Zero bias anomaly out of equilibrium

TL;DR

This paper investigates the non-equilibrium zero bias anomaly in diffusive metals, deriving an effective action and showing how inelastic scattering smooths out the singularities seen in equilibrium.

Contribution

It introduces a new effective action framework for out-of-equilibrium fluctuations and explains the smoothing of the ZBA due to inelastic processes.

Findings

Inelastic scattering removes the singular behavior of the equilibrium ZBA.

Derived an effective action describing virtual fluctuations out-of-equilibrium.

Provided insights into the non-equilibrium tunneling density of states.

Abstract

The non-equilibrium zero bias anomaly (ZBA) in the tunneling density of states of a diffusive metallic film is studied. An effective action describing virtual fluctuations out-of-equilibrium is derived. The singular behavior of the equilibrium ZBA is smoothed out by real processes of inelastic scattering.

Figures

Equations

$$iS[Q,\varphi ]=i\mathrm{Tr}\{{\varphi }^T(U^{-1}+{\nu }_0){\sigma }_1\varphi \}$$

$$-\frac{\pi {\nu }_0}{4}\mathrm{Tr}\{D(\nabla Q{)}^2-4{\partial }_{t}Q\}-i\pi {\nu }_0\mathrm{Tr}\{{\varphi }_{\alpha }{\gamma }^{\alpha }Q\},$$

$$\int d{t}_{1}Q(\mathbf{r},t,t_1)Q(\mathbf{r},t_1,t^{\prime })=\delta (t-t^{\prime }){\sigma }_0,$$

$$iS[Q]=-\frac{\pi {\nu }_0}{4}\left(\mathrm{Tr}\{D(\nabla Q{)}^2-4{\partial }_{t}Q\}\right)$$

$$-\frac{i(\pi {\nu }_0{)}^2}{2}{\mathrm{Tr}}_K\{{\sigma }_{1}Q\}{\left(U^{-1}+{\nu }_0\right)}^{-1}{\mathrm{Tr}}_K\{Q\}.$$

$$Z[{\phi }_1,{\phi }_2]=\exp \left(iS[Q]\right)\exp (\frac{i{\nu }_0}{1+{\nu }_{0}U}$$

$$\displaystyle\left.\times\bigg{[}2\varphi_{1}\varphi_{2}-\pi\varphi_{1}{\rm Tr}_{K}\{Q\}-\pi\varphi_{2}{\rm Tr}_{K}\{\sigma_{1}Q\}\bigg{]}\right)$$

$$\displaystyle Q({\bf r},\epsilon,t)=e^{\frac{i\sigma_{1}}{4}\bar{f}({\bf r},\epsilon,t)}\left(\!\!\begin{array}[]{cc}1&2-4f({\bf r},\epsilon,t)\\ 0&-1\end{array}\!\!\right)e^{-\frac{i\sigma_{1}}{4}\bar{f}({\bf r},\epsilon,t)}.$$

$$Q(\mathbf{r},t,t^{\prime })=e^{\frac{i{\sigma }_1}{2}\bar{f}(\mathbf{r},t)+\frac{i}{2}f(\mathbf{r},t)}{\Lambda }_{\mathbf{r},t-t^{\prime }}{e}^{-\frac{i}{2}f(\mathbf{r},t^{\prime })-\frac{i{\sigma }_1}{2}\bar{f}(\mathbf{r},t^{\prime })},$$

$$\displaystyle\Lambda_{{\bf r},t-t^{\prime}}=\left(\begin{array}[]{cc}1&2-4n({\bf r},t-t^{\prime})\\ 0&-1\end{array}\right)\>,$$

$$iS[f,\bar{f}]=\frac{{\nu }_0}{2}\int (d\omega )(dq)$$

$$\displaystyle\times\bigg{[}\bar{f}(-\omega,-q)\>\omega\left(-Dq^{2}+\frac{i\omega}{1+\nu_{0}U(q)}\right)f(\omega,q)$$

$$\displaystyle-Dq^{2}T(\omega)\bar{f}(-\omega,-q)\bar{f}(\omega,q)\bigg{]}\,.$$

$$T(\omega )={\int }_{-\infty }^{\infty }dϵn(ϵ)[2-n(ϵ-\omega )-n(ϵ+\omega )],$$

$$\displaystyle{1\over 2}\langle\big{[}\hat{\rho}({\bf r},t),\hat{\rho}(0,0)\big{]}_{+}\rangle_{\omega,q}=-\frac{1}{4}\frac{\partial^{2}Z}{\partial\phi_{2}(\omega,q)\partial\phi_{2}(-\omega,-q)}$$

$$=\frac{1}{4}{\left(\frac{{\nu }_0}{1+{\nu }_{0}U}\right)}^2{\omega }^2⟨ff{⟩}_{\omega ,q}.$$

$$i⟨[\hat{\rho }(\mathbf{r},t),\hat{\rho }(0,0){]}_{-}\theta (t){⟩}_{\omega ,q}=\frac{1}{2i}\frac{{\partial }^{2}Z}{\partial {\varphi }_1(\omega ,q)\partial {\varphi }_2(-\omega ,-q)}$$

$$=\frac{i}{2}{\left(\frac{{\nu }_0}{1+{\nu }_{0}U}\right)}^2{\omega }^2⟨f\bar{f}{⟩}_{\omega ,q}+\frac{{\nu }_0}{1+{\nu }_{0}U}.$$

$$⟨ff{⟩}_{\omega ,q}=\frac{4D{q}^{2}T(\omega )}{{\nu }_0{\omega }^2{\left|D{q}^2+\frac{i\omega }{1+{\nu }_{0}V(q)}\right|}^2},$$

$$⟨f_{\omega ,q}{\bar{f}}_{-\omega ,-q}⟩\equiv ⟨f\bar{f}{⟩}_{\omega ,q}=\frac{2}{{\nu }_0}\frac{1}{\omega \left(D{q}^2-\frac{i\omega }{1+{\nu }_{0}U(q)}\right)},$$

$$\displaystyle{1\over 2}\langle\big{[}\hat{\rho}({\bf r},t),\hat{\rho}(0,0)\big{]}_{+}\rangle_{\omega,q}=\frac{\nu_{0}Dq^{2}T(\omega)}{|Dq^{2}[1+\nu_{0}U(q)]+i\omega|^{2}}\,,$$

$$i⟨[\hat{\rho }(\mathbf{r},t),\hat{\rho }(0,0){]}_{-}\theta (t){⟩}_{\omega ,q}=\frac{{\nu }_{0}D{q}^2}{D{q}^2[1+{\nu }_{0}U(q)]-i\omega }.$$

$$\nu (ϵ)=\frac{i}{2\pi }\int (dp)\left(G^R(ϵ,p)-G^A(ϵ,p)\right)$$

$$\nu (ϵ)=\frac{\nu }{2}{\int }_{-\infty }^{\infty }dt{e}^{-iϵ(t-t^{\prime })}⟨{\mathrm{Tr}}_{K}Q(\mathbf{r},t^{\prime },t){\sigma }_z{⟩}_Q.$$

$$\frac{\nu (ϵ)}{{\nu }_0}=1+2i{\int }_{-\infty }^{\infty }dtn(t)e^{iϵt-\frac{1}{4}{I}_{ff}(t)}\sin \left(\frac{1}{2}{I}_{f\bar{f}}(t)\right),$$

$$I_{ff}(t)=\int (d\omega )(dq)(1-\cos \omega t)⟨⟨ff⟩{⟩}_{\omega ,q},$$

$$I_{f\bar{f}}(t)=\int (d\omega )(dq)\sin \omega t⟨⟨f\bar{f}⟩{⟩}_{\omega ,q}.$$

$$⟨⟨f\bar{f}⟩{⟩}_{\omega ,q}$$

$$⟨⟨ff⟩{⟩}_{\omega ,q}=\frac{4D{q}^{2}T(\omega )(2+{\nu }_{0}U)U}{|D{q}^2-i\omega {|}^2|D{q}^2(1+{\nu }_{0}U)-i\omega {|}^2}.$$

$$n(ϵ,x)=a{n}_0(ϵ-eV/2)+(1-a)n_0(ϵ+eV/2),$$

$$T(\omega )=[a^2+(1-a{)}^2]T_{\mathrm{eq}}(\omega )$$

$$+a(1-a)[T_{\mathrm{eq}}(\omega +eV)+T_{\mathrm{eq}}(\omega -eV)],$$

$$T_{\mathrm{eq}}(\omega )=\omega \coth (\omega /T)\underset{T\to 0}{\to }|\omega |.$$

$$\frac{\nu (ϵ)}{{\nu }_0}=a\exp \left(-\frac{1}{8{\pi }^2{\nu }_{0}D}\log \frac{t_{-}}{\tau }\log (D^2{\kappa }^4\tau t_{-})\right)$$

$$+(1-a)\exp \left(-\frac{1}{8{\pi }^2{\nu }_{0}D}\log \frac{t_{+}}{\tau }\log (D^2{\kappa }^4\tau t_{+})\right).$$

$$\displaystyle t_{\pm}^{-1}={\rm max}\bigg{\{}\left|\epsilon\pm\frac{eV}{2}\right|,\frac{a(1-a)eV\log\left(\frac{\nu_{0}D^{2}\kappa^{2}}{eV}\right)}{4\pi\nu_{0}D}\bigg{\}},$$

Full text

The suppression of tunneling current at low bias due to electron-electron interaction is known as the zero bias anomaly (ZBA). The theory of ZBA for disordered metals at thermal equilibrium has been developed, on a perturbative level, by Altshuler and Aronov AA ; AA-review . The non-perturbative generalization of this theory was achieved by Finkelstein Finkelstein83 . Measurements of the tunneling density of states (DOS) in biased quasi-one-dimensional wires Pothier call for an extension of the theory to non-equilibrium setups. In this work we study the ZBA for disordered metallic films out of equilibrium, in both the perturbative and the non-perturbative (in interaction) limits.

Besides the experimental motivation, the problem of ZBA in a non-equilibrium system is of fundamental theoretical interest. At equilibrium, the distribution of electrons in phase space has a single edge at the Fermi surface. The Coulomb interaction between the tunneling electron and the electrons in the Fermi sea excites virtual particle-hole pairs around the Fermi edge, leading to the suppression of the tunneling DOS, similarly to the Debye-Waller factor. The suppression gets stronger when the electron energy approaches the Fermi energy. Out of equilibrium, the distribution of particles may have several sharp edges rather than a single one at the Fermi surface, which poses important questions addressed in this work: How will the excitation of electron-hole pairs in this situation affect the tunneling DOS? Will there be an interpaly between the two edges? We show that the two edges are not independent: one edge affects the ZBA near the other via real interaction-induced scattering processes governing the dephasing of electrons in the non-equlibrium regime. From this point of view the problem we are considering is a representive of a class of phenomena that involve renormalization away from thermal equilibrium, such as the Fermi edge singularity Abanin and the Kondo effect Paaske .

What makes the ZBA particularly interesting is its deep connection to various conceptually important phenomenological ideas. At equilibrium, the non-perturbative results Finkelstein83 have been reproduced by quantum hydrodynamical methods Levitov , and, within the framework of the theory of dissipation Nazarov89 , by methods that rely on the fluctuation-dissipation theorem. Our work circumvents this restriction. Starting with the Keldysh non-linear $\sigma$-model, we derive an effective action that accounts for virtual fluctuations in disordered metals away from equilibrium. This action is complementary to the one for kinetics of real fluctuations (such as noise) developed earlier GMG ; bodineau04 ; pilgram04a ; Bagrets . Further, we discuss a connection between our theory and phenomenological methods Levitov ; Nazarov89 ; Devoret91 ; Glazman91 . As a central application of our theory, we analyze the ZBA problem and calculate the tunneling DOS for a two-dimensional (2D) diffusive metallic film subject to external bias.

Consider an electron which tunnels from a tip into a metal, subject to an external bias. The Coulomb interaction $U(q)$ ($U(q)=2\pi e^{2}/q$, $d=2$) causes electrons in the metal to readjust their position, such that they try to screen the added charge. Their motion is impeded by static disorder. For a sufficiently good metal, characterized by the dimensionless conductance $g=\epsilon_{F}\tau\gg 1$, ($\epsilon_{F}$ is the Fermi energy and $\tau$ – the elastic mean free time) the electron motion is diffusive. Low energy excitations are accounted for by the non-linear Keldysh $\sigma$-model Kamenev-Andreev

[TABLE]

where $Q({\bf r},t,t^{\prime})$ is a $2\times 2$ matrix field (in the Keldysh space) describing slow electronic modes, $\phi({\bf r},t)$ is a 2-vector in the Keldysh space representing the Coulomb interaction, and ${\rm Tr}$ denotes the trace both over Keldysh indices (${\rm Tr}_{K}$), spatial, and temporal coordinates. The $Q$ field is subject to a non-linear constraint,

[TABLE]

$\gamma_{1}\equiv\sigma_{0}$ is a $2\times 2$ unit matrix, $\gamma_{2}\equiv\sigma_{1}$ the first Pauli matrix, $\nu_{0}$ is the DOS at the Fermi energy in the absence of interaction, and $D$ is the diffusion coefficient. The theory can be rewritten in terms of the field $Q$ only. Integrating the $\phi$ field out, we derive the following action:

[TABLE]

Physical observables can be found by differentiating the generating functional

[TABLE]

over the source fields $\varphi_{1}$ and $\varphi_{2}$.

So far all low energy excitation in the problem have been kept indiscriminately. The price one pays for this is an extreme technical complexity of the theory. In some cases, this description is excessive and the theory can be simplified by singling out a particular subspace of the $Q$ matrices. The resulting theory is less general, but more suitable for tackling a specific class of problems. The noise statistics in disordered systems is a remarkable example of such a simplification. An extension of the Boltzman-Langevin approach KTF to high order correlators is achieved on the subspace of matrices that are diagonal in the Wigner representation,

[TABLE]

The theory pilgram04a ; GMG ; Bagrets resulting from the $\sigma$-model (Zero bias anomaly out of equilibrium), reduced to the subspace (7), accounts for real fluctuations in agreement with the cascade idea of Nagaev Nagaev .

Below we use a similar ideology, “projecting” the $\sigma$-model on the subspace appropriate for the ZBA problem. According to works on ZBA at equilibrium, these are virtual fluctuations of gauge fields that dominate the suppression of the tunneling DOS Finkelstein83 ; Kopietz98 ; Kamenev-Andreev . This suggests that gauge-type fluctuations constituting local-in-time rotations of the saddle point $\Lambda_{t-t^{\prime}}$,

[TABLE]

where $n({\bf r},t-t^{\prime})$ is the Fourier transform of the local distribution function $n({\bf r},\epsilon)$, are to be retained. Plugging (Zero bias anomaly out of equilibrium) into the action (Zero bias anomaly out of equilibrium) and expanding up to quadratic order in $f$ and $\bar{f}$, we derive an effective theory of fluctuations in an interacting diffusive conductor.

[TABLE]

Here the effective temperature $T(\omega)$ is defined as

[TABLE]

and $n(\epsilon)$ is a distribution function determined by the Boltzmann equation.

To demonstrate the consistency of our approach, we first show that the effective action (Zero bias anomaly out of equilibrium) reproduces correctly known density correlation functions. The symmetrized density correlation function is given by

[TABLE]

Similarly, the response function can be expressed as

[TABLE]

Calculating the correlation functions entering Eqs. (14), (15) using the action (Zero bias anomaly out of equilibrium),

[TABLE]

we reproduce the known results for for the spectral function of density fluctuations

[TABLE]

and the density-density response function,

[TABLE]

It is worth noting the analogy between our effective action and phenomenological theories that describe ZBA at equilibrium within an effective environment model Glazman91 ; Devoret91 . The latter explain the ZBA as the influence of virtual fluctuations of an electromagnetic field at equilibrium Nazarov89 on the electron tunneling. In view of the fluctuation-dissipation theorem, fluctuations of the electromagnetic field are determined by the complex impedance of the system. In the zero dimensional case, modes of the electromagnetic field can be considered as independent quantum harmonic oscillators. Being suddenly shaken by the incoming electron, they move away from the equilibrium position, reducing the overlap with their original configuration hence suppressing electron tunneling amplitude. The action (Zero bias anomaly out of equilibrium) describes similar processes in a 2D diffusive system and without the assumption of thermal equilibrium, thus keeping the information about both real and virtual processes in a non-equilibrium state.

Now we are ready to apply the theory to the problem of ZBA out of equilibrium. The tunneling DOS

[TABLE]

can be rewritten in terms of $Q$ matrices as

[TABLE]

Plugging Eq. (Zero bias anomaly out of equilibrium) into Eq.(18), we find

[TABLE]

where we used the notations

[TABLE]

There is a subtle point related to the definition of the correlator $\langle\langle ff\rangle\rangle$ and $\langle\langle f\bar{f}\rangle\rangle$ in (Zero bias anomaly out of equilibrium). The ZBA should vanish in the absence of interaction ($\nolinebreak{U=0}$). This physically obvious statement is valid to all orders of the diagrammatic calculations (all potentially singular contributions vanish after the time integration over the Keldysh contour) and is satisfied by the non-linear $\sigma$ model. However, it ceases to be an exact feature of the theory when the full space of $Q$ matrices is reduced to the subspace Eq. (Zero bias anomaly out of equilibrium). This problem is easily cured. Since we are only interested in summing up the leading interaction-induced $\ln^{2}\epsilon$ contributions to the tunneling DOS (to all orders), these are the interaction-induced parts of $\langle ff\rangle$ and $\langle f\bar{f}\rangle$ that we have to keep in Eq. (Zero bias anomaly out of equilibrium) subtraction :

[TABLE]

Analogously,

[TABLE]

Equations (19)–(22) constitute our general result for the non-equilibrium ZBA. At equilibrium they reproduce the known results for the ZBA in the perturbative AA ; AA-review and non-perturbative Finkelstein83 regimes.

To illustrate the effect of non-equilibrium conditions on ZBA we consider a diffusive film of length $L$ connected to two zero-temperature reservoirs with voltage difference $V$. We assume that the energy relaxation in the film can be neglected, i.e. that $L$ is shorter than the energy relaxation length, a condition met at not too high bias. In this case the solution of the Boltzmann equation is a double-step function

[TABLE]

where $n_{0}(\epsilon)$ is the Fermi distribution function, $a=x/L$, and $x$ is the distance from the reservoir. This distribution corresponds to the effective temperature

[TABLE]

where $T_{\rm eq}(\omega)$ is the equilibrium value

[TABLE]

Substituting Eqs. (21), (22) into Eqs.(Zero bias anomaly out of equilibrium),( 19), we find the tunneling DOS out of equilibrium:

[TABLE]

This result is valid in both the perturbative (when the exponentials can be expanded up to the first non-trivial term) and non-perturbative regimes. The DOS consists of two terms corresponding to electrons coming from the left and right reservoir. The energy scales governing the argument of the logarithm in these two terms are

[TABLE]

respectively ($\kappa=2\pi e^{2}\nu$).

The evolution of the ZBA in the tunneling DOS with decreasing conductance (i.e. from the perturbative to the non-perturbative regime) is illustrated in Fig. 1. The DOS has a two-dip shape with minima reached at the energies where the distribution function exhibits discontinuous jumps (i.e. at the positions of the Fermi edge in the left and right leads, $\epsilon=\pm\frac{eV}{2}$). Away from the minima, the DOS is controlled by the energy measured from the corresponding edge. As this energy decreases (we get closer to one of the Fermi edges), the singularity near this edge gets affected by the presence of the other edge. The broadening of the ZBA singularity takes place on a new energy scale,

[TABLE]

The notation introduced in Eq. (28) stresses the analogy with the equilibrium dephasing rate $\tau_{\phi}^{-1}(T)$ governing the infrared cutoff of the interference phenomena such as the weak localization AA-review . The emergence of $\tau^{-1}_{\phi}(V)$ shows that inelastic scattering processes (responsible for dephasing) lead to smearing of the ZBA singularity.

Qualitatively the results can be explained as follows. Virtual fluctuations of the gauge fields act on a quasiparticle and suppress the tunneling DOS. Their influence is limited by the quasiparticle life time. An external bias enhances fluctuations of the electromagnetic field, which in turn dephase the tunneling quasiparticle. This results in a competition between the virtual processes establishing the ZBA and the real fluctuations (noise) cutting it off.

Let us stress the crucial difference with the equilibrium situation. At equilibrium, the ZBA singularity in 2D is cut off by temperature (which enters via the distribution function); specifically, the dephasing rate $\tau_{\phi}^{-1}(T)$ does not affect the ZBA in any essential way, since $\tau_{\phi}^{-1}(T)\ll T$. It is thus a distinct feature of the strongly non-equilibrium regime that the ZBA is smeared by the inelastic scattering (dephasing) rate.

In conclusion, we have developed an effective theory for virtual fluctuations in disordered metals out of equilibrium. Using this theory, we studied the ZBA in a 2D metallic film biased by external voltage. We have found that out of equilibrium the tunneling DOS has a double-dip structure with minima reached at the ”edges” of the particle distribution. The ZBA near any of the ”edges” is influenced by the other one. The suppression of DOS is smoothed out by the real processes of inelastic scattering (dephasing) with characteristic energy scale $\tau_{\rm\phi}(V)$. Further applications and extensions of our theory include, in particular, the ZBA in quasi-one-dimensional and strictly one-dimensional (Luttinger liquid) wires Mishchenko out of equilibrium; these results will be reported elsewhere.

We thank D. Bagrets, N. Birge, A. Finkelstein, D. Maslov, A. Shnirman, and A. Shytov for useful discussions. This work was supported by NSF-DMR-0308377 (DG), US-Israel BSF, ISF of the Israel Academy of Sciences, and DFG SPP 1285 (YG), DFG Center for Functional Nanostructures, EC Transnational Access Program at the WIS Braun Submicron Center (ADM), and Einstein Minerva Center.