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Finite-temperature Fermi-edge singularity in tunneling studied using random telegraph signals

We show that random telegraph signals in metal-oxide-silicon transistors at millikelvin temperatures provide a powerful means of investigating tunneling between a two-dimensional electron gas and a single defect state. The tunneling rate shows a peak when the defect level lines up with the Fermi energy, in excellent agreement with theory of the Fermi-edge singularity at finite temperature. This theory also indicates that defect levels are the origin of the dissipative two-state systems observed previously in similar devices.

In small electrical devices, noise signals are often seen which reflect the transitions of a single atom or electron between two or more metastable states. These 'random telegraph signals' (RTSs) [1], where the conductance jumps randomly in time between certain discrete values, are being increasingly exploited as a means of investigating a diverse range of tunneling phenomena, such as dissipative tunneling of two-state systems [2], electromigration [3], hopping conduction [4], and tunneling between quantum Hall edge channels [5]. In the present work we use them for the first time to study the dynamics of electrons tunneling between a defect state and a two-dimensional electron gas (2DEG) at millikelvin temperatures. We find that a noninteracting electron picture cannot explain the behaviour of the system. This is not very surprising, because the interaction phenomena of Coulomb blockade [6] and the Kondo effect [7] are known to strongly influence the tunnel conductance through a single defect level [8,9] or a quantum dot [10]. However, what is surprising is that the dominant interaction effect in the present system is another, to which attention has only recently been drawn [11]. It is the interaction of the 2D gas electrons with the defect potential, which produces a peak in the tunneling rates near the Fermi level at low temperature T . At T = 0 this peak becomes a Fermi-edge singularity, whose origin is the same as that of the X-ray absorption edge singularity in metals [12]. This singularity has already been invoked to explain sharp peaks observed in the differential conductance of small tunnel barriers [9], but the interpretation in that situation was hampered by the lack of equilibrium due to the large voltage bias, and by the simultaneous presence of other anomalous structure in the device characteristics. In our RTS experiments we can measure the tunneling rates for an isolated defect directly and in thermal equilibrium. We find remarkable agreement with the theory for the finite temperature generalization of the Fermi-edge singularity, which has not been tested experimentally before. Using the same theory we are then able to attribute the weakly coupled dissipative two-state systems observed previously [13] in similar devices to an electron tunneling between a defect level and a bound state of the defect potential. We also find that the effects of a magnetic field B on an RTS are fully consistent with the electron-trapping defect scenario.
The measurements were made on two-terminal Si metal-oxide-semiconductor field-effect transistors (MOS-FETs) with highly doped contact regions, oxide thickness d ox = 240Å, and channel dimensions 0.6 or 0.8 µm.
Care was taken not to stress the devices electrically, and the threshold gate voltage was about 2.0 V in a dilution refrigerator at 100 mK. The resistance was sampled at up to 5 kHz using a standard constant-current lock-in technique with a Brookdeal 5004 ultra-low noise voltage preamplifier. The low-temperature peak mobility was around 0.2 m 2 V −1 s −1 , corresponding to a transport scattering length of ∼ 300Å. This rather high disorder leads to quantum interference effects (see later) which fortuitously make the RTSs big enough to allow the use of signal levels not significantly larger than kT /e (to avoid resistive heating) down to 100 mK. The mean time spent in each resistance state of an RTS under fixed conditions was found by averaging over several hundred transitions, giving a standard error of a few percent.
Previous work at T ≥ 4.2 K has shown that most RTSs in n-channel MOSFETs result from defect levels situated in the oxide close to the Si/SiO 2 interface [1]. Fig. 1(a) shows a schematic band diagram illustrating the situation at low T , together with a section of a typical RTS. A positive voltage V g is applied to the gate, sufficient to create a degenerate 2DEG, represented by the shaded area at the interface. E d is a defect level (indicated as being in the oxide, though its location is not important here), E F is the Fermi level, and E 0 is the bottom of the lowest 2D subband. We denote the conductance of the device with the defect level empty (state 1) or occupied (state 2) by G 1 or G 2 respectively, and the reciprocals of the mean times spent in the two states by rates γ 1 and γ 2 , as indicated in the figure. The ratio of these rates satisfies the detailed balance condition, The lower graphs in Fig. 1(b) show ln(γ 1 /γ 2 ) plotted against V g at two temperatures. The variation is almost linear, implying that where η (the 'sensitivity') and V g0 (the 'balance' gate voltage) are constants. The straight solid lines in the figure correspond to η = 0.019 and V g0 = 6.662 V. The linearity results from the energy-independent density of states in the inversion layer and the linear sensitivities of E d and E 0 to the oxide electric field over the relevant range of V g . Differences in η between individual RTSs can be attributed to different defect locations. The deduced values of E d − E F for RTS#1 are plotted along the top axes in Fig. 1(b). In a noninteracting picture the individual rates, deduced from the golden rule, are given by where f (E) is the Fermi function, D is the electron density of states, and ∆ is the tunneling matrix element. The lines on the upper graphs in Fig. 1(b) are plots of Eqs. (3), where (2π/h)D∆ 2 = 280 s −1 is the only fitting parameter. At T = 1.2 K the measured values of γ 1 and γ 2 follow them rather well; hence the picture of tunneling without interactions seems to suffice. However, when T is reduced to 0.5 K a distinct peak appears in the rates in the region of E d = E F . To account for this peak, which occurs to some degree for every RTS, we are forced to go beyond the noninteracting picture. Matveev and Larkin [11] have recently pointed out that one should take into account the consequences of the change in the defect potential seen by electrons in the 2DEG when an electron tunnels. Most of the relevant theory was developed in the context of the X-ray absorption edge [12], and at T = 0 it predicts a power-law singularity in the transition rate, where θ(E) is the unit step function. The singularity arises because the tunneling electron can easily lose a small amount of energy to low-energy electron-hole pairs which are created by the sudden change in the defect potential. The finite temperature generalization of Eq. (4) is [14] where C is a constant and α is equal to the zerotemperature exponent. Fig. 2 shows the transition rates for RTS#2, found in the same device as RTS#1 at lower gate voltage (V g0 = 3.0863 V and η = 0.018). The solid lines are plots of Eq. (5) using α = 0.21 ± 0.01 and C = 10.2 ± 0.2 s −1 (with T in degrees Kelvin). The very good fits at both T = 145 mK and 360 mK are convincing evidence that these measurements directly probe the Fermi-edge singularity at finite temperature.
In the theory, α depends only on the defect potential. For simplicity, let us assume the potential before capture, V , is attractive and radially symmetric, while the potential after capture is zero. The one-electron eigenstates can be classified by the perpendicular component of the angular momentum m = 0, ±1, ±2 . . ., together with spin and valley indices. In the 2-DEG at B = 0 there are two equivalent conduction band valleys, and for convenience we combine spin and valley into a single index s = 1 . . . 4. If the phase shift for channel (m, s) at E F in the presence of V is δ m,s , then α is given by [15] Note that if V = 0, δ m,s = 0 for all m and s, α = 1, and with the correct value of C Eq. (5) reduces to the noninteracting result, Eq. (3). Charge neutrality forces the phase shifts to obey the Friedel sum rule [16], m,s δ m,s = π, and we can use this together with Eq. (6) to obtain a lower limit on α. This limit is reached for pure s-wave scattering, when δ m,s = 0 for all m = 0. In 2-D an attractive potential always has a bound state. When screening is strong, ie, for large E F , the bound state may be occupied by four electrons simultaneously [17] and one finds α ≥ 3/4. For weaker screening (small E F ) the bound state is occupied by only one electron and one finds α ≥ 0. The limit α = 0 is reached when the electron tunnels directly to the defect level from the bound state while all extended states remain completely unaffected, ie, all the screening is done by the single electron in the bound state. Then δ m,s is zero for all channels except the one in which the bound state was destroyed, whose phase shift is δ 0,1 = π.
The value α = 0.21 obtained for RTS#2 is only consistent with a singly occupied bound state, with π/2 < δ 0,1 < π and fairly small phase shifts in other channels. On the other hand, the values α ∼ 0.7 and 0.9 for RTSs #1 and #3 respectively allow the possibility of multiple occupancy of the bound state. At this point it is interesting to reconsider the results of some earlier RTS experiments in MOSFETs [13,18]. In Ref. [13] the data were fitted using an expression identical in form to Eq. (5) but derived from the theory of two-state systems (TSSs). This remarkable identity is in fact no coincidence, because an electron tunneling between a defect level and a single bound state in the defect potential constitutes a TSS. If one could vary α from 0 to 1 one could in principle transform the system continuously from a noninteracting TSS into a noninteracting defect level. While the defects in the present work are in the intermediate regime, those in Ref. [13] were found to have α ≪ 1, corresponding to TSSs very weakly coupled to the environment (the 2DEG). In this limit Eq. (5) takes the dramatically different form of a narrow Lorentzian centered at E d = E F . We suggest that for such defects, which are seen only in electrically stressed devices, the 2DEG is locally depleted out due to potential fluctuations at the interface. The extended states therefore remain unaffected on electron capture ¿from the bound state because they are distant from the defect/bound-state system. For these defects we can also offer a resolution of a paradox that would arise if the defect were located in a metallic region, namely, that a small value of α implies very weak scattering of electrons at E F , while a large RTS amplitude appears to require strong scattering. If the defect actually lies in a depleted region then electrons at E F may be able to tunnel across this region via the bound state when the defect is ionized. Hence the conductance decreases by as much as e 2 /h when the defect captures an electron, because the bound state is destroyed, even though all extended-state wavefunctions are unaltered.
Finally, we examine the effects of a magnetic field on another RTS, RTS#3 (seen in a different device from RTS#1), which lend further support to the basic picture of electrons tunneling between a defect and the 2DEG. For RTS#3, α ∼ 0.9, so the deviations from the noninteracting result, Eq. (3), are moderate. In Fig. 3, trace (i) is the variation of G 1 up to 12 T. The edge of a quantum Hall plateau is visible at the highest field, while only universal conductance fluctuations [19] are seen at lower B. Trace (ii) shows the corresponding variation of V g0 , which above about 4 T undergoes oscillations commensurate with the vertical dotted lines marking points at which n Landau levels are full. This confirms our expectation that the tunneling process should be sensitive to the modulations of the density of states in the 2DEG, although the nature of this sensitivity is complex and will be the subject of future work. Inset trace (iii) is an expanded section of a magnetic field sweep made whilst sampling the conductance at 1 kHz, illustrating how G 1 and G 2 undergo distinct reproducible fluctuations. Trace (iv) shows ∆G 1 , which is the deviation of G 1 from the smooth classical magnetoresistance background present in trace (i). Trace (v) shows the RTS amplitude δG 21 ≡ G 2 − G 1 over the same field range. As can be seen, the fluctuations in δG 21 and ∆G 1 are qualitatively similar and exhibit virtually the same correlation length B c [19]. In the range 0 < B < 5 T we find B c = 0.04 ± 0.01 T, giving an estimate of A φ = (B c e/h) −1 = 0.1 µm 2 for the typical phase-coherent area, compared with the device area A ∼ 0.5 µm 2 . The rms amplitudes of the fluctuations in the two quantities are σ 1 = 10 ± 1 µS and σ 21 = 3.6 ± 0.4 µS. Hence σ 21 /σ 1 ∼ 0.36 ± 0.06. Allowing for the uncertainties in A and A φ , this is consistent with the prediction σ 21 /σ 1 ∼ (A φ /A) 1/2 ∼ 0.45 for the removal of a single strong scatterer from a disordered 2D system at finite temperature [20,2]. The dashed line superimposed on trace (v) was obtained by smoothing δG 21 over a range of 2 T. Notice that it is always above zero, ie, on average G 2 > G 1 . This helps to justify our earlier simplifying assumption that on electron capture a scattering center is removed from the 2DEG, ie, that the defect goes from positive to neutral.
In conclusion, using random telegraph signals we have shown that electronic tunneling between a defect level and a 2DEG exhibits the finite-temperature counterpart of the Fermi-edge singularity, with exponent α ranging from nearly zero to nearly one. Defects which have very small values of α behave as two-state systems.
We thank C. Barnes