Charge spin and heat shot noises in the absence of average currents Conditions on bounds at zero and nite frequencies Ludovico Tesser Matteo Acciai Christian Sp ansl att Juliette Monsel and Janine Splettstoesser

2025-04-27 0 0 762.34KB 17 页 10玖币

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Charge, spin, and heat shot noises in the absence of average currents:

Conditions on bounds at zero and ﬁnite frequencies

Ludovico Tesser, Matteo Acciai, Christian Sp˚ansl¨att, Juliette Monsel, and Janine Splettstoesser

Department of Microtechnology and Nanoscience (MC2),

Chalmers University of Technology, S-412 96 G¨oteborg, Sweden

(Dated: April 7, 2023)

Nonequilibrium situations where selected currents are suppressed are of interest in ﬁelds like

thermoelectrics and spintronics, raising the question of how the related noises behave. We study

such zero-current charge, spin, and heat noises in a two-terminal mesoscopic conductor. In the

presence of voltage, spin and temperature biases, the nonequilibrium (shot) noises of charge, spin,

and heat can be arbitrarily large, even if their average currents vanish. However, as soon as a

temperature bias is present, additional equilibrium (thermal-like) noise necessarily occurs. We show

that this equilibrium noise sets an upper bound on the zero-current charge and spin shot noises, even

if additional voltage or spin biases are present. We demonstrate that these bounds can be overcome

for heat transport by breaking the spin and electron-hole symmetries, respectively. By contrast, we

show that the bound on the charge noise for strictly two-terminal conductors even extends into the

ﬁnite-frequency regime.

I. INTRODUCTION

Fluctuations, or noise, in physical observables dis-

close important properties of small electronic conductors.

While equilibrium charge noise relates a conductor’s tem-

perature to its dc conductance according to the Nyquist-

Johnson relation [1,2], nonequilibrium noise oﬀers addi-

tional opportunities. Most prominently, shot noise—or

partition noise—which arises from the granularity of the

electric charge, has in the last decades emerged as a ubiq-

uitous tool for characterizing nanoscale systems [3–5]. It

has, e.g., been used to reveal the charge of fractional-

ized quasiparticles [6,7], Cooper pairs [8,9] as well as

Bogoliubov quasiparticles [10] in superconductors. Be-

sides, analyzing and understanding nonequilibrium noise

in nanoscale thermoelectric devices is crucial as it limits

their performances [11–19].

More recently, charge noise as a response to a temper-

ature gradient and in the absence of a voltage bias—

dubbed delta-Tnoise—has been investigated in sev-

eral theoretical [20–30] and experimental [31–37] studies.

These studies demonstrate that delta-Tnoise oﬀers ad-

ditional insights beyond the traditional shot noise, e.g.,

for quantifying local temperature gradients [25,32] or

as a potential tool for extracting scaling dimensions and

exchange statistics of anyons [23,28,29]. Still, the full

scope of delta-Tnoise remains to be understood.

A key feature of delta-Tnoise is that, for energy in-

dependent transport through the system, the absence of

voltage bias causes the average charge current to vanish.

Yet, the conductor partitions the opposing, equal cur-

rent contributions emanating from the reservoirs, which

results in detectable shot noise. It was, however, pointed

out in Ref. [27] that noise in the absence of a current is

a broader concept than delta-Tnoise. More speciﬁcally,

one can imagine experimental setups with generic trans-

missions, where temperature and voltage biases are care-

fully combined such that the average current vanishes.

This is particularly relevant for characterizing thermo-

electric properties of nanodevices: A zero charge current

situation indeed occurs at the stopping voltage, also re-

µL, TLµR, TR

I= 0,Σ=0

µL

µR

D(E)

fL(E)

fR(E)

(a) E

µL, σLµR, σR

I= 0

σL

µLµR

D(E)

(b)

µL, σLµR, σR

Σ=0

∆µ

µL

µR

D(E)

µL, TLµR, TR

JL= 0

µLµR

∆µJ

D(E)

(d)

Figure 1. Zero-current conditions obtained for charge (I), spin

(Σ), and heat (JL) currents for the case of uniform trans-

mission D(E)=1/2 (depicted in purple). (a) Using only

a temperature bias, both average charge and spin currents

vanish. (b) The zero-charge-current condition is achieved by

using only a spin imbalance σLon the left contact. (c) The

zero-spin-current condition is achieved by using only a volt-

age bias ∆µbetween the contacts. (d) The zero-heat-current

condition, JL= 0, is achieved using both a voltage and a

temperature bias. The voltage bias can be replaced by a spin

imbalance.

arXiv:2210.06051v3 [cond-mat.mes-hall] 5 Apr 2023

ferred to as the thermovoltage. Under open circuit condi-

tions, the zero-current condition is exactly that at which

a device’s thermopower—the ratio between the thermo-

voltage and the temperature bias—is extracted. More-

over, zero-current nonequilibrium noise is not limited to

charge transport, but can also be considered for currents

of, e.g., heat [38] or spins [39], see Fig. 1.

In this paper, we focus on zero-current ﬂuctuations and

compare how nonequilibrium ﬂuctuations behave com-

pared with their equilibrium-like counterpart. We extend

the analysis of Ref. [27], for a coherent, mesoscopic con-

ductor, characterized by a transmission function D(E),

which is connected to two macroscopic reservoirs (equiv-

alently terminals, or contacts). Within a scattering ap-

proach [4], we compute the noise of charge, heat, and

spin currents by combining external biases and transmis-

sion functions such that one or more average currents

vanish. More precisely, we always consider a vanishing

average current in a given contact for the same trans-

ported quantity as for the studied noise, e.g., heat shot

noise with zero average heat current in one contact.

First, we broaden the scope of delta-Tnoise by show-

ing that, even in the simple case of constant transmission

D(E) = D, charge, spin and heat shot noises can arise

under zero-current conditions and have similar functional

forms. These zero-current shot noises can become arbi-

trarily large with increasing magnitude of the bias caus-

ing the nonequilibrium situation. Furthermore, we dis-

cuss how the nonequilibrium noise is related to the ﬂow

of excitations between the reservoirs, under the condi-

tion that the applied biases are large with respect to the

energy scale of the base temperature.

Next, when a temperature is comparable to the energy

scales set by the biases, it makes sense to ask how large

the nonequilibrium ﬂuctuations can be with respect to

the thermal component of the noise. We tackle this prob-

lem for an arbitrary transmission function D(E). For a

spin-degenerate system, it was shown in Ref. [27] that

the zero-current charge shot noise is always bounded by

its thermal counterpart, independently of the conductor’s

transmission function. Here, we extend this result in sev-

eral ways:

(i) We obtain an even stricter upper bound, given in

Eq. (22), than the one previously derived in Ref. [27], see

Eq. (18), which is valid at arbitrary reservoir tempera-

tures TL, TRand transmission function D(E).

(ii) We show that the new bound (22) also applies to

the zero-current spin noise in the presence of spin and

temperature biases.

(iii) We demonstrate that the zero-current charge noise

remains bounded at ﬁnite frequency [see Eq. (28)], pro-

vided the noise is measured in the colder reservoir. By

contrast, if the noise is measured in the hotter reservoir,

the bound (28) does not hold.

Our ﬁndings in this paper highlight several important

features of zero-current nonequilibrium noise underlining

that it could be used as a future noise spectroscopic tool,

in particular, to probe nanoscale gradients of tempera-

ture [25,32] or spin polarization.

The remainder of this paper is structured as follows.

In Sec. II, we introduce the here employed scattering-

based formalism. In Sec. III we extend the concept of

delta-Tnoise to diﬀerent kinds of bias and apply it to

charge, spin, and heat currents. In Sec. IV, we demon-

strate how to achieve unbounded zero-current nonequilib-

rium heat noise, and present an improved bound for the

charge noise, which also holds for the spin noise. The

bound on charge shot noise is further extended to the

ﬁnite-frequency noise in Sec. V. In Sec. VI, we address

the experimental prospects to verify our bounds for the

zero-current charge noise.

II. SCATTERING APPROACH TO NOISE

We study steady-state transport in a coherent quan-

tum conductor connected to two macroscopic reser-

voirs, labeled by α= L,R . The conductor is char-

acterized by a spin-independent transmission function

D(E) = |sLR(E)|2= 1−|sLL(E)|2, obtained from a spin-

preserving scattering matrix

s(E) = sLL(E)sLR(E)

sRL(E)sRR(E).(1)

The electronic occupations in the reservoirs are governed

by Fermi distribution functions

fατ (E) = 1

1 + eβα(E−µατ ),(2)

where βα= (kBTα)−1are the inverse temperature scales,

and kBis the Boltzmann constant. When the spin de-

generacy for τ=↑,↓in the reservoirs is broken, we write

the spin-dependent electrochemical potentials as

µατ =µα−(−1)δτ↓σα

2,(3)

where δττ 0is the Kronecker delta and the spin-splitting in

reservoir αis given by σα. We are interested in nonequi-

librium situations, where the distributions of the two

reservoirs diﬀer due to any of the three biases ∆µ=

µL−µR, ∆T=TL−TR, or ∆σ=σL−σR.

The average charge (I), heat (J) and spin (Σ) currents,

which can possibly ﬂow out of the left contact in response

to these three biases and their combinations, are given

by [4]

XL=1

τZ∞

−∞

dE xD(E)[fLτ(E)−fRτ(E)] ,(4)

with x→ {−e, E −µLτ,(−1)δτ↓~/2}for X→ {I, J, Σ}

and analogously for XR. Here, e > 0 is the elementary

charge (the electron charge is thus −e), and h≡2π~

is the Planck constant. All energies are measured with

respect to a reference electrochemical potential (e.g.,

µ0= (µR+µL)/2) and, unless otherwise speciﬁed, all

energy integrals in the remainder of the paper are to be

understood as RdE =R∞

−∞ dE.

We deﬁne the noise at frequency ωassociated with the

current Xas [4]

αβ(ω) = Z∞

−∞h{δˆ

Xα(t), δ ˆ

Xβ(0)}ieiωtdt, (5)

where δˆ

Xα=ˆ

Xα−Xαis the ﬂuctuation of the operator

Xαaround its thermal average value Xα≡ h ˆ

Xαi, and

{ˆ

Xα,ˆ

Yβ}=ˆ

Xαˆ

Yβ+ˆ

Yβˆ

Xαis the anticommutator. In

the following, we study the left autocorrelator SX(ω)≡

LL(ω), which corresponds to measuring the current ﬂuc-

tuations in the left contact. In the ﬁrst part of the paper,

we will focus on the zero-frequency regime, ω= 0, and

analyze the noise for various types of biases and asso-

ciated currents. Here, the noise of a conserved current,

in our case both charge and spin current X→ {I, Σ},

satisﬁes SX

LL(0) = SX

RR(0) = −SX

LR(0) = −SX

RL(0). By

contrast, these conservation laws do not hold for heat

noise, nor for noise at ﬁnite frequency [4]; in these spe-

ciﬁc cases, the noise depends on the contact in which

it is measured. The analysis of ﬁnite-frequency noise is

reported in Sec. Vand focuses on the charge noise.

At ω= 0, the noise can be written in a compact form

and is straightforwardly separated into two contributions,

SX(0) = SX

th(0) + SX

sh(0), with [4,40]

th(0) = X

ατ ZdE 2x2

hD(E)fατ (E)[1 −fατ (E)] ,(6a)

sh(0) = X

τZdE 2x2

hD(E)[1 −D(E)][fLτ(E)−fRτ(E)]2.

(6b)

Here, SX

th(0) is thermal-like noise to which each reservoir

contributes individually, even at equilibrium, fLτ=fRτ.

By contrast, SX

sh(0) is the so-called shot noise which is

nonzero only under nonequilibrium conditions, i.e., when

fLτ6=fRτ. It contains the characteristic partitioning

factor D(E)[1 −D(E)] and thus vanishes in the limits of

perfect, D(E) = 1, or completely suppressed transmis-

sion, D(E) = 0. Note that the factors of 2 in Eq. (6)

come from the anticommutator in Eq. (5).

Of main interest for our work is the shot noise,

Eq. (6b), under the zero-current condition

XL= 0.(7)

Depending on the transmission function D(E) and on

the type of current that should vanish, a combination

of biases ∆µ, ∆Tand ∆σis required. Note that for a

conserved current (charge and spin) XL= 0 =⇒XR=

0. This conservation does not hold for the heat current,

which can be made to vanish only in one contact at a

time. Here, we impose XL= 0, consistent with our choice

to study the noise correlator in the left contact SX(ω)≡

LL(ω).

III. CHARGE, SPIN, AND HEAT NOISES AT

ZERO AVERAGE CURRENT

We begin by presenting charge, spin, and heat ﬂuc-

tuations in the absence of the related average currents

under nonequilibrium conditions determined by diﬀerent

kinds of biases, see Fig. 1. We focus here on the situation

where the possible applied biases—the potential bias ∆µ,

temperature bias ∆T, or spin bias ∆σ—are large, result-

ing in large shot noise. Concretely, this means for any of

the applied biases, ∆µ, kB∆T, ∆σkBTR, such that we

can eﬀectively set TR→0 and ∆T=TLin the present

section. In Appendix A, we present results for the zero-

current heat noise in the opposite regime of weak biases,

thus complementing the literature for the zero-current

charge shot noise in this regime [32].

To start with, we consider a uniform transmission func-

tion, D(E) = D. This simple choice results in electron-

hole as well as spin symmetry in the scattering process.

Consequently, shot noise at vanishing average charge or

spin currents can be obtained in the presence of a single

type of bias. Concretely, zero current is here obtained

when the applied bias does not break the symmetry re-

lated to the transported observable: temperature and

spin biases do not break electron-hole symmetry, result-

ing in zero charge current; temperature and voltage bi-

ases do not break the spin symmetry, resulting in zero

spin current. We show that in these cases, current can-

cellation results from incoming ﬂuxes of opposite sign,

which, however, sum up to a nonvanishing contribution

to the nonequilibrium (shot) noise. In contrast, to reach

the zero-heat-current condition at constant transmission,

it is necessary to have at least two biases, one of which

being the temperature bias. The reason for this is that

any of the three biases breaks the symmetry with respect

to the excess energy transported into the left contact and

needs hence to be (nontrivially) compensated by a sec-

ond bias. Only when the transmission has an appropriate

energy-dependence can a zero heat current be reached

by the application of a single bias. All diﬀerent possible

settings, which we present in this section, are listed in

Tab. I.

A. Delta-Tcharge and spin current noises

First, we consider a spin-degenerate setup in which

only a temperature bias between the contacts generates

the nonequilibrium noise, as illustrated in Fig. 1(a). This

is the situation in which the delta-Tnoise was studied in

Refs. [32,33,35]. In this case, transport can be un-

derstood in terms of negatively (electrons) or positively

(holes) charged excitations ﬂowing from the hot contact

to the cold one. Therefore, both the average charge and

I= 0

Sec. III A

I= 0

Sec. III B

Σ = 0

Sec. III C

Σ = 0

Sec. III A

JL= 0

Sec. III D

JL= 0

Sec. III D

∆µ∆T∆σ

SΣ

Table I. Realizations of generalized zero-current shot noise,

for energy-independent transmissions D(E) = D(purple).

spin currents vanish

I=−eX

τ=↑,↓

(jeτ−jhτ)=0,(8a)

Σ = ~

i=e,h

(ji↑−ji↓)=0.(8b)

Here, we have deﬁned the inﬂuxes of spin-resolved exci-

tations from the left contact as

jeτ=D

hZ∞

µR

dEfLτ(E),(9a)

jhτ=D

hZµR

−∞

dE[1 −fL¯τ(E)],(9b)

where the notation ¯τrefers to the opposite spin of τ. In

Eq. (8a), we classify the excitations according to their

charge, while we classify them according to their spin in

Eq. (8b).

Unlike the currents, the noise is ﬁnite, and importantly

it contains a nonvanishing shot noise component

SI(0) = SI

sh(0) + SI

th(0) (10)

=kBTL

4e2

hD(1 −D)(2 ln 2 −1) + kBTL

4e2

hD.

When the conductor is opaque, namely D1, the

charge-current ﬂuctuations are given by

SI(0) ≈8e2

hDkBTLln 2 = 2e2j. (11)

Here, we recognize that the charge current noise is pro-

portional to the inﬂux of excitations jﬂowing to the low-

temperature contact [41], namely

j=X

τ=↑,↓

(jeτ+jhτ) = X

i=e,h

(ji↑+ji↓).(12)

Importantly, this total ﬂow of excitations includes all par-

ticles (electrons and holes), irrespective of their charge

and spin. The opacity of the conductor makes transport

happen via uncorrelated single-particle tunneling events,

which means that, in this limit, the current ﬂuctuations

simply count how many excitations per unit time, elec-

trons or holes, travel from the hot to the cold contact [42–

45]. The factor kBTLln 2, which was associated to the

degeneracy of the transported excitations in Ref. [35],

stems from the inﬂuxes jiτ , all of which take the same

value for a temperature bias because the electron-hole or

spin symmetry is not broken. This factor takes the role of

an eﬀective noise temperature when recognizing Eq. (11)

as a generalized ﬂuctuation-dissipation theorem [20,35].

The spin-current ﬂuctuations are essentially identical

to the charge current ﬂuctuations in Eq. (11), the dif-

ference being the quantity transported. This leads to a

diﬀerent prefactor, namely

SΣ(0) ≈8

h~

22

DkBTLln 2 = 2 ~

22

j. (13)

We highlight that the ﬂow of excitations jaccounts for

both spin-↑and spin-↓excitations ﬂowing from left to

right, irrespective of their spin.

B. Charge current noise due to spin bias

Instead of generating the nonequilibrium noise with a

temperature bias, here we set TL=TR→0 and choose a

spin bias. To this end, we consider a spin-nondegenerate

setup in which the two spin populations in each reservoir

αhave a ﬁnite energy separation σαbetween spin-↑and

spin-↓electrons. Such setups can be realized by inject-

ing spin currents in a normal metal using ferromagnetic

contacting [46]. The occupation probability of electrons

injected into the conductor is then spin-dependent and is

given by Eqs. (2) and (3). The energy separation σαacts

as a spin-dependent chemical potential and allows the

driving of spin currents through the conductor. The zero-

charge-current condition can for instance be achieved as

illustrated in Fig. 1(b), with ∆σﬁnite and σR= 0. Then,

the ﬂow of spin-↑excitations above µL(electrons) is per-

fectly balanced by the ﬂow of spin-↓excitations below

µL(holes). Hence, the average charge current is zero

I=−ePτ=↑,↓(jeτ−jhτ) = 0, whereas the spin current

is ﬁnite. By using Eq. (6b), we then ﬁnd that the charge

noise in the absence of charge current becomes

SI(0) ≡SI

sh(0) = 2 e2

hD(1 −D)|∆σ|

= 2e2(1 −D)j, (14)

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Charge,spin,andheatshotnoisesintheabsenceofaveragecurrents:ConditionsonboundsatzeroandnitefrequenciesLudovicoTesser,MatteoAcciai,ChristianSpanslatt,JulietteMonsel,andJanineSplettstoesserDepartmentofMicrotechnologyandNanoscience(MC2),ChalmersUniversityofTechnology,S-41296Goteborg,Sweden(Dated:Apr...

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Charge spin and heat shot noises in the absence of average currents Conditions on bounds at zero and nite frequencies Ludovico Tesser Matteo Acciai Christian Sp ansl att Juliette Monsel and Janine Splettstoesser

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