization field) on the conduction electrons. Therefore, the interaction between LO phonons and conduction electrons affects the carrier characteristics in ionic crystals. Please calculate the average number of virtual phonons excited around the electron.
|
[] |
Expression
|
{"$\\langle N_{\\text{ph}} \\rangle$": "average number of virtual phonons", "$\\mathbf{q}$": "phonon wave vector", "$\\mathbf{k}$": "electron wave vector", "$H_{ep}$": "electron-phonon interaction Hamiltonian", "$\\varepsilon_{\\mathbf{k}}$": "electron energy with wave vector k", "$\\varepsilon_{\\mathbf{k}-\\mathbf{q}}$": "electron energy with wave vector k-q", "$\\hbar$": "reduced Planck constant", "$\\omega_L$": "longitudinal optical phonon frequency", "$e$": "elementary charge", "$F$": "electric field or force constant", "$m$": "electron mass", "$\\alpha$": "dimensionless coupling constant in the expression"}
| |||
500 |
Superconductivity
|
Effective interaction of current electronic exchange virtual phonons
\begin{equation*}
H_{\text{eff}} = \frac{1}{2} \sum_{\substack{\mathbf{k}_1, \mathbf{k}_2 \\ q_1, q_2}} V_{\mathbf{k}_1, \mathbf{q}} C^\dagger_{\mathbf{k}_1 + \mathbf{q}, q_1} C^\dagger_{\mathbf{k}_2 - \mathbf{q}, q_2} C_{\mathbf{k}_2, q_2} C_{\mathbf{k}_1, q_1} c
\end{equation*}
The interaction coefficient is:
\begin{equation*}
V_{\mathbf{k}_1, \mathbf{q}} = |D_{\mathbf{q}}|^2 \frac{2\hbar\omega_{\mathbf{q}}}{(E_{\mathbf{k}_1 + \mathbf{q}} - E_{\mathbf{k}_1})^2 - (\hbar\omega_{\mathbf{q}})^2}
\end{equation*} Analyze the situation near the Fermi surface and describe the interaction when the attractive potential is greater than the screened Coulomb potential, and elaborate on the approximation method of BCS theory.
|
[] |
Expression
|
{"$E_{\\mathbf{k}_1+\\mathbf{q}}$": "energy of electron state with momentum $\\mathbf{k}_1 + \\mathbf{q}$", "$E_{\\mathbf{k}_1}$": "energy of electron state with momentum $\\mathbf{k}_1$", "$\\hbar \\omega_{\\mathbf{q}}$": "energy of phonon with momentum $\\mathbf{q}$", "$\\hbar \\omega_D$": "energy corresponding to Debye frequency", "$\\omega_D$": "Debye frequency of phonons", "$V_{\\mathbf{k}_1,\\mathbf{q}}$": "effective potential for electron interaction", "$q$": "momentum transfer", "$\\lambda$": "screening length", "$e$": "elementary charge", "$V_{\\text{net}}$": "net potential after considering electron interaction", "$\\sigma_1$": "spin of first electron", "$\\sigma_2$": "spin of second electron", "$V$": "constant attractive interaction potential in BCS approximation"}
|
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