Table Of ContentTemperature dependent third cumulant of tunneling noise
C. W. J. Beenakker,1 M. Kindermann,1 and Yu. V. Nazarov2
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2Department of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
(Dated: January 2003)
3
0 Poissonstatisticspredictsthattheshotnoiseinatunneljunctionhasatemperatureindependent
0 third cumulant e2I¯, determined solely bythe mean current I¯. Experimental data, however, show a
2 puzzling temperature dependence. We demonstrate theoretically that the third cumulant becomes
n strongly temperature dependent and may even change sign as a result of feedback from the elec-
a tromagnetic environment. In the limit of a noninvasive (zero-impedance) measurement circuit in
J thermalequilibriumwiththejunction,wefindthatthethirdcumulantcrossesoverfrome2I¯atlow
4 temperatures to −e2I¯at high temperatures.
2
PACSnumbers: 73.50.Td,05.40.Ca,72.70.+m,74.40.+k
]
l
l
a Shot noise of the electrical current was studied a cen-
h tury ago as a way to measure the fundamental unit of Z , T
- 1 1
s charge[1]. Todayshot noise is used for this purpose in a
e wide range of contexts, including superconductivity and
m ∆
thefractionalquantumHalleffect[2]. Alreadyintheear- V+ V
I+∆I
. liest work on vacuum tubes it was realized that thermal V
t 0
a fluctuationsofthecurrentcanmaskthefluctuationsdue
m
to the discreteness of the charge. In semiconductors, in
- particular, accurate measurements of shot noise are no-
d
toriouslydifficultbecauseoftherequirementtomaintain Z , T
n 2 2
a low temperature at a high applied voltage.
o
c Until very recently, only the second cumulant of the
[ fluctuating current was ever measured. The distribution FIG. 1: Two resistors in series with a voltage source. The
of transferred charge is nearly Gaussian, because of the fluctuatingcurrent and voltage are indicated.
1
law of large numbers, so it is quite nontrivial to extract
v
6 cumulants higher than the second. Much of the exper-
7 imental effort was motivated by Levitov and Reznikov’s ment is supposed to be noninvasive. The temperature
4 prediction[3]that oddcumulants ofthe currentthrough independent result for the third cumulant of tunneling
1 a tunnel junction should not be affected by the thermal noise is recovered only if the measurement circuit has
0
noise that contaminates the even cumulants. This is a both negligible impedance and negligible temperature.
3
direct consequence of the Poisson statistics of tunneling The circuit is shown schematically in Fig. 1. Two re-
0
events. The third cumulant should thus have the linear sistors(impedances Z , Z andtemperatures T , T )are
/ 1 2 1 2
t dependence on the applied voltage characteristic of shot connected in series to a voltage source (voltage V ). We
a 0
m noise,regardlessofthe ratioofvoltageandtemperature. will specialize later to the case that resistor1 is a tunnel
Incontrast,thesecondcumulantlevelsoffatthethermal junction and that resistor 2 represents the macroscopic
-
d noise for low voltages. measurement circuit, but our main results hold for any
n Thefirstexperimentsonthevoltagedependenceofthe two resistors. We disregard possible Coulomb blockade
o
third cumulant of tunnel noise have now been reported effects on fluctuations [8, 9, 10], which is justified if the
c
[4]. The pictures are strikingly different from what was impedances at frequencies of order eV/¯h are small com-
:
v expected theoretically. The slope varies by an order of pared to h¯/e2 [11].
i
X magnitudebetweenlowandhighvoltages,andforcertain Wehavecalculatedthetemperaturedependenceofthe
samples even changes sign. Such a behavior is expected third cumulant by two altogether different methods, the
r
a foradiffusiveconductor[5],butnotforatunneljunction. Keldysh formalism [12] and the Langevin approach [13].
Althoughthedataisstillpreliminary,itseemsclearthat The equivalence of the two methods has already been
aninputofnewphysicsisrequiredforanunderstanding. demonstrated for a single resistor in the absence of any
It is the purpose of this paper to provide such input. measurement circuit [5]. Likewise, we have obtained the
We willshow thatthe thirdcumulantof the measured same results in both calculations of the backaction from
noise (unlike the second cumulant [6]) is affected by the the measurement. We choose to present the Langevin
measurement circuit in a nonlinear way. The effect can approach in this paper, because it can be explained in
be seen as a backaction of the electromagnetic environ- elementary terms and provides an intuitive physical in-
ment [7]. We have found that the backaction persists sight.
even in the limit of zero impedance, when the measure- Starting point of the Langevin approach is the sep-
2
aration of the fluctuation ∆I of the current through Eq.(5)appliestoatimeindependent meanvoltageV.
i
resistor i = 1,2 into an intrinsic fluctuation δI plus a Foratimedependentperturbationv(t)onehas,tolinear
i
term induced by a fluctuation ∆V of the voltage over order,
i
the resistor: ∆I = δI + ∆V /Z . At low frequencies
i i i i
∆I1 = ∆I2 ≡ ∆I and ∆V1 = −∆V2 ≡ ∆V. Upon sub- h∆X(ω)∆Y(ω′)iV+v =h∆X(ω)∆Y(ω′)iV
stitution we arrive at the two equations
d
′
+v(ω+ω ) S (V). (7)
XY
Z∆I =Z δI +Z δI , Z∆V =Z Z (δI −δI ), (1) dV
1 1 2 2 1 2 2 1 We will use this equation, with v =∆V, to describe the
effect of a fluctuating voltageoverthe resistors. This as-
where Z =Z +Z is the total impedance of the circuit.
1 2 sumes a separation of time scales between ∆V and the
Forsimplicity we assumethatZ is realandfrequency
i intrinsic currentfluctuations δI , so that we can firstav-
independent in the frequency range of the measurement. i
erage over δI for given ∆V and then averageover ∆V.
All formulas have a straightforward generalization to i
Turning now to the third cumulant, we first note that
complex Z (ω). We do not need to assume at this stage
i
atfixedvoltagethe intrinsiccurrentfluctuationsδI and
that the current-voltage characteristic of the resistors is 1
δI are uncorrelated, with third moment
linear. If it is not, then one should simply replace 1/Z 2
i
by the differential conductance evaluated at the mean
voltage Vi over the resistor. hδIi(ω1)δIi(ω2)δIi(ω3)iV =2πδ(ω1+ω2+ω3)Ci(3)(Vi).
The mean voltages are given by V = (Z /Z)V ≡ V (8)
1 1 0
and V2 = V0−V. The intrinsic current fluctuations δIi The spectral density Ci(3) vanishes for a macroscopic re-
are driven by the fluctuating voltageVi =Vi+∆Vi, and sistor. For a tunnel junction it has the temperature in-
thereforedependinanonlinearwayon∆V. Thenonlin- dependent value [3]
earityhastheeffectofmixinginlowerordercumulantsof
δIi inthecalculationofthep-thcumulantof∆I,starting C(3)(V )=e2V /Z =e2I¯, (9)
from p=3. i i i i
Before addressing the case p=3 we first consider p= with I¯the mean current.
2,whenallaveragesh···i canbeperformedatthemean
V We introduce the nonlinear feedback from the voltage
voltage. At low frequencies one has
fluctuations through the relation
hδI (ω)δI (ω′)i =2πδ(ω+ω′)C(2)(V ). (2)
i i V i i h∆X ∆X ∆X i=h∆X ∆X ∆X i
1 2 3 1 2 3 V
Ttohre. WnoeisgeivpeowtweroCeix(2a)mdpepleesn.dIsnoanmthaecrmosocdoeplicforrestihsetorretshise- + h∆Xj∆V(ωk+ωl)iV ddV SXkXl(V). (10)
cXyclic
shot noise is suppressed by electron-phonon scattering
and only thermal noise remains:
The variable X stands for I(ω ) or V(ω ) and the sum
j j j
is over the three cyclic permutations j,k,l of the indices
C(2) =2kT /Z (3)
i i i 1,2,3. These three terms account for the fact that the
samevoltagefluctuation∆V thataffectsS alsocor-
at temperature T , independent of the voltage. (The XkXl
i relates with X , resulting in a cross-correlation.
noise poweris a factor oftwo largerif positive and nega- j
Eq. (10) has the same form as the “cascaded average”
tivefrequenciesareidentified.) Inatunneljunctionboth
through which Nagaev introduced a nonlinear feedback
thermal noise and shot noise coexist, according to [2]
intotheLangevinequation[13]. Inthatworkthenonlin-
earity appears because the Langevin source depends on
C(2)(V )=(eV /Z )cotanh(eV /2kT ). (4)
i i i i i i the electron density, which is itself a fluctuating quan-
tity — but on a slower time scale, so the averages can
From Eq. (1) we compute the correlator
be carriedout separately,or “cascaded”. In our case the
′ ′ voltage drop ∆V over the resistors is the slow variable,
h∆X(ω)∆Y(ω )i =2πδ(ω+ω )S (V), (5) i
V XY relative to the intrinsic current fluctuations δI .
i
where X and Y can represent I or V. The result is Eq.(10)determinesthecurrentandvoltagecorrelators
S = Z−2[Z2C(2)(V)+Z2C(2)(V −V)], (6a)
II 1 1 2 2 0
S = Z−2(Z Z )2[C(2)(V)+C(2)(V −V)], (6b) h∆X(ω1)∆Y(ω2)∆Z(ω3)i=2πδ(ω1+ω2+ω3)CXYZ(V).
VV 1 2 1 2 0 (11)
S = Z−2Z Z [Z C(2)(V −V)−Z C(2)(V)].(6c) We find
IV 1 2 2 2 0 1 1
3
d
C = Z−3[Z3C(3)(V)+Z3C(3)(V −V)]+3S S , (12a)
III 1 1 2 2 0 IV dV II
d
C = Z−3(Z Z )3[C(3)(V −V)−C(3)(V)]+3S S , (12b)
VVV 1 2 2 0 1 VV dV VV
d d
C = Z−3(Z Z )2[Z C(3)(V)+Z C(3)(V −V)]+2S S +S S , (12c)
VVI 1 2 1 1 2 2 0 VV dV IV IV dV VV
d d
C = Z−3Z Z [Z2C(3)(V −V)−Z2C(3)(V)]+2S S +S S . (12d)
IIV 1 2 2 2 0 1 1 IV dV IV VV dV II
We apply the general result (12) to a tunnel barrier
(resistor number 1) in series with a macroscopic resis-
tor (number 2). The spectral densities C(2) and C(3) are
1 1
given by Eqs. (4) and (9), respectively. For C(2) we use
2
Eq. (3), while C(3) = 0. From this point on we assume
2
linear current-voltage characteristics, so V-independent
Z ’s. We compare C ≡ C with C ≡ −C /Z3.
i I III V VVV 2
The choice ofC is motivated by the typicalexperimen-
V
tal situation in which one measures the current fluctua-
tions indirectly through the voltage over a macroscopic
series resistor. From Eq. (12) we find
e2I¯ 3(sinhucoshu−u)
C = 1+
x (1+Z2/Z1)3 (cid:20) (1+Z1/Z2)sinh2u
T g
2 x
× −cotanhu , (13)
(cid:18)T u (cid:19)(cid:21) FIG. 2: Voltage dependence of the third cumulants CI and
1
CV ofcurrentandvoltageforatunneljunction(resistanceZ1)
with g =1, g =−Z /Z , and u=eV/2kT . inserieswithamacroscopicresistorZ2. Thetwosolidcurves
InthIeshotnVoiselim1it(e2V ≫kT )werecov1erthethird areforZ2/Z1 →0andthedashedcurvesforZ2/Z1 =1. The
1 curves are computed from Eq. (13) for T1 = T2 ≡ T. The
cumulant obtained in Ref. [7] by the Keldysh technique:
high voltage slopes are the same for CI and CV, while the
low voltage slopes havetheopposite sign.
1−2Z /Z
C =C =e2I¯ 2 1 . (14)
I V (1+Z /Z )4
2 1
In the opposite limit of small voltages (eV ≪ kT1) we ThislimitisalsoplottedinFig.2,forthecaseT2 =T1 ≡
obtain T ofthermalequilibriumbetweenthetunneljunctionand
the macroscopic series resistor. The slope then changes
CI = e2I¯1+(Z(21/+Z1Z)(22/TZ21/)T41−1), (15) afrtomhigdhCvVo/ltdaI¯ge=s.−Teh2eamt ilnoiwmuvmoltaCgVes=to−d1C.7Ve/kdTI¯/Z=1 e=2
1−Z /Z −2T /T −0.6e2I¯is reached at eV =2.7kT.
C = e2I¯ 2 1 2 1. (16)
V (1+Z /Z )4
2 1 In conclusion, we have demonstrated that feedback
WeconcludethatthereisachangeintheslopedC /dI¯ from the measurement circuit introduces a temperature
x
dependenceofthethirdcumulantoftunnelingnoise. The
fromlowto highvoltages. Ifthe entiresystemisinther-
temperatureindependentresulte2I¯ofanisolatedtunnel
mal equilibrium (T =T ), then the change in slope is a
2 1
factor ±(Z −2Z )(Z +Z )−1, where the + sign is for junction [3] acquires a striking temperature dependence
1 2 1 2
in an electromagnetic environment, to the extent that
C and the − sign for C . In Fig. 2 we plot the entire
I V
the third cumulant may even change its sign. Precise
voltage dependence of the third cumulants.
predictions have been made for the dependence of the
The limit Z /Z →0 of a noninvasive measurement is
2 1
of particular interest. Then C = e2I¯has the expected noise on the environmentalimpedance and temperature,
I
which can be tested in ongoing experiments [4].
resultforanisolatedtunneljunction[3],butC remains
V
affected by the measurement circuit: We gratefully acknowledgediscussionswith B.Reulet,
which motivated us to write this paper. Our research is
T 3(sinhucoshu−u)
lim C =e2I¯ 1− 2 . (17) supportedbytheDutchScienceFoundationNWO/FOM.
Z2/Z1→0 V (cid:18) T1 usinh2u (cid:19)
4
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