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Dynamic nuclear polarisation in biased quantum wires with spin-orbit interaction

We propose a new method for dynamic nuclear polarisation in a quasi one-dimensional quantum wire utilising the spin-orbit interaction, the hyperfine interaction, and a finite source-drain potential difference. In contrast with current methods, our scheme does not rely on external magnetic or optical sources which makes independent control of closely placed devices much more feasible. Using this method, a significant polarisation of a few per cent is possible in currently available InAs wires which may be detected by conductance measurements. This may prove useful for nuclear-magnetic-resonance studies in nanoscale systems as well as in spin-based devices where external magnetic and optical sources will not be suitable.

The ability to locally manipulate spin has been the subject of intense research recently. In particular, local control of nuclear polarisation in the vicinity of quantum dots and in wires has diverse uses such as in nuclear magnetic resonance (NMR) studies [1] or for control of nuclear spins in devices where external magnetic fields and/or optical sources will not be suitable or convenient as for instance in non-magnetic spin-filtering [2] and qubits [3]. In some very beautiful recent experiments related to spin-based qubits, dynamically-generated nuclear polarisation has been used to control the decoherence of electron spins in quantum dots [4,5]. Nuclear polarisation can also be used to manipulate the electron spin-splitting in quantum wires [6] without using a magnetic field. Local control of nuclear polarisation in quantum wells using a gate contact has now been demonstrated [7]; however, the polarisation itself involved the use of an optical source. In this Letter, we propose a new method for dynamically generating and controlling nuclear polarisation, locally and without using magnetic and optical sources, in a nonequilibrium quasi one-dimensional quantum wire with spin-orbit interaction. By local we mean the nuclear polarisation is within the wire and not in the bulk surroundings. We argue that the role of spin-orbit interaction in creating a spin-polarised electron distribution cannot in this scheme be replaced with an external magnetic field. We estimate the magnitude and build-up time for the polarisation as around 2.5% and 200s respectively in typical InAs quantum wires which compares favourably with other methods. We discuss possible ways in which the nuclear polarisation may be detected and measured, as for instance through a measurement of the two-terminal conductance [6,8].
The Overhauser effect [9] is used for dynamically polarising nuclei through their hyperfine coupling with electrons using a magnetic field and an external radio source tuned to the electron Zeeman splitting. The electron spin resonance (ESR) inducing radio source tends to equalise (saturate) the numbers of spin up and down electrons even though they have different Zeeman energies -this implies that the chemical potentials of the two spin species must be different and the spin distribution is not in equilibrium. The electron spins at the higher chemical potential ultimately relax with a spin flip to equalise the chemical potentials; and some of these spins can be exchanged with the nuclei. In this manner, a nonzero nuclear polarisation is built up. The key requirement for the Overhauser scheme is the creation of a nonequilibrium electron spin distribution. This usually involves pumping and saturap-1 tion of the electron spins. In the presence of a magnetic field, saturation may be attained by, besides ESR, hot electrons [10], or optically through unpolarised light [11]. Hot electrons have been used to generate nuclear polarisation in InSb [12] subjected to a Zeeman field, and more recently, in GaAs [13]. In the so-called optical Overhauser effect, electron spin pumping is achieved using circularly polarised light [11]. However these methods all involve the use of physically large external sources of magnetic field and light making local control of nuclear polarisation difficult, which is the main motivation of our work.
Spin-orbit interaction offers one way where the electron spin degeneracy can be lifted without using an external field. The lifting of electron spin degeneracy by a spin-orbit interaction differs in many ways from that in a magnetic field. In particular, no electron spin polarisation σ e can be created solely through spin-orbit interaction in a single sub-band quantum wire carrying a current [14]. However, it can be shown that in a quasi one-dimensional model where the spin-orbit coupling mixes different subbands, it is possible to obtain a finite σ e by passing an electric current through the wire [15]. This will be the model we shall consider. Spin-orbit interaction in semiconductor devices is usually of the Rashba or Dresselhaus kind. For simplicity, our analysis focuses on a Rashba interaction, but our results will hold even in the presence of a Dresselhaus interaction.
For strong spin-orbit coupling, the left and right moving charges in the lowest two sub-bands of transverse momentum quantisation can become, at sufficiently large wavevectors, completely spin-polarised with opposite spin orientations for the two directions [15]. In the same multiband model, we show that a small (partial) σ e occurs even when the spin-orbit coupling is weak, typical in quantum wires. As the left and right-moving electrons have overall opposite polarisations, an applied potential difference creates a nonequilibrium electron spin distribution. Thereupon we show how nuclear polarisation develops in the quantum wire due to the hyperfine coupling of the nuclei with the nonequilibrium electrons.
It is important to note that merely polarising the conduction electron spins does not in general lead to dynamic nuclear polarisation. We discuss later that dynamic nuclear polarisation is not possible by simply using an external magnetic field to polarise the conduction electrons instead of spin orbit interaction.
Let us first discuss how spin-orbit coupling in a quasi one-dimensional wire can lead to electron spin polarisation. Consider a two-dimensional gas (2DEG) of electrons in the (x, z) plane. The geometry of the quantum wire is such that the electrons are confined in the y and z directions and the transport "channel" is along the x axis. We assume a hard wall confinement at z = 0 and z = W, and at y = 0 and y = δ. The Hamiltonian of the 2D electrons is is the Rashba spin-orbit interaction arising from the 2DEG's asymmetric confinement in the y-direction. Its interaction strength, k SO , may be controlled by an external gate voltage [16]. σ z and σ x are Pauli matrices in the space of electron spin.
In the absence of the Rashba interaction, the scattering states are labelled by the sub-band index n of transverse momentum quantisation: where k (n) z = nπ/W, n = 1, 2, . . . , and the energy eigenvalues are E (2) respects this sub-band quantisation but the σ x p z term mixes sub-bands whose indices differ by an odd number as well as the spin eigenstates of σ z p x . It is convenient to regard this mixing term as a perturbation, and the rest as the bare Hamiltonian. Then, the spatial part of the bare eigenstates has the same form (Eq. (3)) as that in the absence of spin-orbit interaction. The matrix elements of the second term in the spin-orbit interaction are The mixing is strongest for sub-bands whose indices differ by one; and so, it is sufficient to consider the two lowest sub-bands [17]. In this approximation, the truncated Hamiltonian is expressible as a 4 × 4 matrix in the basis |nσ >= |1 ↑ , |2 ↓ , |2 ↑ , |1 ↓ (in that order); where The eigenvectors are also easily found. In the spin basis | ↑ , | ↓ , we have  Figure 1: Electron spin-polarisation σe at large wave-vectors in quantum wires with Rashba interaction -dispersion curves in the two-band approximation for spin-orbit interaction strength kSOW = 0.5 (a) and kSOW = 2 (b). Dotted curves: spin-orbit split bands in the absence of sub-band mixing (Eq. (6)). Solid curves: sub-band mixing lifts the degeneracy at kx = 0 intersections. (a) Weak coupling: n = 1 and n = 2 sub-bands intersection occurs at a wave-vector much larger than that at the bottom of the n = 2 sub-band. Below this wave-vector, the lowest two sub-bands 1a and 1b are oppositely polarised and σe is small. (b) Strong coupling: the n = 1 and n = 2 sub-bands (dotted) intersect near the bottom of the n = 2 subband. When kx > 3π 2 /(4kSOW 2 ) (dotted intersection), electrons moving in a given direction in the lowest two sub-bands 1a and 1b will have the same orientation of spin polarisation.
where the "mixing" functions g 1a and g 1b are Similar scattering states have also been obtained for more general Rashba interactions and a parabolic confining potential [18]. Figure 1 explains how nearly complete electron spinpolarisation is possible in quantum wires transmitting the lowest sub-bands under strong Rashba interaction.
We show that any Rashba interaction results in a finite σ e . The electron spin polarisation σ e = σ z (E) at a given energy E for the 1a and 1b sub-bands is The mixing functions g 1a,b (k x ) have been re-expressed as functions of energy E as σ e is to be calculated at a fixed E rather than a fixed k x . Clearly, in the absence of inter sub-band mixing, σ z (E) is zero. When the spin-orbit interaction is weak (k SO W < 1), and the energy is far from the region of sub-band mixing, the mixing functions g 1a(b) will be small compared to unity and σ z (E) ≈ 2[(g 1b (E, k SO )) 2 − (g 1a (E, k SO )) 2 ]. In this regime, the energy-wavevector relations for the 1a and 1b sub-bands to Here k (1D) F = 2mE/ 2 − π 2 /W 2 is the 1D Fermi wavevector in the n = 1 sub-band in the absence of Rashba coupling. It follows that for weak spin-orbit coupling, the electron spin polarisations for right (R) movers and left (L) movers at energy E are respectively Note that unlike an external magnetic field, the right and left movers have an opposite net polarisation for both strong and weak Rashba coupling. Thus, by imposing a net electrical current which imbalances the left and rightmoving electrons, we are able to satisfy one of the requirements for the Overhauser effect; namely, we have a nonequilibrium distribution of electron spin. Spin-orbit interaction mediated electron spin polarisation can be controlled non-magnetically also by applying a local strain [19]; however, this will not fulfil our requirement of compact external sources. In Figure 2 we show a numerical calculation for the electron spin polarisation | σ z | for the lowest two sub-bands (1a and 1b) as a function of the dimensionless spin-orbit interaction k SO W at different values of the energy E, using Eq. (12). The range of values of energy shown spans, approximately, the range of Fermi energy over which only the two lowest sub-bands (1a and 1b) are occupied. Note that the polarisation undergoes a significant change when k SO W reaches some large value; this is especially true at low energies. One can understand this change by looking at the spin structure of the 1a and 1b sub-bands in Figure  1b. When the energy is small enough, the two sub-bands have opposite spin polarisations, and total polarisation is the difference of the polarisations of the two. As k SO W is increased, the sub-bands move down in energy until, eventually, the energy will exceed the value at which both sub-bands have the same polarisation, which causes a significant increase in | σ z |. Now we show how a nonzero nuclear polarisation can be created with a finite source-drain potential difference p-3 but without using an external magnetic field. Consider the hyperfine interaction of the electron spin density s(r i ) with the host nuclear moments I i at the centre of the quantum wire, where A ≡ (4/W δ)A 3d , where A 3d is the bulk hyperfine constant, 4/W δ is the electron density at the centre of the wire, and s + = (s x + is y )/2, etc. When the Rashba interaction is strong and the Fermi energy is large enough, right movers at the Fermi surface are nearly completely polarised spin down and the corresponding left movers are polarised spin up. When a right-moving electron at the Fermi surface, say in the 1b state (i.e. spin down), exchanges its spin with a nucleus because of the hyperfine coupling, the final state of the electron can either be a higher energy sub-band such as 2b (not shown in Figure  1) with k x in the same direction, or a sub-band such as 1a with k x in the opposite direction with a similar energy. The first process cannot occur with energy conservation, unless there is simultaneous energy absorption from the heat bath; hence it is small (exponentially activated) at low temperature. The latter process involves a backscattering which can occur while scattering from the nucleus. If the chemical potential µ L of the left contact is higher than the chemical potential µ R of the right contact, then after backscattering with a spin flip, the electron can have a higher energy than the chemical potential of the right contact. Spin-flip backscattering of the left-moving electrons is suppressed because µ R < µ L . We thus have a situation where electrons exchanging spins with the nuclei scatter preferentially from a spin down state. The rate of change of transition probability for a hyper-fine mediated scattering, say with ∆m I = −1, is ) is the (1D) density of final states and G = 1 2I+1 I m=−I |I − m−1,m | 2 = 2 3 I(I + 1), is the appropriate average value of |I − | 2 assuming that the nuclear moments are unpolarised. We now obtain an expression for the rate of electron flips w(R ↓; L ↑) from the right-moving | ↓ state to the left-moving | ↑ state following the general prescription of Ref. [20]. The distribution functions of the right-moving and left-moving electrons are . The energies ǫ n,s (k x ) have been obtained in Eq. (7) and Eq.(8). We have not considered magnetic fields here but that can also be incorporated in principle. The rate of electron flips per nucleus (17) Here k min and k max correspond to the minimum and maximum values of k x > 0 where the electrons can be considered to be nearly completely polarised spin-down. k min is approximately given by the value of k x at which the 1 ↑and 2 ↓ bands cross (see Figure 1). We assume that the temperature is low compared to the inter sub-band separation as well as the potential difference and choose the left and right chemical potentials to lie in the energy band where spin polarisation is possible. The energy conservation condition for the above transition is ǫ n,s (k x ) = ǫ n ′ ,s ′ (k ′ x ). Starting with unpolarised nuclei, the initial rate of change of the polarisation I z per nucleus at the centre of the wire is −[w(R ↓; L ↑) − w(L ↑; R ↓)], i.e.
Here scattering from L ↑ to R ↓ is suppressed because µ R < µ L . For I = 1/2 or at long times in general, I z + I will decay exponentially. A spin-down nuclear polarisation is thus built up. When spin-orbit coupling is weak, and the Fermi energy is far from the (avoided) intersection of the sub-bands, we have shown earlier that the right movers in the n = 1 subbands can have either spin orientation with a small excess of spin down electrons. Likewise the left movers have a small excess of spin up electrons. Thus spin-flip backscattering of spin-down right movers is not completely cancelled by the spin-flip backscattering of the spin-up right movers. Forward scattering does not contribute as in this case the two competing spin flips occur at the same rate. The initial rate of change of I z , therefore, is suppressed by a factor | σ z (ǫ F ) |. The initial rate T for nuclear polarisation, Tn, given by the first of Eqs. (19), is shown here as a function of kSOW for dimensionless energies E = 10, 20, 30, and 40, nuclear spin I = 9/2, and a potential difference of 1mV between the left and right leads. The inset plot of log(Tn) versus log(kSOW ) clearly reveals the k 3 SO dependence of the nuclear polarisation for kSOW < 1 obtained analytically in Eq. (19). The example discussed at the end of the paper approximately corresponds to kSOW = 0.3. the polarisation I z of a nucleus is where ∆µ = |µ L − µ R | and ν n,s ≈ m/(2π 2 k (1D) F ). We used Eq. (14) for | σ z (ǫ F ) | ≪ 1, which is the expected experimental situation. We considered only the lowest subbands. At finite temperatures exceeding |∆µ|/k B , the polarisation build-up rate decreases by an amount proportional to the temperature owing to Korringa relaxation. Eq. (19) is the main result of the paper. Figure 3 shows a numerical calculation for T n for a InAs/GaSb quantum wire as a function of k SO W for four different values of the energy at a fixed potential difference of 1mV and δ = 5nm. The log-log plot in the inset shows that the k 3 SO law obtained in Eq. (19) is indeed obeyed. Complete nuclear polarisation is not possible in our scheme when | σ z (ǫ F ) | < 1 per sub band. Consider the worst case scenario, I = 1/2. Nuclear polarisation stops increasing once w(R ↓; L ↑)p ↑ − w(R ↑; L ↓)p ↓ = 0, where p ↑(↓) is the probability that the nucleus is spin up (down). It is easy to see that this amounts to a steady state nuclear polarisation I z ∼ | σ z (ǫ F ) |. However, for larger values of I, a nuclear transition m I → (m I − 1) does not, except in the lowest two states, cause a corresponding increase in the rate of the undesirable (m I − 1) → m I transition. Larger polarisations are thus possible for I > 1/2 : I z /I ∼ I| σ z (ǫ F ) | for small | σ z (ǫ F ) |. Including more sub-bands will further enhance both T −1 n and | σ z (ǫ F ) |. The role of Rashba coupling in the build-up of nuclear polarisation cannot be substituted by an external (Zee-man) magnetic field. In that case, under conditions of a non-zero (but small) bias, one can show that within linear response, the probability of a nuclear spin-flip process involving scattering of a spin-up right-mover into a spindown left mover is identical to the probability of a nuclear spin-flip process involving scattering of a spin-down rightmover into a spin-up left mover. These two processes compensate leading to no net build-up of nuclear polarisation.
The proposed method works even in the presence of a Dresselhaus spin-orbit interaction, say, of the form H D = kD m (p z σ z −p x σ x ). This is most easily seen by going over to a basis consisting of eigenstates of σ x . Then the Pauli matrices transform as (σ x , σ y , σ z ) → (σ z , −σ y , σ x ), and the Dresselhaus interaction changes to a Rashba interaction and our earlier treatment can be repeated. Hence if only Dresselhaus is present, the spins will align along the x direction. If both Rashba and Dresselhaus are present, then the spin polarisation will be aligned along some direction in the x-z plane. If the Dresselhaus interaction also contains a σ y p y term, that will cause a mixing of sub-bands in a direction perpendicular to the 2DEG. Since the confinement in this direction is stronger than the confinement in the plane, the separation of the sub-bands will be large and their mixing will be small compared to the in-plane sub-bands.
Consider a typical InAs/GaSb heterostructure with effective electron mass m = 0.027m e , I As = 3/2 and I In = 9/2, and confinement ratio (W/δ) 2 = 10 (typical ratio of sub-band energy spacings in GaAs devices). The atomic density is 3.6×10 28 m −3 , hyperfine couplings are of the order of 100 µeV per atom (see for e.g. Ref. [21]), and the Rashba coupling from Ref. [16] is k SO = 1.4×10 7 m −1 . The average value of G is 9.5. For a small potential difference ∆µ = 1 meV ǫ F , and using ǫ F = 2 (k , we have I z /I ≈ 2.5 × 10 −2 for In atoms. The build-up rate is very sensitive to the spin-orbit coupling strength and the confinement asymmetry. The small number of polarised spins may pose a considerable problem for detection by conventional NMR. Though the situation can be further improved by having multiple quantum wires parallel to each other, this is not ideal. We would like to suggest two methods of detection. Firstly, because I z = 0, the final term of the hyperfine Hamiltonian (15) produces an effective magnetic field. By applying an in-plane magnetic field as a reference, it is possible to infer the strength of the effective hyperfine field and hence the polarisation from two-terminal conductance measurements [6,8].Secondly, since polarised radioactive nuclei have an anisotropic decay pattern [22], it should be possible to detect polarisation by measuring the beta decay statistics [23]. In this case, 114 In possesses a half-life of 50 days, and may prove suitable.
Dipolar interactions of the nuclei will reduce the steady state nuclear polarisation through spin diffusion out of the boundary of the quantum wire, and spin-flip processes that p-5 do not conserve total spin. Small magnetic fields can be used to strongly suppress such processes [24].
In conclusion, we have shown how a gate-controlled local dynamic nuclear polarisation is possible in a quasi onedimensional wire with spin-orbit interaction subjected to a finite source-drain potential. A significant polarisation of a few per cent can be easily attained in present InAs devices in about 200s, and may be detected by conductance measurements [6,8]. The polarisation is very sensitive to the strength of (gate-controlled) spin-orbit coupling as well as the asymmetry of confinement in the quantum wire. This method may prove useful in NMR studies in semiconductor devices and in the local control of electron spin decoherence in spin-based devices. * * * VT thanks the support of TIFR and a DST Ramanujan Fellowship (sanction no. 100/IFD/154/2007-08). ACHC thanks Trinity college, Cambridge for support. NRC acknowledges support by EPSRC grant GR/S61263/01.