A - Derivation of Eq.(19)

# Shot noise in Weyl semimetals

## Abstract

We study the effect of inelastic processes on the magneto-transport of a quasi-one dimensional Weyl semi-metal, using a modified Boltzmann-Langevin approach. The magnetic field drives a crossover to a ballistic regime in which the propagation along the wire is dominated by the chiral anomaly, and the role of fluctuations inside the sample is exponentially suppressed. We show that inelastic collisions modify the parametric dependence of the current fluctuations on the magnetic field. By measuring shot noise as a function of a magnetic field, for different applied voltage, one can estimate the electron-electron inelastic length .

1

## I Introduction

Weyl semimetals have been a subject of active experimental and theoretical research due to their unusual transport properties (1); (2); (3); (4); (5); (6); (7); (8); (9); (10); (11); (12). A Weyl semimetal is characterized by a three dimensional band structure, where valence and conduction band touch at discrete isolated points in the Brillouin zone. Excitations in the vicinity of the band degeneracy points are governed by the Weyl Hamiltonian .

The non-trivial topology of Weyl semimetal can be revealed by applying an external magnetic field, which leads to the formation of Landau levels (LL). The Weyl nodes result in the emergence of chiral zero Landau levels, protected against scattering for a sufficiently smooth disorder. Remarkably, transport properties in the presence of a magnetic field can be described by the semiclassical Boltzmann equation(13); (14); (15).

 ∂f(p,r)∂t+˙r∂f(p,r)∂r+˙p∂f(p,r)∂p=I[f(p,r)]. (1)

Here is the distribution function, is the momentum, is the collision integral and:

 ˙r =∂ϵp∂p+˙p×Ωp, (2) ˙p =eE+˙r×eB.

The nontrivial topology is reflected in Berry curvature terms that appear in the Liouville operator (16); (17); (18):

 Ωp =∇p×Ap, (3) Ap =i⟨up|∇pup⟩,

where is a periodic part of the Bloch wave function. These terms originate from the chiral anomaly(19) and are absent in topologically trivial matter. They give rise to ballistic propagation(15) and non-local ac conductance(20).

In this work, we study the effect of inelastic processes on the magneto-transport of a quasi-one dimensional Weyl semi-metal, within a semi-classical description. The wire, with cross-section is pierced by a magnetic field , oriented along the wire in direction. We assume that the inter-nodal scattering length is much longer than the system length . This allows us to focus in the vicinity of a single Weyl node with a given chirality. The final answer is given by a sum over all Weyl nodes.

We first note that similar to a metal, the conductance is unaffected by interactions between electrons, as those do not lead to momentum relaxation. Moreover, as the conductance is measured as a response to the difference of total electro-chemical potentials, the results obtained for interacting electrons within self-consistent field approximation, and for non-interacting electrons subjected to a difference in chemical potential, coincide. Consequently, the conductance can be calculated in the limit of non-interacting Weyl fermions, neglecting both the inelastic collision integral and the self-consistent electric field.

Using Eq. (2) for non-interacting Weyl fermions of positive chirality, the density of electric current is given by:

 j(r)=∫d3p(2πℏ)3[∂ϵp∂p+eBcΩp⋅∂ϵp∂p]f(p,r). (4)

Assuming that intra-nodal scattering is short (), we perform a diffusive approximation, , where is a momentum relaxation (transport) time, is an effective mass of an electron. The current density within diffusion approximation is

 j(z)=(D∂zρ−eB4π2νρ), (5)

where the density of electrons

 ρ(z)=ν∫∞−∞dϵf(ϵ,z). (6)

Using the fact that the current density is subject to the continuity equation, which in the static limit reads , we find that the electron density obeys a drift-diffusion equation (13); (14); (15) characteristic of a one dimensional random walk with non-equal probabilities:

 ξ∂2zρ(z)∓Nϕ∂zρ(z)=0. (7)

Here signs correspond to Weyl nodes of opposite chiralities, is the localization length (, with three dimensional density of states and diffusion constant), and

 Nϕ=AeBΦ0=A2πeBℏc. (8)

is the number of magnetic flux quanta piercing the wire, which also marks the imbalance between left and right moving modes.

The validity of Eqs. (5) and (7) was recently established with Keldysh (21) and super-symmetry (22); (23) non-linear sigma model formalism. Eq. (7) shows that while the disorder scattering between different LL disrupts the ballistic propagation of the zero Landau level, the imbalance between the number of left and right moving modes (for a given Weyl node) remains. Eq. (7) should be supplemented with the boundary conditions for the value of distribution function at the leads. For the sample subject to voltage difference, the boundary conditions are ; here is Fermi-Dirac distribution function.

Solving Eq. (7) for the electron density and using Eq. (5) we obtain the conductance

 G(B)=NWe2Nϕ4πℏcoth(L2a), (9)

in agreement with Keldysh (21) and super-symmetry sigma model calculations (22); (23). Here the drift length , and we have multiplied the result by the total number of Weyl nodes . The semiclassical description (7) shows that the magnetic field drives a crossover from diffusive propagation to a ballistic regime, where only the chiral channels contribute to transport, leading in particular, to a positive magneto-conductance (19); (24); (25); (26).

We note that while the semiclassical approximation restricts the strength of magnetic fields , the value of may be large provided that , where is an elastic mean free path and is Fermi velocity. Moreover, in order to ignore localization effects, the sample should be shorter than localization length . Both conditions can be fulfilled simultaneously, provided .

## Ii Fluctuations in disordered Weyl semimetals

We now turn to a computation of current noise. When calculating the low frequency noise, the self-consistent electric field can be neglected. This is because it leads to the replacement of the inverse Thouless time with the inverse Maxwell time as a characteristic frequency below which the zero frequency limit is achieved. The inelastic collisions on the other hand play an important role for non-equilibrium current fluctuations, and the corresponding collision integrals needs to be restored in the kinetic equation (1). Within the diffusive approximation, the latter reads:

 ξ∂2zf(ϵ,z)∓Nϕ∂zf(ϵ,z)−Ie−e[f]−Ie−ph[f]=0, (10)

where and are electron-electron and electron-phonon collision integrals.

For topologically trivial metals, current fluctuations can be computed following a Boltzmann-Langevin approach(27); (28). This approach relates the fluctuation of observable quantities, such as current, to the fluctuation of the occupation in the phase space , with the same applicability as the kinetic equation. We employ this approach, taking into account effects of topology present in Weyl semimetals.

Similar to the current density (4), the density of current fluctuations in a Weyl semi-metal can be related to the fluctuation of phase space density as

 δj(t)=∑p[∂ϵp∂pδf(p,r,t)+eB4π2νδf(p,r,t)]. (11)

The first term in Eq. (11) is affected by the antisymmetric part of the momentum-dependent distribution function, while the second term is proportional to its symmetric part, i.e. the total density of electrons at a given point. This second term is absent in topologically trivial metals and corresponds to the drift current associated with electrons occupying the zero Landau level.

We note that due to particle number conservation, the correlation function of current fluctuation is independent of the lateral position . This allows to compute the correlation of integrated currents:

 δI(t)=1L∫d3rδj(r,t). (12)

The correlation function of current fluctuation is calculated by representing the kinetic theory of fluctuations through a Keldysh field theory(29). This approach is identical to the original Boltzmann-Langevin equation for pair-correlation function but allows to compute current cumulants of any order. While our main focus in this manuscript is on the pair-correlation function only, the calculation presented hereafter enables one to obtain the full kinetic theory of fluctuations in Weyl semimetals.

The generating function of counting statistics(30) can be written as (31); (32); (33)

 κ[λ]=∫DfDδ¯feiS[f,δ¯f,λ]+i∫dtdzλ(t)δI(z,t). (13)

Here the density of particles in the phase space is treated as a field , and the field is its conjugate. The field consists of a mean value, which solves the Boltzmann equation (10), and a fluctuation part . The auxiliary counting field, , is chosen in accordance with the correlation function in question. The effective action consists of two parts

 S[f,δ¯f,λ]=SDyn+SNoise[λ]. (14)

The dynamical part of the action

 iSDyn=2πiνA∫dtdϵdz[δ¯f(ϵ,z,t){∂∂t−D∂2∂z2+eB4π2ν∂∂z+^Ie−ph+^Ie−e}f(ϵ,z,t)], (15)

describes the evolution of the particle occupation in the phase space. The cascade creation (34) of the noise is encoded in

 iSNoise[λ] = −νA∫dtdϵdz(λ+∂δ¯f(ϵ,z,t)∂z)2 (16) × f(ϵ,z,t)(1−f(ϵ,z,t)).

There are two modifications of the action in Eq. (14) as compared with the one for normal metals (31); (32); (33). First, the third term (in curly brackets) in Eq. (15) is absent in normal metals. This term dictates that the dynamics of noise propagation on scales longer than the elastic collision length is of drift-diffusion type, as opposed to diffusion in normal metals. The second is more subtle and is related to the fluctuating current that couples to the source term in Eq. (13). In normal metals is proportional to the velocity (see the first term in Eq. (11)), which is related to spatial gradients of the density within the diffusive approximation. For this reason, in topologically trivial metals it contributes to the mean value of the current only which is determined by gradients of the density, see Eq. (5), and plays no role in its fluctuations, as the source term in Eq. (13) couples to the integrated current. In Weyl semimetals, on the other hand, this term has a drift contribution, encoded in the second term in Eq. (11), which has no gradients. For that reason it contributes to the fluctuations of currents.

Using the action (14) one expresses the current fluctuations

 S2=∫dt⟨δI(t)δI(0)⟩ (17)

through the correlation functions of phase space density operators

 S2=e2L2∫d1d2{δ(1−2)f(1)(1−f(1))−ξ22[a−2DK(1,2)+4a−1DR(1,2)∂z2f(2)(1−f(2))]}. (18)

Where we have used , and similarly for , for brevity; are Keldysh and retarded components of correlation functions. Computing the correlation functions, in the presence of inelastic scattering (see Appendix A for the details) one finds the low-frequency noise spectrum

 S2=e2ξa2∫L0dzT(z)exp(2(L−z)/a)(eL/a−1)2. (19)

The information about non-equilibrium state of the system is encoded in effective temperature

 T(z)=∫∞−∞dϵf(ϵ,z)(1−f(ϵ,z)), (20)

which accounts for the spread of the distribution function determined by Eq.(10) in the presence of all scattering processes.

Eq.(19) is the central result of this work. It extends the standard expression for the noise spectrum in normal metals to Weyl semimetal. While in normal metals the noise is proportional to an integral over effective temperature alone, for a Weyl semimetal it acquires an additional kernel, which depends exponentially on the lateral distance from the leads. The physical reason for that has to do with the chiral anomaly. In normal matter, fluctuations of phase space distribution that occur anywhere in the sample diffusively propagate to the contacts, giving rise to a current noise. For Weyl semimetal, on the other hand, on scales much greater than , the fluctuations in the phase-space occupation move predominantly in a ballistic fashion. Hence, in longe samples, , where the transport is dominated by ballistic (deterministic) propagation, stochastic processes are exponentially suppressed, and the system is noiseless.

We now analyze the noise for a number of limiting cases and start from the limit of large inelastic length.

### ii.1 Elastic scattering

At equilibrium, the distribution function is of Fermi-Dirac form and Eq. (19) yields

 S2=2TG(B), (21)

with from (9), in accordance with the fluctuation-dissipation theorem. In fact, this result holds even in the presence of inelastic scattering, as can be easily checked by imposing thermal equilibrium, which implies a uniform temperature in Eq. (19).

Next, we study the shot noise in the limit , and finite bias . Substituting Eqs. (37) for non-interacting electrons into Eq. (18) one arrives at Eq. (19) that yields

 S2=e3NWNϕV16πsinhx−xsinh4(x/2), (22)

with in agreement with (21); (23). It corresponds to Fano factor ()

 Fel(x)=sinhx−x2sinhxsinh2(x/2). (23)

At short distances , the Fano factor approaches the value of diffusive metals, , while it is exponentially suppressed for , see Fig.1. This suppression indicates that at large sample sizes (or large magnetic fields), , all electronic motion is predominantly ballistic.

### ii.2 Electron-phonon relaxation (L≫le−ph)

We now consider the case when the sample is much longer that electron-phonon inelastic length, yet momentum relaxation is still governed by static disorder, namely, . For electron-phonon inelastic scattering the only conserved quantity is the total density of electron. This implies

 ∫dϵIe−ph[f]=0. (24)

For systems longer that electron-phonon inelastic length the distribution function takes the following form

 f(ϵ,z)=fF(ϵ−μ(z)T), (25)

where the temperature is equal to the temperature of phonon bath, and the chemical potential satisfies

 (−∂2z+a−1∂z)μ(z)=0. (26)

For distribution (25) Eq.(19) yields

 S2=2TG(B), (27)

corresponding to Nyquist noise with phonon-temperature. As in normal metals, if current fluctuations are only carried by fluctuations of chemical potential, they are not sensitive to the external bias.

### ii.3 Electron-electron relaxation (L,a≫le−e)

For a system longer than the electron-electron collision length and with no electron-phonon scattering there are two propagating modes: density and temperature fluctuations. The latter arise due to the energy conservation for electron-electron scattering

 ∫dϵϵIe−e[f]=0. (28)

This implies that the mean value of the distribution function has a form

 f(ϵ,z)=fF(ϵ−μ(z)T(z)). (29)

Plugin this into Eq.(10) one finds

 (−∂2z+a−1∂z)μ(z)=0, (30) (−∂2z+a−1∂z)(12μ2(z)+π26T2(z))=0.

Solving these equations one finds the chemical potential

 μ(z)=μ+eV21+eL/a−2ez/a1−eL/a, (31)

and the effective temperature

 T2(z)=3e2V2π2(eL/a−ez/a)(ez/a−1)(eL/a−1)2, (32)

here we assumed at the leads. Substituting these expressions into the distribution function, we obtain

 S2=√38πe3NWξVLxsinh(x/2)=√38πe3NWNϕVsinh(x/2), (33)

and the corresponding Fano factor

 Fee(x)=√34coshx/2. (34)

Note, that unlike normal metal, where different inelastic regimes result in different numerical values of the Fano factors, in Weyl semimetals different inelastic processes have different magnetic field dependence. For , as expected for normal metals (35); (36). In the limit the shot noise is exponentially suppressed. By changing the magnetic field one should be able to experimentally explore the crossover from diffusive to “ballistic” regimes. Moreover, the different parametric dependence in Eqs. (23) and (34) allows to estimate from measurement of shot noise as a function of magnetic field.

## Iii Conclusions

In this paper we studied the current noise in Weyl semimetals in the presence of inelastic scattering. We constructed a Boltzmann-Langevin approach, taking into account chiral anomaly effects. Similar to a case of elastic propagation (21); (23), we find that chiral anomaly effects dominate the evolution of the occupation in phase space for samples longer than the drift length , experimentally controllable by tuning the magnetic field. This gives rise to deterministic (ballistic) propagation of fluctuations at long samples , in a direction determined by the chirality. As a result, only stochastic processes that occur within a distance from the leads, contribute to the current noise.

We show that inelastic collisions resulting from electron-electron interactions, modify this parametric dependence on the magnetic field. By measuring shot noise as a function of a magnetic field, for different applied voltage, which interpolates between the elastic and inelastic limit, one can estimate the electron-electron inelastic length .

Finally, we show that as in normal metals, the presence of electron-phonon scattering suppresses the shot noise, and the current fluctuations correspond to Nyquist noise, governed by the phonon bath temperature.

## Iv Acknowledgements

This work was supported by ISF (grant 584/14), Israeli Ministry of Science, Technology and Space, and by RFBR grant No 15-52-06009. D. M. acknowledges support from the Israel Science Foundation (Grant No. 737/14) and from the European Union’s Seventh Framework Programme (FP7/2007- 2013) under Grant No. 631064. The discussions with D. Bagrets, P. Ostrovsky, A. Stern are gratefully acknowledged.

## Appendix A - Derivation of Eq.(19)

The action (14) allows to compute the correlation functions with In the static limit the retarded propagator

 [ξ(−∂2z+a−1∂z)+Ie−e+Ie−ph]DR(ϵ,z;ϵ′,z′)=δ(ϵ−ϵ′)δ(z−z′). (35)

The advanced propagator is given by Hermitian conjugation of , understood as a matrix, with respect to energy and spatial coordinates. The Keldysh part of the propagator

 DK(ϵ,z;ϵ′,z′)=−2ξ∫L0dz1∫∞−∞dϵ1DR(ϵ,z;ϵ1,z1)∂z1f(ϵ1,z1)(1−f(ϵ1,z1))∂z1DA(ϵ1,z1;ϵ′,z′). (36)

The key observation is that diffusion propagator that enters into Eq.(18) contains diffusion propagators integrated over energy and . It follows from particle conservation preserved by both electron-electron and electron-phonon scattering, that . This implies that propagators integrated over energy are equal to those in elastic case. Hence the structure of Eq.(19) is preserved in the case of inelastic scattering. We now prove it with more details in some important limiting cases.

### a.1 lee,le−ph≫L

For the system shorter that shortest inelastic length the electron-phonon and electron-electron collisions can be neglected. In this case the evolution is purely elastic propagation, and the energy is conserved.

 Dα(ϵ,z;ϵ′,z′)=δ(ϵ−ϵ′)Dαϵ(z,z′). (37)

Here and we define

 DR(z,z′)=aξ11−eL/a⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩[]ll(ez/a−1)(e(L−z′)/a−1),zz′, (38)

The advanced propagator is obtained from by replacing .

 DKϵ(z,z′)=−2ξ∫L0dz1DR(z,z1)∂z1f(ϵ,z1)(1−f(ϵ,z1)∂z1DA(z1,z′)

Using Eq.(38) and integrating over energy and coordinate we come to Eq. (19).

### a.2 le−ph≪L

The diffusion propagator is

 DR(ϵ,z;ϵ′,z′) =∂μf(ϵ−μ(z)T)DR(z,z′), (39) DA(ϵ,z;ϵ′,z′) =DA(z,z′)∂μf(ϵ′−μ(z′)T),

and can be expressed through via (36). Note that derivatives of distribution function over energies always appear on the outer part of diffusion propagator. The corresponding integral over energies can be easily performed, reproducing Eq. (19).

### a.3 lee≪L and le−ph≫L

In case of strong inelastic electron-electron scattering the propagators are

 DR(ϵ,z;ϵ′,z′)) =∂μfF(ϵ−μ(z)T(z))[1+3π2ϵ−μ(z)T(z)ϵ′−μ(z′)T(z′)]DR(z,z′), (40) DA(ϵ,z;ϵ′,z′) =[1+3π2ϵ−μ(z)T(z)ϵ′−μ(z′)T(z′)]DR(z,z′)∂μfF(ϵ′−μ(z′)T(z′)),

The Keldysh part of the propagator is determined via (36). Using the idenities

 ∫∞−∞ϵ∂fF(ϵ)∂ϵdϵ=0, (41) ∫∞−∞ϵ∂fF(ϵ)∂ϵfF(ϵ)(1−fF(ϵ))dϵ=0,

one reduces Eq.(18) to Eq.(19).

1. preprint:

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