Effect of long-range Coulomb interaction on shot-noise suppression in ballistic transport

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Effect of long-range Coulomb interaction on shot-noise suppression in ballistic transport
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  Effect of long-range Coulomb interaction on shot-noise 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, E-37008 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 shot-noise suppression due to long-range Coulomb interaction insemiconductor devices under ballistic transport conditions. An ensemble Monte Carlo simulator self-consistently coupled with a Poisson solver is used for the calculations. A wide range of injection-rate 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 space-charge effects in the structure.   S0163-1829  97  09735-X  The phenomenon of shot noise, associated with the ran-domness in the flux of carriers crossing the active region of adevice, has become a fundamental issue in the study of elec-tron transport through mesoscopic devices. In particular, thepossibility of shot-noise 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. How-ever, correlations between carriers can reduce the shot-noisevalue, giving     1. In real mesoscopic devices differenttypes of mechanisms resulting in shot-noise suppression canbe distinguished:   i   statistical correlations due to the Pauliexclusion principle   important for degenerate materials obey-ing Fermi statistics  ,   ii   short-range Coulomb interaction  electron-electron scattering  , and   iii   long-range Coulombinteraction   by means of the self-consistent electric poten-tial  . While the first two mechanisms have been extensivelydiscussed in solid-state literature, 1 the last one has receivedless attention, 2 although its role in shot-noise suppression hasbeen known for a long time in vacuum-tube devices. 3 Theonly exception that should be mentioned is the Coulombblockade in resonant-tunneling devices, which can be alsoreferred to as the last mechanism of suppression. The block-ade is provided by a built-in charge inside a quantum wellwhich redistributes the chemical potential, and prevents theincoming carriers from passing through the well, thereby re-sulting in carrier correlation and shot-noise suppression   seethe experimental evidence 4  . The Coulomb blockade is aconsequence of long-range Coulomb interaction, and it actsunder the  sequential  tunneling regime of carrier transport.The main objective of the present paper is to prove theimportance of long-range Coulomb interaction between thecarriers on the shot-noise 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 char-acteristic lengths of the order, or smaller, than the carriermean free path. The current existing theories invoked to in-terpret the experimentally observed shot-noise 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 long-rangeCoulomb interaction between the carriers by considering thecarrier transport in the  self-consistent   potential governed bythe Poisson equation. We show that under the ballistic re-gime this interaction is crucial, and that noise characteristicsare strongly modified depending on whether the carrier cor-relation   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 athermal-equilibrium Maxwell-Boltzmann 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 equa-tions 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 gener-ated 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 elec-tron effective mass  . The electron gas is assumed to be non-degenerate to exclude possible correlations due to the Fermistatistics. For simplicity, the conduction band of the semi-conductor is considered to be spherically parabolic. Both thetime-averaged 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 self-consistently 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 1997-IVOLUME 56, NUMBER 11560163-1829/97/56  11   /6424  4   /$10.00 6424 © 1997 The American Physical Society  For the calculations we used the following set of param-eters:  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 steady-state spatial distributionsof the quantities of interest inside the sample. In a generalcase the carrier concentration is nonuniform, having maxi-mum 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 ampli-tude 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 im-portant 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 impor-tant 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 con-tacts and the applied bias. Therefore,  N   fluctuates in time andwe can evaluate both the time-averaged value    N    and itsfluctuations by means of the Monte Carlo algorithm. One canobserve that    N    is constant at the increasing part of thecurrent-voltage 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 direc-tion 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  ) associ-ated, respectively, with the fluctuations in the mean velocityof electrons  C  V   , the fluctuations in the carrier number  C   N   ,and the velocity-number cross correlation  C  VN   . The corre-sponding 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 low-frequency 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 com-pare the results for different injection-rate densities   differentcontact dopings  . We provide the results for two differentsimulation schemes. The first one involves a  dynamic  PS,which means that any fluctuation of space-charge appeareddue to the random injection from the contacts causes a redis-tribution of the potential, which is self-consistently updatedby solving the Poisson equation at each time step during thesimulation to account for the fluctuations associated withlong-range Coulomb interaction. In the second scheme weuse a  static  PS to calculate only the stationary potential pro-file, 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 steady-state spatial distributions and totalcurrent, but the noise characteristics are different. Severalvalues of   n c   and therefore several injection-rate densities  have been considered. As  n c  increases, space-charge effectsbecome more and more significant, the dimensionless param-eter    being the indicator of their importance.In the static case, by increasing the applied voltage  U   wealways obtain an excellent coincidence with the well-knownformula 10 used to describe the crossover from thermal to shotnoise when carrier correlation plays no role   represented inthe figure by dashed lines  : FIG. 1. Current-noise spectral density  S   I   vs applied voltage  U  calculated by using static   open symbols   and self-consistent  closed symbols, solid line   potentials for several injection-rate den-sities  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 injection-rate 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 forward-bias 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 space-charge,which is dependent on  U  . This agreement supports the va-lidity 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 space-charge is negligible and  V  m → 0  ,  S   I  → 4 qI  s  ,while as    increases  V  m  becomes significant and  S   I   de-creases. When  qU   k   B T  ,  I     I   , the transition from ther-mal 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 dy-namic and static cases is obviously detected. However, forhigher   , when space-charge effects become significant, thepicture is drastically different for the dynamic case. Startingfrom  qU   k   B T   the current noise, instead of increasing, de-creases 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 space-charge effects  .To understand the physical reason for the shot-noise sup-pression, in Fig. 2 we provide the decomposition of   S   I   , cal-culated with static and dynamic PS, into the additive contri-butions  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 contri-butions of   S   N   and  S  VN   to the current noise vanish at equilib-rium ( 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 velocity-number correlations, represented by  S  VN   ,are negative, while for the static case they are positive. Fur-thermore, for the current fluctuations calculated using theself-consistent 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 space-charge limited. As a con-sequence, 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 fluctua-tions 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 in-duced by the self-consistent potential fluctuations is mainlyresponsible, through  S  VN   , for the shot-noise suppression.This velocity-number coupling becomes especially pro-nounced just before the current saturation ( U   7 k   B T   /  q ),when the potential minimum is close to vanishing com-pletely ( V  m → 0), and the fluctuations of the potential barriermodulate the transmission of the more populated states of theinjected carriers   the low-velocity states  . Under saturation FIG. 2. Decomposition of the spectral density of current fluc-tuations  S   I   into velocity, number, and velocity-number contribu-tions 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 severalinjection-rate densities  n c   different values of    .6426 56BRIEF REPORTS  conditions space-charge effects do not modulate the randominjection   no potential minimum is present  , and again bothdynamic and static cases provide the same additive contribu-tions and total noise (2 qI  s ).Finally, in Fig. 3 we present the reduction factor      de-fined as the ratio between  S   I   as calculated with the dynamicPS ( S   I   d  ) and, as given by Eq.   2   neglecting the in-fluence of long-range 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 shot-noise range of applied voltages. Here it is observed how theshot-noise suppression becomes more pronounced as    in-creases. For example, for    30.9 it reaches 0.04. Thus ourself-consistent 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 shot-noise sup-pression 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 in-creasing device length   or lattice temperature   the carriertransport actually goes from the ballistic to the diffusive re-gime, and the shot-noise suppression is washed out. There-fore, the maximum suppression factor predicted by our cal-culations 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, statis-tical   Pauli   correlations between the carriers appear, whichwill be additive   in the sense of shot-noise suppression   tothe Coulomb correlations.It should be emphasized that two essential conditions arenecessary for the strong shot-noise suppression due to long-range 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 re-sults obtained in the present paper extend to much morephysical situations. For example, in a recent experiment byReznikov  et al. 6 the shot-noise level measured in a quantumpoint contact in the pinched-off regime was found to be un-expectedly low   about one-third and less  . In that regime thetransport is controlled by the potential barrier present at thegates, and both conditions for the shot-noise suppressionmentioned above are fulfilled. Hence the results obtained inour calculations strongly support the suggestion of the au-thors 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 velocity-number fluc-tuations, and resulting in shot-noise suppression.In conclusion, we have investigated the influence of long-range Coulomb interaction on shot-noise suppression in bal-listic transport by using an ensemble Monte Carlo simulatorself-consistently coupled with a Poisson solver. We havefound that this suppression is stronger as space-charge ef-fects become more important, and it can be monitored by adimensionless parameter   . More than one order of magni-tude of shot-noise suppression is predicted. The main contri-bution to the suppression is found to srcinate from thevelocity-number correlations induced by the self-consistentfield.This work has been partially supported by the Comisio´nInterministerial de Ciencia y Tecnologı´a through Project No.TIC95-0652. * Present address: Dept. Fı´sica Fonamental, Universitat de Barce-lona, Av. Diagonal 647, E-08028 Barcelona, Spain. 1 See, e.g., recent review M. J. M. de Jong and C. W. J. Beenakker,cond-mat/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 Child-Langmuir  I  - V   dependences typical of vacuum diodes. It is due to the pres-ence 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
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