Effect of longrange Coulomb interaction on shotnoise suppression in ballistic transport
T. Gonza´lez, O. M. Bulashenko,
*
J. Mateos, and D. Pardo
Departamento de Fı´ sica Aplicada, Universidad de Salamanca, Plaza de la Merced s/n, E37008 Salamanca, Spain
L. Reggiani
Istituto Nazionale di Fisica della Materia, Dipartimento di Scienza dei Materiali, Universita` di Lecce Via Arnesano, 73100 Lecce, Italy
Received 19 December 1996
We present a microscopic analysis of shotnoise suppression due to longrange Coulomb interaction insemiconductor devices under ballistic transport conditions. An ensemble Monte Carlo simulator selfconsistently coupled with a Poisson solver is used for the calculations. A wide range of injectionrate densitiesleading to different degrees of suppression is investigated. A sharp tendency of noise suppression at increasinginjection densities is found to scale with a dimensionless Debye length related to the importance of spacecharge effects in the structure.
S01631829
97
09735X
The phenomenon of shot noise, associated with the randomness in the ﬂux of carriers crossing the active region of adevice, has become a fundamental issue in the study of electron transport through mesoscopic devices. In particular, thepossibility of shotnoise suppression has recently attracted alot of attention, both theoretically and experimentally.
1
Atlow frequency
small compared to the inverse transit timethrough the active region
the power spectral density of shotnoise is given by
S
I
2
qI
, where
I
is the dc current,
q
isthe electron charge, and
is the suppression factor. Whenthe carriers crossing the active region are uncorrelated, fullshot noise with
1
Poisson statistics
is observed. However, correlations between carriers can reduce the shotnoisevalue, giving
1. In real mesoscopic devices differenttypes of mechanisms resulting in shotnoise suppression canbe distinguished:
i
statistical correlations due to the Pauliexclusion principle
important for degenerate materials obeying Fermi statistics
,
ii
shortrange Coulomb interaction
electronelectron scattering
, and
iii
longrange Coulombinteraction
by means of the selfconsistent electric potential
. While the ﬁrst two mechanisms have been extensivelydiscussed in solidstate literature,
1
the last one has receivedless attention,
2
although its role in shotnoise suppression hasbeen known for a long time in vacuumtube devices.
3
Theonly exception that should be mentioned is the Coulombblockade in resonanttunneling devices, which can be alsoreferred to as the last mechanism of suppression. The blockade is provided by a builtin charge inside a quantum wellwhich redistributes the chemical potential, and prevents theincoming carriers from passing through the well, thereby resulting in carrier correlation and shotnoise suppression
seethe experimental evidence
4
. The Coulomb blockade is aconsequence of longrange Coulomb interaction, and it actsunder the
sequential
tunneling regime of carrier transport.The main objective of the present paper is to prove theimportance of longrange Coulomb interaction between thecarriers on the shotnoise power spectrum
under the ballisticregime of electron transport
. The ballistic regime is nowaccessible in modern mesoscopic devices like electronwaveguides, quantum point contacts, etc., which have characteristic lengths of the order, or smaller, than the carriermean free path. The current existing theories invoked to interpret the experimentally observed shotnoise suppression insuch devices
5–7
assume that carriers move inside the devicewithout inducing any redistribution of the electric potential.We use a more rigorous approach which includes longrangeCoulomb interaction between the carriers by considering thecarrier transport in the
selfconsistent
potential governed bythe Poisson equation. We show that under the ballistic regime this interaction is crucial, and that noise characteristicsare strongly modiﬁed depending on whether the carrier correlation
mediated by the ﬁeld
is taken into account or not.To this purpose we consider a simple structure: a lightlydoped active region of a semiconductor device sandwichedbetween two heavily doped contacts injecting the carriersinto the active region. The device then acts similarly to avacuum diode, with a relevant difference in the fact thatthere are two opposing currents instead of a single current.Electrons are emitted from the contacts according to athermalequilibrium MaxwellBoltzmann distribution, andthey move ballistically inside the active region
the meanfree path is considered to be much larger than the distance
L
between the contacts
according to the semiclassical equations of motion. The ﬂuctuating emission rate at the contactsis taken to follow a Poisson statistics. This means that thetime between two consecutive electron emissions is generated according to the probability density
P
(
t
)
e
t
,where
12
n
c
v
th
S
is the injection rate, with
n
c
the electrondensity at the contact,
S
the cross sectional area of thedevice, and
v
th
2
k
B
T
/(
m
) the thermal velocity
T
is thelattice temperature,
k
B
Boltzmann constant, and
m
the electron effective mass
. The electron gas is assumed to be nondegenerate to exclude possible correlations due to the Fermistatistics. For simplicity, the conduction band of the semiconductor is considered to be spherically parabolic. Both thetimeaveraged current and the current ﬂuctuations inside theactive region of the device are analyzed for different biasvoltages applied between the contacts. The calculations areperformed by using an ensemble Monte Carlo simulator selfconsistently coupled with a Poisson solver
PS
. By usingthis approach we can analyze much more general situationsthan those studied in previous analytical calculations.
2
PHYSICAL REVIEW B 15 SEPTEMBER 1997IVOLUME 56, NUMBER 115601631829/97/56
11
/6424
4
/$10.00 6424 © 1997 The American Physical Society
For the calculations we used the following set of parameters:
T
300 K,
m
0.25
m
0
, dielectric constant
11.7
0
,sample length
L
2000 Å, and contact doping
n
c
rangingbetween 10
13
and 4
10
17
cm
3
always at least two ordersof magnitude higher than the sample doping
. However, wemust stress that the results we are going to present do notdepend on the particular values of these parameters, but onlyon the dimensionless length
L
/
L
Dc
, where
L
Dc
k
B
T
/
q
2
n
c
is the Debye length corresponding to thecarrier concentration at the contact.Let us discuss brieﬂy the steadystate spatial distributionsof the quantities of interest inside the sample. In a generalcase the carrier concentration is nonuniform, having maximum values at the contacts due to the electron injection anddecaying toward the middle of the sample. Accordingly,without an external voltage bias the potential distribution hasa minimum in the middle of the sample due to the spacecharge. When a positive voltage is applied to the anode, theminimum is displaced toward the cathode, while its amplitude tends to diminish. This minimum provides a potentialbarrier for the electrons moving between the contacts, so thata part of the electrons, not having enough energy to go overthe barrier, are reﬂected back to the contacts. The most important fact is that the transmission through the barrier is
current dependent
, which is crucial in calculating the noisecharacteristics. In the structure the current is limited by thespace charge and increases linearly with the applied voltageup to a certain value of the external bias when the barriervanishes, so that all the electrons emitted from the cathodecan reach the anode. Under the latter regime the current issaturated and becomes independent of the bias.
8
It is important to stress that in our approach we do not impose a ﬁxednumber of electrons
N
to be present inside the sample. Thevalue of
N
is determined by the emission rates of the contacts and the applied bias. Therefore,
N
ﬂuctuates in time andwe can evaluate both the timeaveraged value
N
and itsﬂuctuations by means of the Monte Carlo algorithm. One canobserve that
N
is constant at the increasing part of thecurrentvoltage characteristic, and it decreases with the biasonce the current is saturated.Under a ﬁxed applied voltage the current density in thestructure is given by
I
(
t
)
(
q
/
L
)
i
1
N
(
t
)
v
i
(
t
), where
v
i
(
t
) isthe instantaneous velocity component along the ﬁeld direction of the
i
th particle.
9
The current autocorrelation function
C
I
(
t
)
I
(
t
)
I
(
t
t
)
is evaluated from the sequence
I
(
t
) obtained from the Monte Carlo simulation, where thecurrent ﬂuctuation is given by
I
(
t
)
I
(
t
)
I
. To clarifythe role of different contributions to the current noise wedecompose the current autocorrelation function into threemain contributions
C
I
(
t
)
C
V
(
t
)
C
N
(
t
)
C
VN
(
t
) associated, respectively, with the ﬂuctuations in the mean velocityof electrons
C
V
, the ﬂuctuations in the carrier number
C
N
,and the velocitynumber cross correlation
C
VN
. The corresponding formulas are given by
9
C
V
t
q
2
L
2
N
2
v
t
v
t
t
,
1a
C
N
t
q
2
L
2
v
2
N
t
N
t
t
,
1b
C
VN
t
q
2
L
2
v
N
v
t
N
t
t
N
t
v
t
t
.
1c
Figure 1 shows the lowfrequency value of the spectraldensity of current ﬂuctuations
S
I
2
C
I
(
t
)
dt
normalizedto 2
qI
s
, where
I
s
q
12
qn
c
v
th
S
is the saturation current
notice that this is the maximum current that a contact mayprovide
. This normalization is performed in order to compare the results for different injectionrate densities
differentcontact dopings
. We provide the results for two differentsimulation schemes. The ﬁrst one involves a
dynamic
PS,which means that any ﬂuctuation of spacecharge appeareddue to the random injection from the contacts causes a redistribution of the potential, which is selfconsistently updatedby solving the Poisson equation at each time step during thesimulation to account for the ﬂuctuations associated withlongrange Coulomb interaction. In the second scheme weuse a
static
PS to calculate only the stationary potential proﬁle, and, once the steady state is reached, the PS is switchedoff, so that the carriers move in the
frozen
nonﬂuctuatingelectric ﬁeld proﬁle. We checked that both schemes giveexactly the same steadystate spatial distributions and totalcurrent, but the noise characteristics are different. Severalvalues of
n
c
and therefore several injectionrate densities
have been considered. As
n
c
increases, spacecharge effectsbecome more and more signiﬁcant, the dimensionless parameter
being the indicator of their importance.In the static case, by increasing the applied voltage
U
wealways obtain an excellent coincidence with the wellknownformula
10
used to describe the crossover from thermal to shotnoise when carrier correlation plays no role
represented inthe ﬁgure by dashed lines
:
FIG. 1. Currentnoise spectral density
S
I
vs applied voltage
U
calculated by using static
open symbols
and selfconsistent
closed symbols, solid line
potentials for several injectionrate densities
n
c
in cm
3
, with the corresponding
:
10
13
,
0.15;
2
10
15
,
2.18;
10
16
,
4.88;
2.5
10
16
,
7.72;
10
17
,
15.45;
4
10
17
,
30.9. The static caseis shown to be nicely described by Eq.
2
dashed line
. The dottedlines represent 2
qI
marked for each injectionrate density by thecorresponding symbol
.56 6425BRIEF REPORTS
S
I
2
q
I
I
2
qI
coth
qU
/2
k
B
T
,
2
where
I
I
I
is the total current ﬂowing through thediode, consisting of two opposing currents,
I
I
s
exp
qV
m
/
k
B
T
in the forwardbias direction and
I
I
s
exp
q
(
V
m
U
)/
k
B
T
in the opposite direction,
V
m
being the potential minimum induced by the spacecharge,which is dependent on
U
. This agreement supports the validity of the simulation scheme used for the calculations. For
qU
k
B
T
,
I
I
, thermal noise is dominant and
S
I
4
qI
s
exp
qV
m
/
k
B
T
. Therefore, for the lowest value of
when spacecharge is negligible and
V
m
→
0
,
S
I
→
4
qI
s
,while as
increases
V
m
becomes signiﬁcant and
S
I
decreases. When
qU
k
B
T
,
I
I
, the transition from thermal noise to shot noise takes place and
S
I
2
qI
. Finally,for the highest values of
U
, saturation occurs,
V
m
vanishes,and
S
I
2
qI
s
.For the lowest values of
no difference between the dynamic and static cases is obviously detected. However, forhigher
, when spacecharge effects become signiﬁcant, thepicture is drastically different for the dynamic case. Startingfrom
qU
k
B
T
the current noise, instead of increasing, decreases until the proximity of saturation. Under saturation,the results for both schemes coincide and full shot noise 2
qI
s
is recovered. When compared with the static case the noisesuppression is stronger for higher
more important spacecharge effects
.To understand the physical reason for the shotnoise suppression, in Fig. 2 we provide the decomposition of
S
I
, calculated with static and dynamic PS, into the additive contributions
S
V
,
S
N
, and
S
VN
Eqs.
1
for different appliedvoltages
U
and
7.72 (
n
c
2.5
10
16
cm
3
). The contributions of
S
N
and
S
VN
to the current noise vanish at equilibrium (
U
→
0), since they are proportional to
v
2
and
v
→
0. Thus for small biases (
qU
k
B
T
)
S
I
S
V
, whichmeans that the current noise is thermal noise associated withvelocity ﬂuctuations and is governed by the Nyquist theorem
S
I
4
k
B
TG
, with
G
dI
/
dV
V
0
the conductance. For thiscase the results for the static
Fig. 2
a
and dynamic
Fig.2
b
schemes evidently coincide. However, starting from
qU
k
B
T
the difference becomes drastic. For the dynamiccase the velocitynumber correlations, represented by
S
VN
,are negative, while for the static case they are positive. Furthermore, for the current ﬂuctuations calculated using theselfconsistent potential,
S
N
and
S
VN
are of opposite sign andcompensate for each other, so that
S
I
approximately follows
S
V
as long as the current is spacecharge limited. As a consequence, the current noise, which now corresponds to shotnoise, is considerably suppressed below the value 2
qI
givenby the static case. This result reﬂects the fact that as thecarriers move through the active region, the dynamic ﬂuctuations of the electric ﬁeld modulate the transmission throughthe potential minimum and smooth the current ﬂuctuationsimposed by the random injection at the contacts. Therefore,the coupling between number and velocity ﬂuctuations induced by the selfconsistent potential ﬂuctuations is mainlyresponsible, through
S
VN
, for the shotnoise suppression.This velocitynumber coupling becomes especially pronounced just before the current saturation (
U
7
k
B
T
/
q
),when the potential minimum is close to vanishing completely (
V
m
→
0), and the ﬂuctuations of the potential barriermodulate the transmission of the more populated states of theinjected carriers
the lowvelocity states
. Under saturation
FIG. 2. Decomposition of the spectral density of current ﬂuctuations
S
I
into velocity, number, and velocitynumber contributions vs applied voltage for the case
n
c
2.5
10
16
cm
3
,
7.72calculated by using
a
the static and
b
the dynamic Poissonsolver.FIG. 3. Shot noise reduction factor
vs voltage
U
for severalinjectionrate densities
n
c
different values of
.6426 56BRIEF REPORTS
conditions spacecharge effects do not modulate the randominjection
no potential minimum is present
, and again bothdynamic and static cases provide the same additive contributions and total noise (2
qI
s
).Finally, in Fig. 3 we present the reduction factor
deﬁned as the ratio between
S
I
as calculated with the dynamicPS (
S
I
d
) and, as given by Eq.
2
neglecting the inﬂuence of longrange Coulomb interaction,
S
I
d
/
2
qI
coth(
qU
/2
k
B
T
)
. In the context of our calculations
is more appropriate than the standard suppression factor
S
I
d
/(2
qI
), since it covers both the thermal and shotnoise range of applied voltages. Here it is observed how theshotnoise suppression becomes more pronounced as
increases. For example, for
30.9 it reaches 0.04. Thus ourselfconsistent approach predicts much lower values of thesuppression factor than the previous analytical model of vander Ziel and Bosman,
2
where the dependence of the potentialminimum and its position on the applied voltage was nottaken into account.In principle, the value of the parameter
in our modelhas no lower limit. We observe that it follows asymptoticallythe behavior
k
B
T
/
qU
in the range where shotnoise suppression is more pronounced
qU
k
B
T
,
U
U
sat
.
canreach a value as low as desired by appropriate increasing thesample length and/or the carrier concentration at the contact,provided the transport remains ballistic. However, with increasing device length
or lattice temperature
the carriertransport actually goes from the ballistic to the diffusive regime, and the shotnoise suppression is washed out. Therefore, the maximum suppression factor predicted by our calculations for a system with a given value of the mean freepath
p
would be obtained approximately at
max
p
/
L
Dc
.Moreover, when the carrier concentration at the contact isincreased so that the electron gas becomes degenerate, statistical
Pauli
correlations between the carriers appear, whichwill be additive
in the sense of shotnoise suppression
tothe Coulomb correlations.It should be emphasized that two essential conditions arenecessary for the strong shotnoise suppression due to longrange Coulomb interaction:
i
the presence of a potentialbarrier inside the device which controls the current, and
ii
the carrier transmission through the barrier should depend onthe current. This fact is quite general and, therefore, the results obtained in the present paper extend to much morephysical situations. For example, in a recent experiment byReznikov
et al.
6
the shotnoise level measured in a quantumpoint contact in the pinchedoff regime was found to be unexpectedly low
about onethird and less
. In that regime thetransport is controlled by the potential barrier present at thegates, and both conditions for the shotnoise suppressionmentioned above are fulﬁlled. Hence the results obtained inour calculations strongly support the suggestion of the authors of the experiment that the srcin of the discrepanciesbetween experimental results and theoretical predictions liesin the disregarding of Coulomb interaction between electronspassing through the contact. More precisely, the electronﬂow considerably modiﬁes the potential distribution insidethe contact, yielding the coupling of velocitynumber ﬂuctuations, and resulting in shotnoise suppression.In conclusion, we have investigated the inﬂuence of longrange Coulomb interaction on shotnoise suppression in ballistic transport by using an ensemble Monte Carlo simulatorselfconsistently coupled with a Poisson solver. We havefound that this suppression is stronger as spacecharge effects become more important, and it can be monitored by adimensionless parameter
. More than one order of magnitude of shotnoise suppression is predicted. The main contribution to the suppression is found to srcinate from thevelocitynumber correlations induced by the selfconsistentﬁeld.This work has been partially supported by the Comisio´nInterministerial de Ciencia y Tecnologı´a through Project No.TIC950652.
*
Present address: Dept. Fı´sica Fonamental, Universitat de Barcelona, Av. Diagonal 647, E08028 Barcelona, Spain.
1
See, e.g., recent review M. J. M. de Jong and C. W. J. Beenakker,condmat/9611140
unpublished
.
2
A. van der Ziel and G. Bosman, Phys. Status Solidi A
73
, K93
1982
.
3
D. O. North, RCA Rev.
4
, 441
1940
;
5
, 106
1941
.
4
H. Birk, M. J. M. de Jong, and C. Scho¨nenberger, Phys. Rev.Lett.
75
, 1610
1995
.
5
Y. P. Li, D. C. Tsui, J. J. Heremans, J. A. Simmons, and G. W.Weimann, Appl. Phys. Lett.
57
, 774
1990
.
6
M. Reznikov, M. Heiblum, H. Shtrikman, and D. Mahalu, Phys.Rev. Lett.
75
, 3340
1995
.
7
A. Kumar, L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne,Phys. Rev. Lett.
76
, 2778
1996
.
8
This behavior contrasts with the exponential and ChildLangmuir
I

V
dependences typical of vacuum diodes. It is due to the presence of the opposite current injected at the anode.
9
T. Gonza´lez and D. Pardo, J. Appl. Phys.
73
, 7453
1993
.
10
A. van der Ziel,
Noise in Solid State Devices and Circuits
Wiley,New York, 1986
.56 6427BRIEF REPORTS