Exact results for nonlinear ac-transport through a resonant level model

# Exact results for nonlinear ac-transport through a resonant level model

P. Wang    M. Heyl    S. Kehrein Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience,
Ludwig-Maximilians-Universität, Theresienstrasse 37, 80333 Munich, Germany
July 15, 2019
###### Abstract

We obtain exact results for the transport through a resonant level model (noninteracting Anderson impurity model) for rectangular voltage bias as a function of time. We study both the transient behavior after switching on the tunneling at time and the ensuing steady state behavior. Explicit expressions are obtained for the ac-current in the linear response regime and beyond for large voltage bias. Among other effects, we observe current ringing and PAT (photon assisted tunneling) oscillations.

## I Introduction

The recent advances in nanotechnology created a lot of interest in transport through correlated quantum impurities. While the linear response regime essentially probes the ground state properties of the system, transport beyond the linear response regime explores genuine non-equilibrium quantum many-body phenomena. However, theoretical calculations beyond the linear response regime are challenging since the steady state cannot be constructed via a variational principle like equilibrium states. Even for dc-bias only recently exact numerical methods have been developed that permit such investigations for interacting systems, notably the time-dependent numerical renormalization group AndersPRL101 (), Monte Carlo methods Schmidt2008 (); Millis2009 (), and the time-dependent density matrix renormalization group WhitePRL2004 (); Schmitteckert2004 (). Some of the analytical methods that have been applied successfully are perturbative Keldysh calculations Fuji2003 (), extensions of the renormalization group Rosch2001 (); Schoeller2009 (), flow equations Kehrein_Kondo (), and generalizations of NCA (non-crossing approximation) to non-equilibrium nordlander (); Meir93 (). A comparative review of theoretical methods can be found in Ref. FabianNewJPhys ()

For ac-bias beyond the linear response regime still much less is known since, e.g., the numerical methods cannot easily be generalized to time-dependent bias. Interesting ac-phenomena are for example the photon assisted tunneling effect (PAT) tien () that has been observed in experiments kouwenhoven94 (), or the ”current ringing” after a step-like bias puls wingreen94 (). Non-equilibrium Green’s function methods can be employed wingreen93 (); wingreen94 (); maciejko06 () when the correlation effects are not too strong. In the strongly correlated regime of the Kondo model the non-crossing approximation was found to be reliable nordlander (); plihal (); Goker08 (); nordlander2000 (). At a specific point of the two-lead Kondo model it can be solved exactly hershfield96 (), which permits exact results for the current in the steady state hershfield95 (), after a rectangular pulse hershfield00 () or under sinusoidal bias hershfield96 (). Unfortunately, this special point is not generic for a Kondo impurity that can be derived from an underlying Anderson impurity model, which is experimentally the most relevant situation.

In this paper we study the response of a resonant level model (noninteracting Anderson impurity model) under rectangular ac voltage bias after switching on the tunneling at time . We derive exact analytical results for the transient and steady-state current by diagonalizing the Hamiltonian. This exact solution contains both dc- and ac-bias in and beyond the linear response regime. While dc-results and ac-results with sinusoidal bias have been obtained previously in the literature wingreen94 (), rectangular ac-driving beyond the linear regime seems not to have been studied before. Besides being experimentally relevant, our results are also helpful for exploring the various crossovers in this important model and serve as an exact benchmark for future work.

## Ii Model and diagonalization

The resonant level model coupled to two leads is defined by the following Hamiltonian

 H = ∑kαϵkc†kαckα+∑kαg√2(c†kαd+h.c.),

where denotes the leads. The spin index can be omitted since the model is non-interacting and we work with spinless fermions. All energies are measured with respect to the single-particle energy of the impurity orbital (). We take a wide band limit with a linear dispersion relation, , where denotes the level spacing and an integer number. The hybridization is defined by where . The impurity orbital spectral function in equilibrium is then given by

 ρd(ϵ)=Γπ(ϵ2+Γ2) (1)

Our strategy to obtain exact results is to first diagonalize the discretized Hamiltonian and to then take the thermodynamic limit . We introduce the hybridized basis . It is then straightforward to diagonalize the Hamiltonian

 H=∑kϵkc†k−ck−+∑sϵsc†scs, (2)

where . The inverse transformation is and . The eigenvalues are determined as solutions of the equation

 ϵsg2=πηcotπϵsη. (3)

In the thermodynamic limit

 B2s=g2ϵ2s+Γ2. (4)

From the diagonalization one also derives the following set of equations

 ∑sB2s = 1, ∑sg2B2s(ϵs−ϵk)2 = 1, ∑sB2sϵs−ϵk = 0, ∑sB2s(ϵs−ϵk)(ϵs−ϵk′) = 0,k′≠k.

which will be important for calculating various summations below.

An ac voltage bias leads to time-dependent potentials in the leads and the Hamiltonian takes the form

 H=∑kα(ϵk−uα(t))c†kαckα+∑kαg√2(c†kαd+h.c.). (5)

We suppose that initially (at time ) the left and right lead chemical potential are the same, , the hybridization is switched off and that there is no electron in the dot, . At time the hybridization is switched on and a rectangular voltage bias with period  (see Fig. 2) is applied: for and for .levelV () therefore gives the energy difference of the impurity orbital to the ”average” Fermi energy of the leads for time  (Fig. 2).

The current operator is defined as

 Iα=sαedNαdt=igesα√2∑k(d†ckα−c†kαd), (6)

where denotes the total number of electrons in lead  and , .

In the first half period the Hamiltonian is

 Ha = ∑k(ϵk+V2)c†kLckL+∑k(ϵk−V2)c†kRckR (7) +∑kαg√2(c†kαd+h.c.).

Because the dispersion relation is linear and runs from to (wide band limit), we can simply relabel the fermion operators, . The potentials in the leads are eliminated by this transformation and the Hamiltonian can be diagonalized as before: , where and . Similarly, in the second half period the Hamiltonian is diagonalized as , where and .

In the Heisenberg picture the current operator at time (first half period) can be expressed as

 Iα(t)=(eiHaTeiHbT)NeiHaτIαe−iHaτ(e−iHbTe−iHaT)N, (8)

and in the second half period ()

 Iα(t) = (eiHaTeiHbT)NeiHaTeiHbτIα (9) ×e−iHbτe−iHaT(e−iHbTe−iHaT)N.

To find we first calculate the time evolution of the single fermion operator and under or by expressing and in the hybridized basis, next applying the diagonal time evolution and finally transforming back to the original basis. The calculation is straightforward but one needs to pay attention when encountering summations with respect to the eigenenergies . In the thermodynamic limit the summation can be transformed into an integral when there is no pole in the integrand, e.g., . If there are poles in the integrand we first calculate the time derivative to get rid of the pole terms. Key formulas are

 ∑sB2se−iϵstϵs−ϵk = e−iϵkt−e−Γtϵk+iΓ , (10) ∑sB2se−iϵst(ϵs−ϵk)2 = (1g2+−1−(iϵk−Γ)t(ϵk+iΓ)2)e−iϵkt (11) +e−Γt(ϵk+iΓ)2 .

By using these two formulas we get

 eiH(a,b)Td†e−iH(a,b)T=e−ΓTd†+∑kαg√2W(a,b)kαc†kα (12) eiH(a,b)Tc†kαe−iH(a,b)T=g√2W(a,b)kαd† +∑k′α′(g2(W(a,b)kα−W(a,b)k′α′)2(ϵ(a,b)kα−ϵ(a,b)k′α′)+δα,α′δk,k′eiϵ(a,b)kαT)c†k′α′,

where , and . Employing this formula twice gives the evolution over a full period:

 eiHaTeiHbTd†e−iHbTe−iHaT = e−2ΓTd†+∑kαg√2Dakαc†kα, eiHaTeiHbTc†kαe−iHbTe−iHaT = g√2Dbkαd†+∑k′α′(Kk′α′,kα+δk,k′δα,α′e2iϵkT)c†k′α′, (14)

where

 D(a,b)kα = eiϵ(a,b)kαTW(b,a)kα+e−ΓTW(a,b)kα, Kk′α′,kα = eiϵak′α′Tg2(Wbk′α′−Wbkα)2(ϵbk′α′−ϵbkα)+eiϵbkαTg2(Wak′α′−Wakα)2(ϵak′α′−ϵakα)+g22Wak′α′Wbkα. (15)

We perform the summation over by transforming it into an integral and then employing the residue theorem. Applying the above formula recursively times yields

 (eiHaTeiHbT)Nd†(e−iHbTe−iHaT)N = e−2NΓTd†+∑kαg√2DakαγN(k)c†kα, (eiHaTeiHbT)Nc†kα(e−iHbTe−iHaT)N = ∑kαg√2DbkαγN(k)d†+∑k′α′(αN(k′,k)Kk′α′,kα (16) +δk,k′δα,α′e2NiϵkT+g22βN(k′,k)Dak′α′Dbkα)c†k′α′,

where and the recursion relations are

 αN+1(k′,k) = αN(k′,k)e2iϵk′T+e2NiϵkT, βN+1(k′,k) = βN(k′,k)e2iϵk′T+γN(k), γN+1(k) = γN(k)e−2ΓT+e2NiϵkT.

It is easy to find

 αN = e2NiϵkT−e2Niϵk′Te2iϵkT−e2iϵk′T (17) γN = e2NiϵkT−e−2NΓTe2iϵkT−e−2ΓT (18)

In the first half period the current evaluates to

 Iα(t) = sαeΓh∫dϵknk(∑α′Γ|ξ(1)kα′|2 (19) −2Im(ξ(1)kαe−2NiϵkT−iϵakατ)).

where . is the Fermi-Dirac distribution function. In the sequel we will always specialize to the zero temperature case ( for , for ). In the second half period the current evaluates to

 Iα(t) = sαeΓh∫dϵknk(∑α′Γ|ξ(2)kα′|2 (20) −2Im(ξ(2)kαe−2NiϵkT−iϵakαT−iϵbkατ))

where . To simplify notation in lengthy expressions we will frequently employ as the unit of energy and current, and as the unit of time. In the final results we always reintroduce all dimensionful parameters.

## Iii Buildup of the steady state

There is a transient time regime after the coupling of the dot to the leads is switched on at time before a steady state has built up. Initially, the left lead current is opposite to the right one and the initially empty dot is being charged. We will see that these transient effects decay exponentially (proportional to ) to the steady state.

Let us explicitly look at the two limits of period (dc bias) and (very fast driving). For one finds from Eq. (19)

 Iα(t) = sαeh∫dϵknk(∑α′1+e−2t−eiϵakα′t−t−e−iϵakα′t−t(ϵakα′)2+1 (21) −2Im[1−e−iϵakαt−tϵakα−i]).

The steady limit () is

 I=eΓh∫dϵ(n(ϵ+eV2)−n(ϵ−eV2))Γϵ2+Γ2 (22)

which of course coincides with the well-known result for the stationary dc-current wingreen94 (), e.g. for zero temperature

 I=2eΓharctan(eV2Γ) . (23)

In the fast driving limit we keep invariant and let . According to the Trotter formula, the evolution then becomes equivalent to zero voltage bias Eisler (), . We find

 Iα(t)=sα2eΓe−th∫dϵn(ϵ)e−t−cosϵt−ϵsinϵtϵ2+1. (24)

In Fig. 2 we show the transient currents in the left and right lead for different periods when . The current oscillations are suppressed when the frequency goes to infinity. The -curves gradually change from the dc limit to the high frequency limit described by Eq. (24) when the period decreases. In the fast driving limit the left and right currents are opposite to each other and both decay to zero with increasing time.

## Iv Steady state behavior

When the time is much larger than , the current reaches its steady state behavior. By taking we find this steady state limit given by

 Iα(τ)=sαeΓh∫dϵknk(|~ξkL|2+|~ξkR|2−2Im~ξkα), (25)

where . In the first half period we have

 ~ξkα=1ϵakα−i+sαV(e2iϵT−iϵakατ−τ−eiϵakα(T−τ)−T−τ)(e2iϵT−e−2T)(ϵakα−i)(ϵbkα−i),

and in the second half period

 ~ξkα=1ϵbkα−i+sαV(eiϵbkα(T−τ)−T−τ−e2iϵT−iϵbkατ−τ)(e2iϵT−e−2T)(ϵakα−i)(ϵbkα−i).

From Eqs. (IV) and (IV) one immediately verifies that the steady state current satisfies as expected intuitively, where denotes the opposite lead.

### iv.1 Linear response regime

In the linear response regime of small voltage bias a sinusoidal signal drives a sinusoidal current with the same frequency, and signals with different frequencies can be superimposed linearly. Therefore we can factorize the rectangular signal into a series of sinusoidal components and find the frequency-dependent complex admittance of the system.

In the linear response regime the left lead current is equal to the right lead one and can be expressed as

 limV→0I(τ)V=e2h∫dϵn(ϵ)T(ϵ), (28)

where

 T(ϵ)=2ϵΓ3(ϵ2+Γ2)2−Im[2Γ2eiϵT−iϵτ−τ(eiϵT+e−T)(ϵ−iΓ)2]. (29)

We Fourier transform both the ac-voltage signal and the current. We define , where we use the property , and . Here and is an odd number. The voltage bias is for and for , leading to . By adjusting the frequency can be an arbitrary real number, and the linear response admittance at zero temperature is given by

 G(ω)=e2h⎛⎜⎝arccot−ω−μΓ−arccotω−μΓ2ω/Γ−iΓ4ωln(μ2+Γ2)2((μ+ω)2+Γ2)((μ−ω)2+Γ2)⎞⎟⎠, (30)

where denotes the position of the dot level with respect to the average Fermi energy of the leads, see Fig. 1. Eq. (30) agrees with previous ac-calculations in the linear response regime, see Ref. fu (). Fig. 3 depicts for different level positions . The admittance goes to zero for fast driving, . For one recovers the well-known dc-conductance . For asymmetric dot positions the resonance peak is around , showing the PAT (photon assisted tunneling) effect kouwenhoven94 (): When the frequency of the ac-signal is equal to the energy difference of the dot level from the Fermi energy in the leads, electrons in the leads can absorb a photon and jump into the dot. Notice from Fig. 3 that the symmetric dot always acts like an inductor as already explained in Ref. fu (). For asymmetric dots there is a crossover from capacitive to inductive behavior around  fu ().

### iv.2 Beyond the linear response regime

For a voltage bias beyond the linear response regime it is impossible to calculate by performing a Fourier transformation since the different frequency components interact with each other nonlinearly. Therefore we now depict the behavior of the current as a function of time  during one full period in the steady state situation. Due to the nonlinearities we need to discuss this separately for different driving periods . We will always take zero temperature in the sequel, the generalization to nonzero temperature is straightforward.

We first look at fast driving, . For the symmetric situation the curve becomes triangled: The current decreases from maximum to minimum in the first half period, and then increases from minimum to maximum in the second half period, see Fig. 4. In the opposite slow driving limit , the curve becomes rectangled. The saturated current in each half period is simply given by the corresponding steady dc-current (23). For intermediate driving speed, , we observe ringing oscillations wingreen94 () of the current with period (see Fig. 5).

For asymmetric dot positions the current also has characteristics of PAT and ringing, which are, however, not easily visible in a plot like Fig. 6. Clear signatures can be found in the the differential conductance with respect to the gate voltage, which we denote as gate differential conductance  to distinguish it from the usual definition of differential conductance with respect to the voltage bias between the leads. We define

 Ggateα(ϵ,τ)\lx@stackreldef=dIα(τ)dμ|μ=ϵ (31)

and the current can then be expressed as .

Figs. 7 and 8 shows in the first half period ( in the second half period follows via ). In the linear response regime we find a pair of bright PAT lines at (see Fig. 7). In the regime far from equilibrium, high order PAT lines at () can be observed (see Fig. 8), indicating multiple photon assisted tunneling processes. These PAT lines combine and are replaced by a pair of bright resonance lines at when the period increases. This demonstrates that ac transport for high frequencies is dominated by photon assisted tunneling, and by resonance tunneling for low frequencies.

## V Conclusions

We have investigated a resonant level model driven by rectangular ac-bias in and beyond the linear response regime. Even this simple model shows surprisingly rich behavior in its transport properties. One can observe specific nonequilibrium effects like the buildup of the steady state, current ringing and photon assisted tunneling, and the crossover to the well-studied limiting cases of dc-bias and linear response regime. The results are exact and based on an explicit diagonalization of the Hamiltonian in the first and second half period of the rectangular voltage bias driving. Within the flow equation framework, this approach can easily be generalized to an interacting quantum impurity model exposed to ac-driving beyond the linear regime. Much less is known about such systems, which provides another motivation for this work and will be studied in a subsequent publication.

We acknowledge support through SFB 484 of the Deutsche Forschungsgemeinschaft, the Center for NanoScience (CeNS) Munich, and the German Excellence Initiative via the Nanosystems Initiative Munich (NIM).

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