Noise properties and ac conductance of mesoscopic diffusive conductors with screening

14 Pages·0.72 MB·English

by  Y. Naveh

Noise properties and ac conductance of mesoscopic diffusive conductors with screening
Y. Naveh, D. V. Averin, and K. K. Likharev
Department of Physics, State University of New York, Stony Brook, NY 11794-3800
(February 1, 2008)
8 A theory of non-equilibrium (“shot”) noise and high frequency conductance in diffusive meso-
9 scopic conductors with screening is presented. Detailed results are obtained for two simple geome-
9 tries, for both large and short electron-electron scattering length l , at frequencies of the order of
ee
1
the inverse Thouless time 1/τ . The conductance and the noise are found to exhibit significant
T
n frequency dependence. For L ≪ lee, the high-frequency (ωτT ≫ 1) shot noise spectral density
a S (ω) approaches a finite value between 2eI/3 and 2eI, depending on the screening properties of
I
J thesystem,withtemperaturecorrectionstoS (ω)beinglinearinT. However,whenL≫l ,S (ω)
9 grows asω1/4 (atT =0),isnotupper-boundIby2eI,andhasatemperature-dependentcoemeponIent
1 quadratic in T. As a result, measurements of S (ω,T) can be utilized as a probe of the strength of
I
electron-electron scattering.
]
l
al I. INTRODUCTION are locally thermalized.8,9
h
One of the most important characteristics of classical
-
s Significant attention has recently been focused on the shot noise is that it is white up to very high frequencies.
e In ballistic structures the noise is reduced only at fre-
dynamic electronic properties of mesoscopic systems.
m
These properties include the ac conductance which gives quencies of the order of the inverse time of flight of the
. electronacrossthedevice.10Indiffusiveconductors,there
t the meancurrentresponsetoappliedacvoltage,andthe
a noise, i.e., the deviations of the current from its average areatleastthreeaprioricandidatestotheanalogofthis
m
time constant: the elastic scattering time τ, the (much
value. Due to the fluctuation-dissipation theorem, equi-
d- librium Johnson-Nyquist noise, as measured in the ex- ltahregsear)m“pTleh,oaunldestsh”etMimaexwτTellorfeelalexcattrioonndtiimffues(io4nπσt)h−ro1.ugInh
n ternal leads, does not convey any additional information
o to that obtained from the ac conductance. This is not addition, quantum effects also may be manifested by a
c thecasefornon-equilibrium(’shot’)noise. Here,thecur- frequency-dependentnoisespectrum,increasingthenoise
[ rentfluctuationsaredependentuponthenon-equilibrium atω > eV/¯h,whereV istheappliedvoltage.11 Recently
we have shown12 that even at high voltage, eV ¯h/τ ,
1 distribution function, as wellas on electron-electroncor- ≫ T
v relations. Moreover, the shot noise may be interpreted where quantum effects are negligible, the shot noise in
8 as an indication that the transport mechanism through diffusive structures may exhibit considerable frequency
8 dependenceatfrequenciesaslowas1/τ .13 Inthatwork,
the structure involves discrete transfer of charge, as op- T
1 however, only the zero-temperature case was considered
posed to the continuous charge transfer that takes place
1 in macroscopic conductors.1 and the ac conductance and effects of electron-electron
0
interaction were not explored.
8 Earlier, shot noise in diffusive conductors was calcu-
9 lated in the zero-frequency limit,2–5 with the conclusion The issue of high-frequency noise cannot be separated
/ that the lowfrequency spectraldensity equals1/3 ofthe from that of the ac conductance at the same frequency.
t
a classical Schottky value 2eI, where I is the average (dc) Previous works studied the ac conductance in diffusive
m current in the system. This result was obtained in two structures with ring14–16 or linear17–19 geometries. In
- very different theoretical frameworks,namely in a quan- the first case, non-interacting electrons were considered,
d
tum mechanical transmission approach2 which is gener- andthefrequencydependenceoftheacconductancewas
n
allybasedonthequantumcoherenceofthedifferentelec- found to be similar to the Drude dependence,15 i.e., ap-
o
c tron states, and in a semi-classical approach,3 in which preciable only at ω 1/τ. However, usual conductors
∼
: quantum coherence is neglected, and the only effects on (with electrodes) cannot be considered separately from
v
the noise are due to the single-particle non-equilibrium their electrodynamic environment. For such conductors
i
X distribution function of the electrons. We believe that general expressions for the conductivity were obtained
r this surprising agreement between the two theories was to linear order in the frequency.17–19 Under the assump-
a adequately explained6 by showing that the main ingre- tionofabsolutelocalelectro-neutrality,itwasfoundthat
dient in the transmission approach, the probability dis- this linearcorrection(the “emittance”)vanishes and the
tribution of transmission coefficients, is not affected by conductance is againindependent of the frequency up to
quantuminterferenceindiffusiveconductors,thusestab- ω 1/τ.19 Here we are interested in the case where the
lishing the validity of the semi-classical method.7 The con∼ductor’s length L or its thickness t are comparable
1/3 suppression result is strictly valid only in a non- to the screening length λ, so local charge neutrality may
interacting electron picture. The electron-electron inter- no longer be retained. While we confirm the previous
action slightly increases the zero-frequency noise value, results19 for L,t λ, we show that observable devia-
≫
which reaches (√3/2)eI in the limit when the electrons tions from them may appear already at L/λ 10 or
∼
1
t/λ 10. (“built-in”) electric field E = ε /e. θ is the angle
0 c
∼ −∇
A vast amount of research has been dedicated to the between k and the direction of the current, and Φ(r,t)
effects of weak localization on the dc conductance of is the time-dependent electric potential, so that U(r,t)
mesoscopic diffusive conductors.20 Its effect on the ac is the total instantaneous potential energy of the elec-
conductance21 and the high temperature noise (T > trons. In the above variables,the velocity of the particle
eV)22 lead to corrections to these quantities which are is both position- and time-dependent, v = v(ε,r,t), and
of the order of the quantum unit of conductance e2/h theBoltzmannequationwithintheusualrelaxation-time
(times some characteristic energy in the case of noise). approximationlooks like
Thesecorrectionswillbeneglectedinthiswork,sincewe
will consider the case of conductance much higher than ∂f ∂f ∂U ∂f ∂cosθ
+ U
e2/h, so that deviations of the high-frequency conduc- ∂t ∂ε ∂t − ∂cosθ ∂k ·∇
tance and the noise from their zero-frequency values are f cosθ
+ f v+ a =0 (2.3)
much larger than the weak localization corrections. ∇ · τ
In the present work the noise properties and the ac
conductance of diffusive conductors much shorter than where τ =τ(r) is the local elastic relaxation time. Note
theelectron-phononmeanfreepatharecalculatedatfre- thatinthediffusionapproximationthetermproportional
quenciescomparable1/τT or4πσ,withaccountofscreen- to ∇U is usually neglected, since it is of the order of fa2.
ing. Throughout the work we assume that the electrons However, in our case this term may be linear in f , be-
a
form a degenerate gas with Fermi wavelength λF much cause U hasthe component εc eveninthe absenceof
∇ ∇
smallerthanthe elastic meanfree pathl,while l L,λ. current.
≪
This allows us to use the Boltzmann-Langevinapproach As usual,25 we proceedbyseparatingEq.(2.3)intoits
introducedby KoganandShulman23 (see alsoRef. [24]), symmetricandantisymmetricparts[seeEq.(2.1)]andin
and study both the ac conductance and the noise in a the first order in f we get
a
unified way. In section II we analyze the Boltzmann-
Langevin equation in a non-uniform structure, obtain ∂f ∂f ∂U ∂cosθ
s + s f U + f vcosθ =0, (2.4a)
the boundary conditions for the distribution function at ∂t ∂ε ∂t − a ∂k ·∇ ∇ a·
the conductor-electrode interface, and derive a “drift-
∂ 1
diffusion-Langevin” equation for the current. In section + facosθ+ fs v=0. (2.4b)
∂t τ ∇ ·
III we apply this equation to two specific models of dif- (cid:18) (cid:19)
fusive conductors (a “sandwich” and a conductor over a
In this work we are interested only in the case of fre-
ground plane). In section IV the kernels, which describe
quencies much smaller than 1/τ. In this case Eqs. (2.4)
the response of the system to external voltage, and to
may be combined to give
the randomLangevinsources,arefound. Using thesere-
sponse functions we calculate the conductance and ther-
∂f ∂f ∂U vτ ∂cosθ
malnoise(sectionV),andthenon-equilibriumshotnoise s + s + f U
(sectionVI).SectionVIIpresentsdiscussionoftheresults ∂t ∂ε ∂t ∇ s· cosθ ∂k ·∇
and conclusions. ( fs vτ) v=0. (2.5)
−∇ ∇ · ·
Integrationofthisequationoverthe directionsofkgives
II. BOLTZMANN-LANGEVIN-POISSON
THEORY εkl∂fs 3
[εkl fs] l ⊥fs ⊥U =0 (2.6)
D ∂t −∇· ∇ − 2 ∇ ·∇
Inordertodescribeboththeconductorandelectrodes,
weneedself-consistentequationsforthecurrentinasys- with D = D(r) = l(r)vF(r)/3 and l = l(r) = τ(r)vF(r),
tem which may be substantially non-uniform on a scale vF being the Fermi velocity. ⊥ denotes differentiation
∇
∆r l. In the diffusion approximationthe electron dis- in the plane perpendicular to the current direction x.
trib≫ution function can be written as Eq.(2.6)isageneralizationoftheregulardiffusionequa-
tion for the distribution function in the case when the
f =f(ε,cosθ,r,t)=fs(ε,r,t)+fa(ε,r,t)cosθ, (2.1) potential or the mean free path change substantially in
space.
where f l/L 1 and ε is the total electron energy,
| a|∼ ≪ The random nature of the scattering in the conduc-
ε=εk+εc(r) eΦ(r,t) εk+U(r,t). (2.2) tor may be described23,24 by a stochastic source term
− ≡ Js(r,k,t), with zero average, added to the right hand
Here εk is the kinetic energy of an electron with mo- side of Eq. (2.3). Its correlation function was found by
mentum k while ε (r) is the equilibrium local conduc- KoganandShul’man23 assumingPoissonstatisticsofthe
c
tion band edge, which includes possible band-bending scatteringevents,andtakingintoaccounttheFermicor-
due to mismatch in the local Fermi energies in the non- relationsoftheelectrons. Forthecaseofstrongisotropic
uniform conductor, and hence describes the equilibrium impurity scattering (l L) the result reads
≪
2
Js (r,ε,t)Js(r′,ε′,t′) = where the variables include both the deterministic and
h a a i
6δ(r r′)δ(ε ε′)δ(t t′) stochastic parts.
− τ(r)−(r) − f¯s(ε,r) 1−f¯s(ε,r) , (2.7) The correlation function of the current sources js in
N any direction α follows from equations (2.7) and (2.10):
(cid:2) (cid:3)
where Js is the antisymmetric component of Js, (r) is
a N js(r,t)js(r′,t′) =δ(r r′)δ(t t′) (r) (2.14)
the local density of states at the Fermi level (excluding h α α i − − S
the spin degeneracy), and f¯the ensemble average of the
with the correlator
distribution function.
Equationsforthecurrentcanbeobtainedbyincluding 2 ∞
the sourcetermJs inequations(2.4),andthen integrat- S(r)= 3e2τ(r)N(r)vF2(r) dεf¯s(ε,r) 1−f¯s(ε,r) .
ing them over the electrons’ momenta. It is convenient Z0
(cid:2) (cid:3)
at this stage to change variables in Eqs. (2.4) from ε to (2.15)
εk by using Eq. (2.2). Integrating Eq. (2.4a) over k, we
get the continuity equation Now let us consider a system in which a homogeneous
conductorconnectstwohomogeneouselectrodeswithin-
∂ terfaces at L/2,with the only source of inhomogeneity
ρ(r,t)+ j(r,t)=0 (2.8) ±
∂t ∇· being the band bending due to charge transfer between
the materials [Fig. 1(a)]. Let the interfaces be normalto
withj(r,t)thecurrentdensity,andρ(r,t)theexcesselec- the x axis and much sharper than the screening lengths
trondensity,i.e., thetotalchargedensityminus itsequi- λ,λ in the conductor and the electrodes, respectively.
e
librium value ρ0(r) (which includes the possible charge Wedefinetheinterfaceregionstobetheregionsofwidth
transferwhen twomaterialshavebeen broughtinto con- 2δ around L/2, with l δ λ,λ ,L. A major as-
e
tact). The Langevin term integrates out from this equa- sumption o±f this work is t≪hat t≪he voltage in the system
tion, as expected for particle-conserving scattering pro- drops entirely in the bulk of the conductor, i.e., the re-
cesses. sistances of the electrodes and the electrode-conductor
If, before integration over k, we multiply Eq. (2.4b) interfaces are small compared to that of the conductor.
(including the Langevin term) by v, it yields This is the natural situation when the electrodes are of
highconductivityandwhentheinterfacesaresmoothon
j(r,t)= D(r) [ρ (r)+ρ(r,t)]
0 thescaleofλ ,sonoreflectionsofelectronsoccuratthe
− ∇ F
+[σ(r)+δσ(r,t)][E0(r)+E(r,t)]+js(r,t) (2.9) interfaces.
Let us consider, for example, the interface at L/2 and
ewτit(hr)ρ(σr(,rt))/m(=r). Heτe(rre)ρm0((rr))/ims (trh)e loacnadl effδeσc(triv,te)mas=s vdoefiltnaegefsdcr=opfsin(Lt/h2e−elδe)ctarnoddefsies=nefgsl(igLi/b2le+, fδe).isSijnucset tthhee
s
(for simplicity, the parabolic and isotropic dispersion re- equilibrium Fermi-Dirac distribution
lation was assumed), and
1
fe =f (ε+eV/2)= , (2.16)
js(r,t)=eτ(r) vkJs(r,k,t). (2.10) s 0 1+exp ε+eV/2−µ
T
k
X (cid:16) (cid:17)
At equilibrium, and in the absence of external fluctua- wherethechemicalpotentialµisdefinedastheaverageof
tion sources, the current j(r,t) should vanish. Thus, the the chemical potentials in the two electrodes. Moreover,
built-in electric field satisfies the equation the fact that there is no voltage drop across the inter-
face region implies that fc is also given by Eq. (2.16),
s
D(r) ρ (r)+σ(r)E (r)=0 (2.11) fc = fe. Integrating this equation over the electron’s
0 0 s s
− ∇ momenta and using the relation (r) = 1/4πe2λ2(r)
N
whichmaybeinterpretedastheconstancyoftheelectro- leads to the first boundary condition at the electrode-
chemical potential at V =0 and js =0. conductor interface,
Thetermsproportionaltoδσ(r,t)inEq.(2.9)areneg-
ligible if λ2ρc =λ2ρe (2.17)
e
e Φ (r)+Φ(r,t) ε ε (r), (2.12) with ρc ρ(L/2 δ) and ρe ρ(L/2+δ).
| 0 |≪ F − c ≡ − ≡
Thesecondboundaryconditionisthecontinuityofthe
(where εF is the equilibrium Fermi energy [Fig. 1(a)]), current across the interface,
i.e., if the band bending and the external potential are
smallcomparedtothelocalFermienergy. Underthecon- jc =je (2.18)
ωx ωx
dition(2.12),equation(2.9),togetherwiththeconstraint
(2.11), yield the ’drift-diffusion-Langevin’ equation with jc , je the Fourier components of the transverse
ωx ωx
current density at L/2 δ, L/2+δ, respectively.26
j(r,t)=σ(r)E(r,t) D(r) ρ(r,t)+js(r,t) (2.13) If complemented with−the Poisson equation
− ∇
3
4π
E(r,t)= ρ(r,t), (2.19) Integrationofequations(3.1)and(3.3)providesasim-
∇· ǫ(r) ple relation between the current and the electric field,
where ǫ(r) is the dielectric constant, equations (2.8) and iωε(x)
(2.13 – 2.19) form a closed system which is the basis for Iω(x)= 4π Eω(x)+Iωe. (3.4)
our calculations.
The integration constant Ie has the physical sense of
ω
the current induced deep inside the electrodes (where
III. MODELS E = 0). It can be found from the condition that the
ω
current fluctuations do not affect the voltage V applied
ω
A. The sandwich model to the structure:
∞
4π
We study two analytically solvable models which dif- dx [Ie I (x)]=V . (3.5)
fer in their assumed sample geometry, and hence in Z−∞ iωǫ(x) ω − ω ω
electrostatics.12 In our first, “sandwich” model, which
Inserting equations (3.1) and (3.4) in Eq. (3.2) we get
is schematically shown in Fig. 1(b), a shortconductor of
the basic equation of this model,
length L t is sandwiched between two wide electrodes
≪
(tisthesmallesttransversedimensionoftheconductor).
d2I (x) iω 1
Defining the quantities Φ (x), q (x), E (x), I (x) and ω κ2(x,ω)I (x)= Is(x) Ie, (3.6)
ω ω ω ω dx2 − ω D(x) ω − λ2(x) ω
Is(x) as integrals over the sample’s cross-section of the
ω
temporal Fourier components of Φ(r,t), ρ(r,t), E (r,t),
x with
J (r,t), and js(r,t), respectively, we get from equations
x x
(2.8, 2.13, 2.19) 1 iω
κ2(x,ω)= . (3.7)
λ2(x) − D(x)
dI (x)
ω
iωq (x)+ =0, (3.1)
ω
− dx This equation is valid for both the conductor itself and
the electrodes.
dq(x)
I (x)=σ(x)E (x) D(x) +Is(x), (3.2)
ω ω − dx ω
B. The ground-plane model
dE (x) 4π
ω
= qω(x). (3.3) In the second (“ground plane”) model we consider a
dx ε(x)
long and thin conductor close and parallel to a well-
[Deriving Eq. (3.3), we have neglected the transverse conducting ground plane, L t,d where t is the thick-
≫
derivatives of E(r,t), since, by Gauss’ theorem, they are nessoftheconductor,andditsdistancefromtheground-
proportional to the circumference of the cross-section of plane–seeFig.1(c). ThewidthoftheconductorW (i.e.,
thesample,whilethederivativeinthex-directionispro- its second dimension, parallel to the ground plane) can
portional to the cross-sectionarea.] be arbitrary. As we will show below, this geometry is
more promising for experimental observation of some of
the effects studied in this work.
(a)-L/2 L/2 EEFC 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(ieet.2hqqe.euuI.8n,aa)xettt-aiiqdhoonuinnerda,estsc(aihiot2nmoin.cow1esen3e(w)ivi3nseac.r1yatm,)hnauliaessbcnahemfddosrsro(emd3tpto.eahl2lale)lacet,pehrrddreetiesffhwvgpaeirieorntaceuhdtntsiihttvemheneoeltoyntirrd.oeae-f1TndlDt,shihmveeeevqerPeunfisraoesestiilisodgioosnrnoniaasnns-l,
dients. In this case, the linearity of the Poisson equa-
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tlsoieoccnatilolecnah:dasrgteo,aboltinhΦeaiωnr(txed)ger=paetnAeddqeCωon0(vcxee)ro,tfhtehceopnodtuecnttoiar’lsocn(r3ot.sh8se)-
L
where A is the cross-sectionalarea and C is the specific
0
FIG. 1. Schematic description of the geometries studied. capacitance (per unit length).
(a) The conduction band-edge profile in both models. (b) For a homogeneous cylindrical conductor of radius
The sandwich model. (c) Theground-plane model. R = t/2 and distance h = t/2+ d between its center
4
and the ground plane C assumes the following limiting Combining Eq. (3.2) with Eq. (3.8) provides a
0
values: if the conductor is thick (t λ)27 diffusion-like relationbetween the currentand the linear
≫
charge density,
ǫ
m
C = (R λ), (3.9)
0 2cosh−1(h/R) ≫ dq (x)
I (x)= D′ ω +Is(x), (3.14)
ω
− dx
with ǫ the dielectric constant of the medium between
m
the conductor and the ground plane. In the opposite with
limit, the cylinder is uniformly charged. Taking for sim-
plicity ǫ=ǫ , one finds ′ σA
m D =D+ . (3.15)
C
0
2ǫ
C =
0 1+4ln(h/R)+4I(h/R) With the help of Eq. (3.1) we once again get a simple
2ǫ equation for the current
, (R λ), (3.10)
≃ 1+4ln(2h/R) ≪
d2I (x)
ω κ¯2(ω)I (x)= κ¯2(x,ω)Is(x), (3.16)
with dx2 − ω − ω
1 x with
I(b) dx ln 1+ 1 . (3.11)
≡ − 4b2
Z0 (cid:18) r (cid:19) κ¯2(ω)= iω. (3.17)
−D′
The dependences of C on the ratio R/h are shown in
0
Fig. 2.
Thus, as seen by comparing equations (3.6, 3.7) with
equations (3.16, 3.17), static screening effectively van-
ishes in this model, and only dynamic screening is im-
1.5
portant.
WenowanalyzetheconditionsunderwhichEq.(3.8)is
λ«
t
valid. Expanding Eq. (3.16) in spatial harmonics we see
1 λ»
t that harmonics with wavenumber k κ¯ = ω/D′
ε x ≫ | |
/ contribute negligibly to the current fluctuations I .
0 ω
C Thus, at frequencies ω < τ¯−1 D′/L2 D′/d2p,D′/t2,
0.5 wecanconsideronlythe∼waTvenu≡mbersk≪whicharemuch
x
smallerthand−1,t−1. Fortheseharmonics,thetransver-
salgradientsof the electric field dominate in the Poisson
0 equation, justifying Eq. (3.8) at that frequency range.
0 0.5 1 Equation (3.8) is not valid at distances comparable
to d from the interface with the electrodes. Eq. (3.16)
t/(t+2d) shouldthereforebesolvedonlyinsidetheconductor,with
boundary conditions at L′ = (L δ), where δ is now
± 2 ± 2 −
some distance for which d δ L,1/κ¯. To find these
FIG.2. The dependence of C0 on R/h=t/(t+2d) for an ≪ ≪ | |
boundary conditions we write down Eq. (3.2) (which is
homogeneous cylindrical conductor close to a ground plane.
valid even near the interface) in the form
If the conductor is a uniform strip of width W d,t,
≫ d
then I (x)= σ Φ (x)+4πλ2 q (x) . (3.18)
ω ω ω
− dx
W (cid:2) (cid:3)
C0 = , (t λ), (3.12a) [The sources Is(x) in the small interface region may be
4πd ≫
neglected.] The values of q and Φ at the actual inter-
W ω ω
C0 = , (t λ). (3.12b) faces x= L/2 are found from the boundary conditions
4π(d+t/3) ≪ ±
Eqs. (2.17, 2.18). First note that while q can be arbi-
ω
trarily large on the electrode side, λ q should remain
In the quantum limit, with only the first quantum level e ω
finite,since itis equaltothe totalinterfacechargeinthe
populated, C may still be presented with Eq. (3.12b),
0
electrode. In the case when the screening length in the
though the thickness t must be replaced by an effective
electrodesλ ismuchshorterthanλ,Eq.(2.17)therefore
thickness t . For a square well potential with infinite e
eff
gives (on the conductor side of the interface)
barriers the effective thickness is
3 L
t =t 1+ 1.07t. (3.13) q =0. (3.19)
eff 4π2 ≃ ω ±2
(cid:18) (cid:19) (cid:18) (cid:19)
5
Sincethevoltagedropintheelectrodevanishes,thecon- and resistance negligible in comparison with the resis-
straint of fixed voltage [i.e., the equivalent of Eq. (3.5)] tanceR=L/σAoftheconductor. Wefirstshowthatthe
in this model becomes total noise produced in the electrodes is negligible com-
L V A pared to that originating in the conductor. For equilib-
ω
Φω = . (3.20) riumnoise,thisisadirectconsequenceofthefluctuation-
±2 ∓ 2
(cid:18) (cid:19) dissipationtheorem. The sameis true for shotnoise,be-
Integration of Eq. (3.18) from L/2 to L′/2, and use cause of the fact that the electron distribution function
of Eq. (3.8) at L′/2, now yield±s the req±uired boundary in the electrodes is almost equilibrium. The electrode-
±
conditions, conductor interface is also not an appreciable source of
′ noise since, in the diffusion approximation studied here,
L V Aσ
ω
qω ± 2 =∓ 2D′ . (3.21) the electron distribution at the conductor-side of the in-
(cid:18) (cid:19) terface is the same equilibrium Fermi distribution as in
In this ground-plane model, finite charge densities in the electrode. Moreover, even deviations from this ap-
the conductorcreateimagechargesonthe groundplane. proximation would result at the most in a few inelastic
Thus, at any finite frequency ω some parts of the in- scattering events in the electrodes, leading to thermal-
terface currents are responsible for periodic re-charging ization of hot electrons arriving from the conductor. As
of the ground plane [see Fig. 1(c)]. Therefore, the cur- longas l L,the number ofthose events per transfered
rentIe measuredinthe electrodesatadistancefar from electroni≪smuchsmallerthanthenumberofelasticevents
ω
conductor-electrode interface may be different from the [ (L/l)2] the electron experiences in the conductor, so
∼
currentwhichflowsthroughthisinterface(notealsothat the thermalization process at the interface can also be
the two interface currents are not necessarily equal). In neglected as a source of noise.
an experimental scheme symmetric with respect to the
In this situation, the solutions to equations (3.6) and
conductor, the current Ie flowing into the external cir-
ω (3.16) can thus be presented in the form
cuit is the symmetric component of the two currents:
1 L L
Ie = I +I . (3.22) L
ω 2(cid:20) ω(cid:18)−2(cid:19) ω(cid:18)2(cid:19)(cid:21) Iω(x)=Y(x;ω)Vω + L1 −2L K(x,x′;ω)Iωs(x′)dx′. (4.1)
While the currents Iω( L/2) in this expression are the Z 2
±
currents at the electrode side of the interfaces, due to
Eq.(2.18)they are equalto I ( L′/2). Of course,if the
ω
± Eq. (4.1) shows that the current at frequency ω at any
leads connecting the sample to the measurement instru-
point x is composed of two components. The first is the
ment have some mutual capacitance, the current in the
instrument will be less than Ie(ω) given by Eq. (3.22), responsetotheappliedvoltageacrosstheconductor,and
the second is the response to the random Langevin cur-
but this loss factor may be taken into account by the
rent sources inside the conductor.
standard circuit theory methods.
′
The response functions Y(x;ω) and K(x,x;ω) are
found by solving the equations for the current with
IV. THE RESPONSE FUNCTIONS Is(x) = 0 and V = 0, respectively. They can be
ω ω
presented in a compact form by defining the follow-
Fromnowonwewillconsiderthe mostnaturalcaseof ing auxiliary functions, with η = κλ2/λ , u = κL/2,
e
well-conducting electrodes of size much larger than λ , sinch(u)=sinh(u)/u, Ω=ω/4πσ, and χ= x L/2:
e
| |−
1
(ω)= , (4.2)
D cosh(u)+ηsinh(u)
u [sinh(u κx)+ηcosh(u κx)] (x < L),
E±(x,ω)=(cid:26) 11−−uiiΩΩηexp(−κ±eχ) ± (||x||> L22), (4.3)
1−iΩD(ω)cosh(κx) (x < L),
(x,ω)= [1−iΩ][1−iΩD(ω)sinch(u)] | | 2 (4.4)
F ( 1−iΩ[1[−1−iΩη]D[1(−ωi)ΩsiDnh(ω(u)s)ienxcph((−u)κ]eχ)] (|x|> L2),
iΩ ±(x,ω)
±(x,ω)= E , (4.5)
G sinh(u)+ηcosh(u)
±(x,ω)=iΩ (ω)[ (x,ω)+ ±(x,ω)]. (4.6)
H D F E
6
For the sandwich model, the response functions are 1 cosh(κ¯x)
Y(x;ω)= . (4.11)
R sinch(u¯)
′ ′
K(x,x;ω)= (x,ω) ±(x,ω)sinh(κx)
F ±G At low frequencies, and in both models, the response
′
−H±(x,ω)cosh(κx) (4.7) functions tend to constant values:
and ′ 1
K(x,x;0)=1, Y(x;0)= . (4.12)
R
1 iΩ
Y(x;ω)= − (x,ω), (4.8) Fig. 3 shows the response functions for the sandwich
R F
modelatintermediateandhighfrequencies. Atωτ 1
T
≫
the responses are exponentially close to the source, i.e.,
where, in Eq. (4.7), the upper sign should be used for
x′ >x and the lower sign for x′ <x. to x′ in the case of K(x,x′;ω) and to L/2 in the case
±
of Y(x;ω). At any frequency and position, and for any
In the ground-planemodel, the functions K andY in-
value of L/λ,
side the conductor are found to be the same as for the
sandwich model (but with D D′) in the limit of van- L
ishing conductivity of the con→ductor, i.e., in the formal 1 2 K(x,x′;ω)dx′ =1. (4.13)
limit η2 Ω (which also implies κ=κ¯): L −L
∼ →∞ Z 2
cosh(u¯ κ¯x′) This general result is a manifestation of the constraint
′
K(x,x;ω)=cosh(u¯ κ¯x) ∓ (4.9) V = 0, as can be seen by assuming a uniform current
± sinch(2u¯) ω
source in Eq. (4.1). Then, the only solution of the prob-
lem which maintains the constraint of fixed voltage is a
with u¯ = κ¯L/2 and with the upper sign used for x′ > x
uniform current everywhere, I (x) = Is, leading imme-
and the lower for x′ <x. The response in the electrodes ω ω
diately to Eq. (4.13).
Ke(x′;ω)to a fluctuation atx′ is obtainedwith the help
The relation between the spectral density of the cur-
of Eq. (3.22), rentnoise S (x,ω) and the response function K(x,x′;ω)
I
′ is made clear through the identity
cosh(κ¯x)
Ke(x′;ω)= . (4.10)
sinch(u¯) S (x,ω)δ(ω)=2 I (x)I∗(x) . (4.14)
I h ω ω i
The response to voltage Y(x,ω) in this model is iden- With Eq. (4.1), and using the condition of fixed voltage
tical to Ke(x;ω)/R, andthelocalityofthecurrentcorrelator[Eq.(2.14)],this
expression becomes
4 L
S (x,ω)= 2A 2 K(x,x′;ω)2 (x′)dx′. (4.15)
!) 2 !(cid:28)T =100 !(cid:28)T =10 I L2 Z−L2 | | S
;
0 x The noise power deep inside the electrodes is given by
x; 0
(
K
S (ω) S ( ,ω). (4.16)
-2 !(cid:28)T =10 !(cid:28)T =100 I ≡ I ∞
(a) (b)
L=(cid:21)=10 L=(cid:21)=0:1 The dynamic conductance is the response in the elec-
trodes to external voltage,
)
U. !(cid:28)T =10
!(cid:28)T =10 Y(ω) Y( ,ω). (4.17)
m. 1 ≡ ∞
r
o
N !(cid:28)T =100
(0.5 !(cid:28)T =100
!) L=(cid:21)=10 V. RESULTS: CONDUCTANCE AND
(x; 0 (c) L=(cid:21)=0:1 (d) EQUILIBRIUM NOISE
Y
-1 -0.5 0 0.5 -0.5 0 0.5 1 A. Sandwich model
Position (1=L)
FIG. 3. Position dependence of the real (solid lines) and For the sandwichmodel we have fromequations (4.4),
imaginary (dashed lines) parts of the response functions for (4.8) and ( 4.17)
thesandwichmodelathighfrequencies,andfortwovaluesof
L/λ. Re[Y(x;ω)]isinunitsof1/R. Im[Y(x;ω)]isinunitsof 1 iΩ 1
−Ω/R. x′=0.3 here. Y(ω)= −R 1 iΩ (ω)sinch(u). (5.1)
− D
7
If the screening length in the electrodes is very small As in Eq.(5.2), the emittance in this case canbe viewed
compared to L then ηu 1, and we get asthesumoftheintrinsicemittanceoftheconductorand
≫
thatofthecapacitorformedbytheelectrodes. Then,the
Y(ω)= 1−iΩ =R−1 iωC, C = A , (5.2) total emittance (5.4) is entirely due to the parallel-plate
R − 4πL capacitance, and the emittance of the “conductor itself”
equals zero, in agreement with the result of Ref. [19].
for any frequency ω. Equation 5.2 allows a simple inter- However,asseenfromFig.4(a),thisresultdoesnothold
pretation: Y(ω) is just the complex admittance of the for a relatively short conductor, which means that the
conductance R−1 coupled in parallel to the capacitor C simple view of an additive emittance does not generally
formed by the two electrodes. hold.
Itisimportantto notethatthe calculationsleadingto
1
Eq. (5.1) were not dependent on the distribution func-
) 0.8 (a) Re tion of the electrons. Therefore, the correction to Y(ω)
u
( isdueonlytothescreeningpropertiesofthesystem,and
h 0.6
c does not depend on thermalization or phase-breaking of
n
si 0.4 Im the electrons. Equation (5.1) is thus also valid if the in-
)
! elastic scattering length is smaller than L. Despite its
(
D 0.2
mesoscopic nature [i.e., the fact that Y(ω) assumes its
0 ordinary value Eq. (5.2) for large enough L], the correc-
tion discussed here should not be confused with other
0.8 mesoscopic corrections to the conductivity of diffusive
(b) 0.01
L=(cid:21)= wires.28
0.1
) 0.6
! Re 1
(
eF 0.4 10
Im 100 B. Ground-plane model
0.2
0 For the ground-plane model equations (3.22) and
0.01 0.1 1 10 100 1000 (4.11) lead to the following expression for the conduc-
tance of the system
Frequency (4(cid:25)(cid:27))
1 L L
FIG. 4. The real and imaginary parts of the correction Y(ω)= κ¯ cth κ¯ . (5.5)
to the conductance of the sandwich model. The various R 2 2
(cid:18) (cid:19) (cid:18) (cid:19)
curves are for different values of L/λ, and for λ = λ.
e
(a) D(ω)sinch(u). The low-frequency value of this quan- This expression is identical to the conductance of a
macroscopic wire of resistance R coupled to the ground
tity is equal to one plus the emittance of the system. (b)
Fe(ω)=Y(ω)/[(1−iΩ)/R]. plane via capacitance per unit length of
1 1 −1
However, already at L/λ =10 the correction
e C = + , (5.6)
eff
C C
(cid:18) 0 S(cid:19)
1
e(ω)= (5.3) with
F 1 iΩ (ω)sinch(u)
− D
A
to this simple result is significant. Figure 4 shows the CS = 4πλ2. (5.7)
real and imaginary parts of this correction for different
values of L/λ, for the case λ = λ (for example, the The boundary conditions for this model, Eq. (3.21), can
e
conductor and the electrodes are made of the same ma- also be rewritten in terms of C as
eff
terial, but the electrodes have much fewer impurities).
For L/λe >1 the correction term is insensitive to an in- q L′ = VωC . (5.8)
ω eff
crease of λ, so the appropriate curves of Fig. 4 also cor- ± 2 2
(cid:18) (cid:19)
respond to the case of a low-density conductor between
metal electrodes. Fig. 4(a)shows the term (ω)sinch(u) Thus,theground-planemodelcanbedescribedbythe
D equivalentcircuitshowninFig.5. ThecapacitanceC is
whichappearsinthe denominatorofEq.(5.1). Thelow- S
frequency value of this term is equal to 1 4πσRE(0), due tothe factthatinthinenoughwiresthe screeningis
− notefficient,sothecurrentisdeterminedbythegradient
with E(0) idY/dω being the “emittance”. For a
long conduc≡tor (L/λ|ω=10) ofthefullelectrochemicalpotentialϕ=Φ+Aµ/e,rather
≫ than by Φ = Aq /C alone. In other words, additional
ω 0
A 1 charging of the wire increases not only its electrostatic
E(0)= = . (5.4) energy ( C−1), but also its internal energy ( C−1),
4πL C ∝ 0 ∝ S
8
because of the necessary rise in the Fermi level. In 2D At zero frequency Eq.(4.12) gives
conductors, Eq. (5.7) reduces to the well-known result29
forthetwo-dimensionalelectrongas. Thereisalsoavery 4T
interestinganalogybetweenEq.(5.7)andthe expression SIeq(x,0)= R , (5.13)
L−1 = A/4πλ230 for the specific kinetic inductance of a
0
superconductor (in this case λ is London’s penetration
asexpected. At highfrequencies (Ω,ωτ¯ 1), the equi-
depth). T ≫
librium noise inside the conductor is given by
~
R/L Vω R/L Seq(x,ω)= 2T ωτ¯ /2[f(x)+f( x)] (x <L/2)
C0 C0 I R T − | |
CS CS p (5.14)
with
FIG. 5. Equivalent circuit for the geometry of the
ground-planemodel.
f(x)=2e−κ1L cosh(u κx)2 e2κ1x e−κ1L (5.15)
| − | −
WhenC C thehighfrequency(ωτ¯ 1)conduc-
s ≪ 0 T ≫ (cid:2) (cid:3)
tance is given by
and with κ = Re(κ), so κ L = ωτ¯ /2. f(x) = 1/2
1 1 T
1 L throughout the conductor, except at a narrow layer of
Y(ω)= (5.9) p
R2 iD/ω width 1/κ1 near the edges, where it approaches its lim-
iting values f( L/2)=0, f(L/2)=2. The position and
which its absolute value is jpust the dc conductance of a frequency depe−ndence of Seq is shown in Fig. 6(a) for
I
conductor of the same conductivity σ, but with length the case L/λ=L/λe =10. The generalfeatures here do
equal to twice the diffusion distance in time 1/ω. Thus, not depend on screening.
carriersinjected at eachof the electrodes arediffusing in
and out of the conductor, without reaching the opposite
electrode, and without affecting it by electric fields (the
suppression of the longitudinal electric fields is the only )
R T (cid:29)eV (a)
role of the ground plane in this limit). = 80
T
4 60
(
40
y
C. Equilibrium noise – both models sit 20
n
e
D
Equilibrium thermal noise is related to the conduc- al 1000
tance by the fluctuation-dissipationtheorem(we assume ctr -0.5 0 10 100
T ¯hω) pe 0.5 1 Frequency
≫ Seq(ω)=4TRe[Y(ω)]. (5.10) S Position (L) (1=(cid:28)T)
I
However,since Y(x,ω) gives the current response to the
external voltage, the local noise Seq(x,ω) is not directly
I
related to it, and must be calculated independently. )
I T (cid:28)eV (b)
e
At zero voltage, the average distribution function in 2
( 20
the conductor is the Fermi-Dirac distribution given by y 15
t
Eq. (2.16). Using this distribution in Eq. (2.15) gives si 10
n 5
e
D
=2σT (5.11)
S al
r 1000
for the correlator of the Langevin sources. The spec- ct -0.5 100
e
tral density for the equilibrium noise is thus found from p 0 10
S 0.5 1 Frequency
Eq. (4.15),
Position (L) (1=(cid:28)T)
L
Seq(x,ω)= 4T 2 K(x,x′;ω)2dx′; (5.12) FIG.6. Position and frequency dependence of the current
I RL −L | | noise intensity inside the conductor. (a) Equilibrium noise.
Z 2
(b) Shot noise. Here L/λ = L/λ = 10, but the general
e
thisequationinfactexpressesthefluctuation-dissipation features ofthisfigurearenot sensitivetothescreening prop-
theorem for both the local and external fluctuations. erties.
9
The method presented here for calculating the con- This rise is due to the highly non-equilibrium distribu-
ductance does not depend on the form of the distribu- tion of carriers, Eq. (6.1). Specifically, at any frequency
tion function [other than the diffusion approximation, ω the current fluctuations at position x are due only to
Eq. (2.1)]. Therefore the results apply also to the case electronsatdistances D¯/ω fromx. Thesmallerthis
∼
whenstrong(lee L,withlee theelectron-electronscat- range, the smaller is the smoothing of the singularity in
≪ p
tering length) electron-electron scattering is present in the energydistributionofthe electronsinthe range,and
the conductor. The equilibrium noise is alsonotaffected the larger the noise.
by the e-e processes since this scattering does not affect
theequilibriumFermi-Diracdistribution,whichisthein- The noise in the external electrodes is found by us-
put in Eq. (2.15). ingtheexternalresponsefunctionKe(x′;ω)inEq.(6.2).
WheneverKe(x′;ω)=1,i.e.,atlowfrequencies,orwhen
λ,λ L in the sandwich model, Eq. (6.2) reduces to
e
≪
VI. RESULTS: NON-EQUILIBRIUM NOISE
In this section we will present the results on non- 4 2 1 eV
equilibrium noise for the ground-plane model; these re- SI(ω)= T + eVcth (6.5)
R 3 6 2T
sults are also applicable for the sandwich model in the (cid:20) (cid:18) (cid:19)(cid:21)
regime λ,λ L. In the opposite limit (λ,λ L)
e e
≫ ≪
the noise in the sandwich model is white and is equal to
the zero-frequencynoiseinthe ground-planemodel.12 In
as was found by Nagaev.3 However, in the general case,
contrasttothecaseofequilibriumnoise,theshotnoiseis
the noise in the electrodes exhibits strong frequency de-
very sensitive to the strength of electron-electron inter-
pendence ona frequency scale ofthe orderof the inverse
action in the conductor, so we analyze this noise in the
effective Thouless time 1/τ¯ , which is also affected by
two limits of weak and strong e-e scattering. T
the screening lengths λ, λ .12 Figure 7(a,b) shows the
e
dependence of the noise on frequency and temperature
A. Weak electron-electron scattering (L≪lee) in the ground-plane model. At ωτ¯T ≫1
When L l the electron distribution function is
ee
found as a s≪teady-state solution of Eq. (2.6). Under the 2T eV 1
S (ω)= ωτ¯ /2 1 +eIcth +O .
condition (2.12) and for current perpendicular to the in- I R T − 2T √ω
terfaces, it reads:3 (cid:16)p (cid:17) (cid:18) (cid:19) (cid:18) (cid:19)
(6.6)
1 x 1
f¯(E,x)= + f E+ eV
s 0
2 L 2
(cid:18) (cid:19) (cid:18) (cid:19)
1 x 1
+ f E eV . (6.1)
2 − L 0 − 2 1 T/eV=0
(cid:18) (cid:19) (cid:18) (cid:19) 0.9 0.001
0.01
Eq.(4.15)togetherwithequations(2.15)and(2.16)now 0.8 0.1
give a general expression for the non-equilibrium noise, 0.7 (a) (b)
=0
SI(x,ω)= R2LZ−L2L2 dx′|K(x,x′;ω)|2(cid:20)S++S−4Lx2′2(cid:21) sity(2eI) 0000....3456 L(cid:28)lee L(cid:28)le!e(cid:28)(cid:22)T10246800000
(6.2) en 1 10 100 1000 0 0.1 0.2 0.3 0.4 0.5
D
1
al
with tr 0.9
c
e 0.8
p
eV eV S 0.7 (c) (d)
±(T,V)=T cth . (6.3) =0
S ± 2 2T 0.6 !(cid:28)(cid:22)T20
(cid:18) (cid:19) T/eV=0 40
0.5 0.1 60
0.2 80
Fig. 6(b) shows the spatial and frequency dependence 0.4 0.3 100
oftheshotnoiseforthecaseL λ,λe andT eV. Itis 0.3 L(cid:29)lee L(cid:29)lee
remarkable that inside the con≫ductor the high≪-frequency 1 10 100 0 0.1 0.2 0.3 0.4 0.5
noise is large even at zero temperature: Frequency (1=(cid:28)(cid:22)T) T=eV
FIG. 7. Frequency and temperature dependence of the
eI ωτ¯T 4x2 spectral density of the non-equilibrium noise in the limits of
S (x,ω)= 1 , (ωτ¯ 1). (6.4)
I 2 2 − L2 T ≫ weak (a,b) and strong (c,d) electron-electron scattering.
r (cid:18) (cid:19)
10