^{1}and Eran Rabani

^{1,a)}

### Abstract

We study steady state transport through a double quantum dot array using the equation-of-motion approach to the nonequilibrium Green functions formalism. This popular technique relies on uncontrolled approximations to obtain a closure for a hierarchy of equations; however, its accuracy is questioned. We focus on 4 different closures, 2 of which were previously proposed in the context of the single quantum dot system (Anderson impurity model) and were extended to the double quantum dot array, and develop 2 new closures. Results for the differential conductance are compared to those attained by a master equation approach known to be accurate for weak system-leads couplings and high temperatures. While all 4 closures provide an accurate description of the Coulomb blockade and other transport properties in the single quantum dot case, they differ in the case of the double quantum dot array, where only one of the developed closures provides satisfactory results. This is rationalized by comparing the poles of the Green functions to the exact many-particle energy differences for the isolate system. Our analysis provides means to extend the equation-of-motion technique to more elaborate models of large bridge systems with strong electronic interactions.

This work was supported by the US-Israel Binational Science Foundation and by the FP7 Marie Curie IOF project HJSC. T.J.L. is grateful to the Azrieli Foundation for the award of an Azrieli Fellowship.

I. INTRODUCTION

II. THEORY

A. Model Hamiltonian

B. Equation of motion

1. Approximation-1

2. Approximation-2

3. Approximation-3

4. Approximation-4

C. Master equations

III. RESULTS AND DISCUSSION

A. Symmetric bridge

B. Asymmetric bridge

IV. CONCLUDING REMARKS

### Key Topics

- Quantum dots
- 26.0
- Many body systems
- 10.0
- Transport properties
- 9.0
- Green's function methods
- 7.0
- Kondo effect
- 6.0

## Figures

A sketch of the double QD bridge. See main text for the definition of all quantities.

A sketch of the double QD bridge. See main text for the definition of all quantities.

Plots of the differential conductance versus the bias voltage for the symmetric bridge (ɛ_{α↑} = ɛ_{α↓} = ɛ_{β↑} = ɛ_{β↓} = 0.35*U*) for *V* = 0. Upper left, upper right, lower left, and lower right panels correspond to *h* = 0.1*U*, 0.3*U*, 0.5*U*, and 0.7*U*, respectively. Black curves correspond to results based on the ME. Red (circles), green (diamonds), blue (triangles), and magenta (stars) correspond to the results obtained by approximation schemes 1–4, respectively. The notation |*i*⟩ → |*j*⟩ indicates that the conductance peak calculated by means of ME corresponds to a transition form the *n* _{ i }-particle states to any of the *n* _{ j }-particle states. The remaining model parameters were , , and β^{−1} = *U*/40.

Plots of the differential conductance versus the bias voltage for the symmetric bridge (ɛ_{α↑} = ɛ_{α↓} = ɛ_{β↑} = ɛ_{β↓} = 0.35*U*) for *V* = 0. Upper left, upper right, lower left, and lower right panels correspond to *h* = 0.1*U*, 0.3*U*, 0.5*U*, and 0.7*U*, respectively. Black curves correspond to results based on the ME. Red (circles), green (diamonds), blue (triangles), and magenta (stars) correspond to the results obtained by approximation schemes 1–4, respectively. The notation |*i*⟩ → |*j*⟩ indicates that the conductance peak calculated by means of ME corresponds to a transition form the *n* _{ i }-particle states to any of the *n* _{ j }-particle states. The remaining model parameters were , , and β^{−1} = *U*/40.

Same as Figure 2 but for *V* = 0.8*U*.

Plots of the differential conductance versus the bias voltage for the asymmetric bridge (ɛ_{α↑} = ɛ_{α↓} = 0.15*U* and ɛ_{β↑} = ɛ_{β↓} = −0.2*U*) for *V* = 0. Upper left, upper right, lower left, and lower right panels correspond to *h* = 0.1*U*, 0.3*U*, 0.5*U*, and 0.7*U*, respectively. Black curves correspond to results based on the ME. Red (circles), green (diamonds), blue (triangles), and magenta (stars) correspond to the results obtained by approximation schemes 1–4, respectively. The notation |*i*⟩ → |*j*⟩ indicates that the conductance peak calculated by means of ME corresponds to a transition form the *n* _{ i }-particle states to any of the *n* _{ j }-particle states. The remaining model parameters were , , and β^{−1} = *U*/40.

Plots of the differential conductance versus the bias voltage for the asymmetric bridge (ɛ_{α↑} = ɛ_{α↓} = 0.15*U* and ɛ_{β↑} = ɛ_{β↓} = −0.2*U*) for *V* = 0. Upper left, upper right, lower left, and lower right panels correspond to *h* = 0.1*U*, 0.3*U*, 0.5*U*, and 0.7*U*, respectively. Black curves correspond to results based on the ME. Red (circles), green (diamonds), blue (triangles), and magenta (stars) correspond to the results obtained by approximation schemes 1–4, respectively. The notation |*i*⟩ → |*j*⟩ indicates that the conductance peak calculated by means of ME corresponds to a transition form the *n* _{ i }-particle states to any of the *n* _{ j }-particle states. The remaining model parameters were , , and β^{−1} = *U*/40.

Same as Figure 4 but for *V* = 0.8*U*. Results obtained from approximation 1 are only presented for the case *h* = 0.1*U* (upper left panel) as we could not converge it for higher values of *h*.

Same as Figure 4 but for *V* = 0.8*U*. Results obtained from approximation 1 are only presented for the case *h* = 0.1*U* (upper left panel) as we could not converge it for higher values of *h*.

## Tables

(Left column) Location of the poles of the unperturbed system's GF as calculated using the 2nd approximation. (Right column) The differences in energy between many-particle states that differ by one electron, such that Δ*E*(*N*) = *E*(*N*) − *E*(*N* − 1). Here , and .

(Left column) Location of the poles of the unperturbed system's GF as calculated using the 2nd approximation. (Right column) The differences in energy between many-particle states that differ by one electron, such that Δ*E*(*N*) = *E*(*N*) − *E*(*N* − 1). Here , and .

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