Physica E 1 (1997) 241–244
Resonant tunneling through a single-level quantum dot Jorg Schmid a;b;∗ , Jurgen Konig b , Herbert Schoeller b , Gerd Schon b a b
Max Planck Institut fur Festkorperforschung, Heisenbergstrae 1, 70569 Stuttgart, Germany Institut fur Theoretische Festkorperphysik, Universitat Karlsruhe, 76128 Karlsruhe, Germany
Abstract We study resonant tunneling through a single-level quantum dot in the presence of strong Coulomb interaction. This system is described by the Anderson model, which at low temperatures for a low-lying level shows a peak in the dierential conductance at zero-bias voltage (zero-bias anomaly). This so-called Kondo resonance can be understood as a coherent coupling between the state on the dot and those in the reservoirs, which enhances transport. At higher temperatures and bias voltages, this coherence is destroyed. We obtain an approximate solution of the problem for arbitrary parameters by a diagrammatic expansion technique. Recently, Ralph and Buhrman interpreted their experiments as transport through one such impurity level. We compare our results with their experiment. ? 1997 Elsevier Science B.V. All rights reserved. Keywords: Anderson model; Kondo resonance; Quantum dot
1. Model and Hamiltonian To describe electron transport through a quantum dot we use a model rst introduced by Anderson  for localized magnetic states in metals. We consider a spin-degenerate level, which is coupled to two reservoirs via tunneling barriers. The interaction suppresses the double occupancy of the level by electrons of either spin. This model has been studied in equilibrium by renormalization-group techniques , by the non-crossing approximation , and recently in non-equilibrium [4–7]. Following Ref.  we present a real-time diagrammatic expansion technique to get quantitative results in non-equilibrium and ∗
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1386-9477/97/$17.00 ? 1997 Elsevier Science B.V. All rights reserved PII S 1 3 8 6 - 9 4 7 7 ( 9 7 ) 0 0 0 5 1 - 9
compare them with an experiment of Ralph and Buhrman . The system and relevant energies are depicted in Fig. 1.The Hamiltonian is H = HR + HL + HI + HT ;
where Hr stands for the electrons in the right or left reservoir, r = R; L, reservoir, Hr =
rks a+ rks arks :
The island is described by HI which consists of the energies of the electrons on the island, including the Coulomb interaction U between them HI =
s cs+ cs + Uc↑+ c↓+ c↓ c↑ :
J. Schmid et al. = Physica E 1 (1997) 241–244
Fig. 2. Contributions to a diagram in our expansion.
Fig. 1. Energy scheme of the model. The spin degeneracy of the level is symbolized by a double line and can be lifted by applying a magnetic eld.
Finally, HT describes tunneling of electrons between the impurity and the reservoirs P (4) HT = Trks a+ rks cs + h:c: rks
In the following, we assume the reservoirs to be in equilibrium characterized by Fermi distributions. The tunneling matrix elements and the density of P states are combined to the parameter r (!) = 2 k |Trks |2 (! − rks ) ≈ 2Nr |Tr |2 which is taken as a constant describing the tunneling barrier. 2. Expansion and diagrammatic representation To calculate the current through the system, we start from the general expression for the expectation value of an operator A Z0 i dt HT (t) hA(0)i = tr (−∞)T˜ exp ˜ −∞ Z0 i ×A(0) T exp − dt HT (t) : ˜ −∞
(5) The operators are taken in the interaction picture, with the tunneling part of the Hamiltonian taken as the perturbation. We expand the exponentials and trace over the reservoir degrees of freedom. Each term can be represented by a diagram. Here we do not go into details (see Ref. ) but give some examples of such diagrams.
Three contributions to such diagrams are shown in Fig. 2. In each example, the upper and the lower line are the forward and backward propagators in time (on the Keldysh contour). Each section is marked by the occupation of the level. Tunneling vertices (dots), arising from the tunneling part of the Hamiltonian, are arranged on the propagators. They are connected in pairs by tunneling lines (dashed lines) which stand for a contraction of a pair of reservoir electron operators. The tunneling lines carry the information on the spin s (up, down), reservoir r (right, left) and energy ! of the transferred electron. The operator of interest, A, is inserted to the right of such a diagram; at this point the forward and backward propagator are connected. If the coupling to the leads is weak the system can be described by sequential tunneling, i.e., with transition rates calculated by Fermi’s golden rule. Examples of the corresponding diagrams are shown in the rst diagram of Fig. 2. They are characterized by the condition that the tunneling lines do not overlap in time. The shaded part of the diagram describes the transition to an empty dot. If this state is energetically unfavorable, at low temperatures this diagram is exponentially suppressed by a factor ∝ exp(−=kT ). In situations where sequential tunneling is suppressed (Coulomb blockade), the higher-order contributions become important. One example is cotunneling, where one electron leaves and another one enters the island within one process (see second picture in Fig. 2). At low temperatures quantum uctuations described by these higher-order processes (which are only suppressed by energy denominators and not exponentially), and in general processes of arbitrary order, have to be taken into account. In a conserving approximation we include all contributions, where a vertical line will cut at most two tunneling lines (resonant tunneling approximation). The third diagram shows another part of a contribution which is included in the summation according to this rule. The upper tunneling line, which starts
J. Schmid et al. = Physica E 1 (1997) 241–244
and ends on the same propagator does not change the occupancy of the level, but is a contribution to the self-energy of an electron which is just tunneling. The real and imaginary part of the self-energy account for renormalization and broadening eects, respectively. Furthermore, the self-energy leads to an additional peak in the spectral density at the Fermi energy of the reservoirs (see Fig. 5). 3. Comparison with experiments In the framework of the resonant tunneling approximation, we can setup a system of self-consistency equations, from which the spectral densities and the current through the system can be obtained. These equations could be solved numerically for arbitrary spin splitting. For strong Coulomb interaction, we can, in addition to the approximation introduced in the previous section, neglect double occupancy of the dot. The Coulomb interaction then introduces a cuto. We compare the results with the experiments of Ralph and Buhrman  in Figs. 3 and 4. The non-linear conductance is presented in Fig. 3, which shows good qualitative agreement between theory and experiment.The resonant peak for positivebias voltage is not reproduced; it appears too sharp to be explained by the same level that is responsible for the resonant peak at negative bias voltage. We think that it is due to either a dierent level or due to the doubly-occupied dot state, which was neglected in the calculation. In Fig. 3 we also see that the approximately logarithmic temperature dependence of the Kondo resonance is nicely reproduced. The splitting of the Kondo resonance with magnetic eld (Fig. 4) is reproduced well. However, in the experiment the split peaks show a stronger broadening than in our theory.
Fig. 3. Dierential conductance for zero magnetic eld. is the assymmetry in the capacitive coupling to the leads.
Fig. 4. Splitting of the zero-bias maximum with magnetic eld.
Fig. 5. Spectral function of the system, with the double occupation taken into account, for dierent values of the chemical potential .
4. Double occupation of the dot level In the results presented in the previous section  we have neglected the double occupancy of the dot. The strong Coulomb interaction was accounted for by a cuto. In Fig. 5 we show new results for the spectral density when double occupancy is included. In the absence of Coulomb interaction (U = 0), where the exact solution is a simple Lorentzian, our approximation yields the exact solution. This is due to the fact that all terms which are not included in our summation cancel exactly. This is not the case if the Coulomb interaction is turned on. We see from the gure that the analysis beyond Fermi’s golden rule reveals much new structure. We expect that this system which has been studied extensively since it was introduced for the equilibrium case by Anderson in 1961  will receive more attention in the future.
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