Invited Paper
Shot noise enhancement and suppression in single and multiple barrier diodes Reggiani L. a, Aleshkin V.Ya. b and Reklaitis A. c aNational Nanostructure Laboratory of INFM and Dipartimento di Ingegneria dell' Innovazione, Universita' di Leece, Via Arnesano sfn, 73100 Leece, Italy bDepartment Semiconductor Physics, Institute for Physics of Microstructures, 603950, Nizhny Novgorod, GSP-105 Russia c Semiconductor Physics Institute, Goshtauto 11, 2600 Vilnius, Lithuania ABSTRACT We survey a theoretical investigation of shot-noise in single and multiple barrier diodes. Several mechanisms responsible of the suppression of shot noise are rewieved together with the conditions for obtaining shot noise enhancement. The coherent versus sequential tunneling model for the double barrier resonant diode is discussed in the light of existing experiments. Keywords: coherent, heterostructure, mesoscopic, Monte Carlo, sequential shot-noise, tunnel 1. INTRODUCTION
Shot-noise is the electrical fluctuation due to the discreteness of the charge that provides direct information on the correlation of different current pulses crossing a device. As such, its determination is of basic complement to a current voltage (I-V) measurement. A convenient analysis of shot-noise is usually performed by introducing the dimensionless Fano factor 1 2:: 0 defined as 1 = SI(0)/(2qi), S1(0) being the spectral density of current fluctuations at low frequency, I the current flowing in the device and q the elementary quantum of charge determining I (For a recent review see Ref. [1]. In the absence of correlation between current pulses it is 1 = 1, and this case corresponds to full shot-noise. Deviations from this ideal case is a signature of existing correlations between different pulses and the two possibilities of suppressed (i.e. 1 < 1) and enhanced (i.e. 1 > 1) shot-noise are in principle possible, as confirmed by experiments.2- 9 This paper reviews recent theoretical investigations developed by the authors concerning shot noise suppression and enhancement in single and multiple barrier diodes made by semiconductor heterostructures. 1o-19 The results obtained by different approaches based on sequential and coherent tunneling model as well as Monte Carlo (MC) simulations will be discussed. The role played by Coulomb repulsion, Pauli principle and scattering mechanisms will be detailed. In particular, the effect of temperature will be addressed quantitatively. Theory provides a good qualitative and quantitative interpretation of available experiments and set the basis for new experimental confirmations.
2. SEQUENTIAL TUNNELING APPROACH In this approach we follow the theory developed in Refs. [20-22]. Accordingly the two terminal device is considered as a black box (the active region of the device) connected with two ideal contacts (emitter on the left and collector on the right) acting as ideal thermal reservoirs at given electrochemical potentials, see Fig. 1. From the contacts carriers are emitted with given distributions in time (typical Poissonian) and velocity (typically Maxwell-Boltzmann) and injected (extracted) to (from) the active region perpendicular to the surfaces limiting the active region itself as denoted 1 and 2 in Fig. 1. Under steady state, transport and noise are described in terms of the instaneous current I(t) flowing through the emitter (or the collector) interface which is determined by the probability P(N) of finding N(t) electrons in the device at time t. This probability obeys a master equation which is governed by four rates, denoted as g1,2, r1, 2 , describing the input or generation (output or phone +39.0832.320259, Fax +39.0832.320525 e-mail
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Noise and Information in Nanoelectronics, Sensors, and Standards, L. B. Kish, F. Green, G. Iannaccone, J . R. Vig, Editors, Proceedings of SPIE Vol. 5115 (2003) © 2003 SPIE · 0277-786X/03/$15.00
rl
...
E
.......
gl 1
g2
.........
~
DUT
c
r2
Jlo..
....
2
Figure 1. Schematic of the sequential approach with the emitter {E) and collector (C) contacts exchanging carriers with the DUT through the interfaces 1 and 2 by means of the corresponding generation (g) and recombination (r) rates. The fluctuations of the total number of carriers inside the DUT are controlled by the differential rates at the interface v1,2.
recombination) of a carrier from the contacts into (out) the active region of the device under test (DUT) (see Fig. 1) and providing the stationary current:
(1) From the above four rates, two differential rates, denoted as v1 , 2 , can be constructed. These describe the decay of the carrier number fluctuations in the DUT through the interfaces. We note that the 91 ,2, r 1,2 are positive definite quantities, while the v1 ,2 can be positive (damping of fluctuation) or negative (enhancement of fluctuation). The sum 11 = 111 + 112 can be interpreted as the rate of relaxation for number fluctuations, oN, of the corresponding Langevin equation. The spectral density of current fluctuations at low frequency takes the form: (2) The master equation is coupled with the Poisson equation to account for space charge reaction. Before applying the theory developed above to concrete cases of electron transport in barrier structures, let us analyze the general properties associated with the Fano factor expressed by Eqs (1) and (2) when the applied voltage is high enough to neglect 92 so that shot noise reginle is achievable: Thus, by defining a= 112/(111 + 112), the Fano factor is conveniently written as: (3) The whole scenario for the possible values of the Fano factor is illustrated in Fig. 2. Here, the plane v 1 , v2 is divided into the stability and instability regions according to the conditions 111 + 112 > 0, 111 + 112 $ 0, respectively. In the stability region, one region of suppressed shot noise and two regions of enhanced shot noise are shown for a given value of r2/9 1 < 1. For v2 > 0, the change of the current flow to the collector due to oN promotes the enhancement of the damping rate for this fluctuation, and shot noise is suppressed. For 112 < 0, the change of the current flow to the collector due to oN promotes the suppression of the damping rate for this fluctuation and shot noise is enhanced. Remarkably, we note that full shot noise (i.e. 1 = 1) is recovered in two cases. The former corresponds to the condition 112 = 0, when the current contains only the Poissonian contribution and 1 = 1. The latter corresponds to the condition r 1 ,2 = Nv1,2, when particles cross the device independently and 1 = 1. At the borders of the suppressed region, where a = 0 and a = r 2 / 9 1 there is full Poissonian noise and 1=1.
2.1. Single barrier heterostructure The physical system we analyze is a single barrier structure, as depicted in Fig. 3. By taking a one-dimensional x-space and a three dinlensional momentum space, it is assumed the presence of an applied voltage high enough so that carrier injection from the collector is negligible. To describe the electron transport through this structure we use the following analytical model which is supported by Monte Carlo calculations. 1 First we suppose that
°
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Figure 2. Lifetime plane of different transport regimes and shot-noise behaviors of a two terminal device with current controlled by number of particles in it under high voltages when g2 = 0. Io is the emitter injection current.
the distribution function of the particles injected into the structure from the left contact taken as an ideal thermal reservoir and acting as an emitter is of Maxwell type with the concentration and temperature independent of applied voltage. Second we assume that in the region between the emitter and the barrier there are two groups of particles. One group contains ballistic particles, which do not perform any scattering with the lattice. The other group contains thermalized particles which performed at least one scattering. The assumption of a distribution function composed by ballistic and thermalized particles is a reasonable one. Indeed, after a single optical phonon emission, a ballistic electron is converted into a thermalized one because it cannot return to the emitting contact, and thus it spends a relatively long time in the well when the transparency of the barrier is much smaller than unity. This means that most part of electrons which emit at least one phonon are converted to thermalized electrons. 2.1.1. Results
Here we report numerical calculations with the objective of providing a complete analysis of the electrical and noise properties of semiconductor single barrier heterostructures. The theory developed so far concerns with a model material of static dielectric constant "' and parabolic conduction band with effective mass m. In calculation we use everywhere values for m and "' corresponding to GaAs: m = 0.067mo, r;. = 12.9, mo is free electron mass, being this a material appropriate for an experimental validation of present results. The theory is based on five external parameters, respectively: the contact carrier concentration nb, the applied voltage U, the probability of non-scattering associated with the motion of ballistic particles from the emitter /3, the temperature T, and the length of the DUT L . In the following sub-section the I-V characteristics and the associated Fano factor are analyzed systematically for the case of a triangular barrier structure. Then, for the sake of completeness the next subsection will consider also the case of a barrier with constant transparency. The results so presented are intended to predict general trends and should be of interest to address an experimental verification. In some cases, they enable a direct interpretation of MC simulations. 2.1.2. Triangular barrier transparency
According to the general model presented here, in this section we consider the case of a tunneling described by a quasi classical triangular barrier. First we will discuss the I-V characteristics and then the noise behavior. The current voltage characteristics and the corresponding Fano factors for different values of the parameters n b, U, /3, T, L are presented in Figs. 4 to 6. From these figures one can see the role which is individually played 118
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L
0
X
Figure 3. Sketch of the band diagram of the single barrier structure considered here. Contact resistance at the terminals is neglected for simplicity.
by all the parameters. As a general trend, the current increases monotonically at increasing voltages to finally saturates at sufficiently high voltages. We note, that if we account for the contact resistance then there is no current saturation due to the voltage drop on the contact resistance. However, since we are interested in the operation mode of the active region of the device, contact resistance will be neglected here. In the increasing region, the current exhibits a strong super-Ohmic behavior due to tunneling processes. In the saturation region the value of the current equals that of the current injected from the emitter. The reason for such a value of current saturation is in the absence of any current flow to the emitter coming from both: {i) the reflection of ballistic particles by the barrier, and {ii) the thermalized particle. Indeed, because of the high voltages ballistic particles pass over the barrier, and thermalized particles remain confined in the potential well just before the barrier. The value of the voltage corresponding to the onset for current saturation rises with the increase of nb, U, /3 and L, and with the decrease ofT as shown in Figs. 4 to 6. In the current saturation region, the ratio of the ballistic current to the total current is independent of nb and it is determined only by /3. From Figs. 4 to 6 it is clear that the increase of nb, U, /3, Lor the decrease ofT leads to the increase of the maximum value of the differential conductance di/dV towards current voltage characteristics of S-type as reported in Fig. 5 for L = 2000 A. The dependence of the Fano factor on voltage is reported together with the I-V characteristics in Figs. 4 to 6 where one can see that 'Y depends on voltage for a large variety of parameters and exhibits minima and maxima. In all cases, at the lowest and highest voltages {above current saturation) 'Y = 1. At the lowest applied voltages, when qu «: U, the damping of §N is provided by the induced increase of the flow of thermalized particles to the emitter. Since the back flow to the emitter is much more important than the flow to the collector it is 111 » 112 and 'Y = 1. At the highest applied voltages, in the saturation current region, the damping of 8N is provided by the corresponding change of the flow of thermalized particles to the collector. Accordingly, 112 » 111 and 91 = r2, because the flow to the emitter is absent, and therefore again it is 'Y = 1. In the intermediate region of voltages for the considered values of injection concentration 1017 :5 nb :5 2 x 1018 em - 3 'Y exhibits both a maximum, corresponding to shot noise enhancement, followed by a minimum corresponding to shot noise suppression. The maximum value of the Fano factor exhibits a dramatic increase with increasing nb while the minima remain practically the same with a value around 0.5. Deviations from full shot noise are interpreted in terms of the voltage dependence of the lifetimes which in turn control the value of a in Eq. (3). To underst and the reason for the enhanced shot noise we note that such an enhancement is related to v2 < 0. In the presence of high enough voltages, just before current saturation, ballistic particles pass over the barrier and this positive Proc. of SPIE Vol. 5115
119
1.00 -----· n0=1011
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Fig. 3 for different (3 (a) and for different T (b).
feedback is washed out. Thus, shot noise is suppressed as evidenced by the minima of 'Y in Figs. 4 and 5. From Figs. 4 and 5 one can see that the increase of nb, U, {3, Lor the decrease ofT leads to a remarkable increase of the maximum value of 'Y and also of the maximum of the differential conductivity, i.e. to an increase of the positive feedback. The general trends are that an increase of nb and U increases the particle accumulation near the barrier and thus Coulomb correlation. The increase of {3 increases the role of ballistic particles in the current and thus enforces the effect of shot noise enhancement.
2.1.3. Constant barrier transparency In the framework of the present model, it is interesting to analyze the situation, when the barrier transparency is independent of energy (this condition can be realized by transparency engineering). To this purpose, in Fig. 6 we report the I-V characteristic and 'Y for the structure with nb = 5 x 1017 cm-3 , L = 600 A, T = 300 K, {3 = 0.5, and D = 0.01, 0.05, 0.5. Here one can see that under high enough voltages the current saturates, but at values smaller than those of the injected current. The reason for such a saturation is the absence of the flow of thermalized particles to the emitter. In particular, the value of the current of ballistic particles reflected from the barrier to the emitter Is is given by:
{4} with A the cross sectional area of the device and Vt the value of the injected thermal velocity. Thus Is is increasing with both: the increase of the barrier transparency D and/or the decrease of {3. It is important to remark, that contrary to the case of the triangular barrier structures, in the saturation region g1 fr 2 = 1/[1- {32 {1- D)] > 1. 120
Proc. of SPIE Vol. 5115
1.00
- - ---· L=300 A --L=600A ·········L=2000A
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Accordingly, in the current saturation region 'Y takes the form: 1 + {J2(1- D) .82 (1- D)
'Y = 1 -
(5)
~ 1
For these kind of barriers the current saturation region is placed in the second region of enhanced noise of Fig. 2, where 111 :;?: 0. The reason for noise enhancement here is the dominant role played in the current noise by the third term in the right hand side of Eq. {3) which is connected with number fluctuations. We found that the value of 'Y increases in this region with the increase of .B and the decrease of D.
2.1.4. Monte Carlo results The realistic modelling of enhanced shot noise was provided by MC simulations performed on a single barrier diode consisting of: a GaAs emitter (n-doped with nb = 5 x 1018 cm- 3 ), a GaAs well (50 nm long and ndoped with n = 1016 cm- 3 ), an Al(0.25)Ga(0.75)As barrier (50 nm long and undoped), a GaAs layer {10 nm long and n-doped with n = 1016 cm-3 ), and a GaAs collector (n-doped with nb = 5 x 1018 cm-3 ). The doping of the emitter/collector contacts is thus sufficiently high to provide space-charge effects always remaining below degenerate conditions. The n-GaAs and n-AlGaAs material parameters are summarized in Table I for the sake of completeness. We remark the essential importance of inelastic scattering processes which, through optical phonon emission, are responsible for the confinement of carriers in the potential well preceding the barrier. Such a confinement is important to achieve a carrier density high enough to sustain a significant space charge effect necessary for the the charge fluctuations resulting in a positive feedback. The maximum current I 8 (saturation current) that a contact can provide is given by Is/A= (1/2)lqlncvo = 8.59 x 106 Afcm 2, where vo = (KT/2-;rm) 112 = 2.12 x 107 emfsis the average velocity of injected electrons. Here, Pauli exclusion principle as well as bot-phonon effects are neglected. Figure 7 (lp) reports the I- U characteristic of the device (continuous curve and left scale) together with the voltage dependence of the transmission coefficient D averaged over the carrier distribution function10 (dashed curve and right scale). In the whole range of voltages considered here the device exhibits a strong super Ohmic characteristic because of a transport controlled by tunneling. In any case, we have not observed bistability phenomena and thus noS-type negative differential conductance (NDC) is present. The current density at the highest voltage considered of U = 2.5 Vis 1.5 x 106 Ajcm2 , i.e., the saturation is not reached due to scatterings and electron reflection from the heterobarriers. Figure 7 (rp) reports the Fano factor as a function of the applied voltage in the range of values of the I - U characteristics of Fig. 7(lp). For the purposes of analyzing quantitatively the effects of the space-charge on Proc. of SPIE Vol. 5115
121
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tunneling, calculations performed with a dynamic (static) Poisson solver are reported as continuous and circles (dashed and squares) curves. For the dynamic case, we have found that at low voltages (0.7 < U < 1 V), being D « 1, very few carriers can tunnel the barrier thus the tunneling space-charge feedback is negligible and 'Y-+ 1 as expected by simple partition noise. We remark that simulations are stopped below U = 0.5 V because the extremely small value of the average transmission coefficient (D ~ w- 4 ) makes the noise computation no longer affordable due to the very small number of carriers in the active region of the structure. Thus, the calculations do not reproduce the thermal Nyquist value. At intermediate voltages (1 < U < 2 V) the positive feedback between space-charge and tunneling is active and found to provide a systematic enhancement of shot-noise with a maximum of about a factor of 7 at a bias of 1.5 V. Here carrier confinement in the well is ensured by the fact that the inelastic scattering length is comparable with the width of the potential well. At the highest volt ages (U > 2 V) the barrier becomes transparent and full shot-noise behavior is recovered because of the Poissonian statistics used in the modelling of the injecting contacts and of the efficiency of scattering mechanisms. Indeed, in the GaAs layer between the emitter and the barrier most of electrons reaches the barrier ballistically. However, a small fraction of them emits an optical phonon and remains confined at the hetero-interface. In the barrier region, the electric field is very high (of the order of 300 k V/em). Therefore, here most of the electrons are transferred to the upper X and L valleys. Thus, electrons suffer a very strong int ervalley scattering in the barrier region. When a static Poisson solver is considered neither the barrier height nor the charge concentration are allowed to fluctuate, which corresponds to neglect completely space-charge effects and thus the feedback between space charge and tunneling. Accordingly, as shown by the dashed curve in Fig. 7(rp), nearly full shot-noise is observed in the whole range of voltages, apart from an enhancement at the smallest voltages due to the smooth cross-over between shot and Nyquist noise.
2.2. Double barrier heterostructure The structure here investigated is the standard symmetric double well reported in Fig. 8. As typical values we take the thickness of each barrier d = 100 A, the energy of the resonant level as measured from the center of the potential well cr = 50 meV with the partial width of the resonant level due to the tunnelling through the left and right barriers r = rL + rR = 5 meV. We consider the case of sequential tunneling when there is only one resonant state and we assume that the resonant tunneling diode has unit square contacts. Different temperatures are considered ofT= 4.2 and 77 Kanda carrier concentration of n = 5 x 1016 cm-3 in the emitter and collector regions. 122
Proc. of SPIE Vol. 5115
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o'=".s-~--: 1) we find two separate regions of shot noise suppression matched by a region of full shot noise, and one region of shot noise enhancement. Both suppression minima occur in the positive differential conductance (PDC) region and reach a minimum value "' = 0.5, evidencing the onset of a repulsive correlation between different current pulses. Enhanced shot noise occurs in the bistable region. Strictly speaking, near to the borders of the bistable region the Fano factor goes to infinity, as expected. To provide a microscopic interpretation of the voltage dependence of the Fano factor, in Figs. 9{rp) we present a detailed analysis of the different parameters entering the definition of"'· Figure 9 {rpa) reports the relevant dimensionless parameter a and the normalized current Ij(qg1) vs the applied voltage. As it follows from Fig. 2, we confirmed the scenario of the shot noise behavior with 'Y < 1 when a< If(qg1) and with 'Y > 1 when a> If(qg1). Note that almost at all voltages, except when qV/kBT < 1, Ij(qg1) = 1. It means that the back flow from the resonant state to the emitter is absent. This is a consequence of the degeneracy condition (i.e. Pauli blockade) assumed here, which implies that the total current equals the injected one. Figure 9 (rpb) reports the dependencies of v1 ,2 on voltage. We can see that v1 exhibits a complicate structure at increasing voltages with two positive peaks in the PDC region and one negative peak in the bistable region. By contrast, v2 is constant at almost all voltages since sooner or later tunneling from the resonant state to the collector will occur. At the smallest voltages, V2 decreases due to Pauli principle. At increasing voltages, the condition v1 = 112 gives"'= 0.5 in analogy with the general model of two equal resistors with shot noise sources in series.23 From Fig. 9 (rpb) we argue that all peculiarities of the Fano factor are controlled in essence by the behavior of v1. In the context of the scenario reported in Fig. 2. we remark, that while in the single barrier structure the instability is associated with an S-type I-V characteristic and connected with a negative value of v2, here it is associated with a Z-type I-V characteristic and due to a negative value of v 1 • The case of a high temperature of 77 K is reported in Figs. 10. Here we assist to a vanishing of the effects Proc.ofSPIEVol.5115
123
L
Figure 8. Sketch of the band profile of the double barrier structure considered here under typical operation conditions. Here F., FQw, Fe are the electrochemical potential in the emitter, the quantum well, and the collector regions, respectively. associated with Pauli exclusion principle, thus only Coulomb correlation is of major significance. The I - V characteristics lose the Z-type shape in favor of the standard N-type NDC, see Fig. 10(alp). Furthermore, the contribution of r 1 to the total current becomes of significance. The Fano factor exhibits a suppressed behavior in the PDC region, and an enhanced behavior in the NDC region. Both, suppression and enhanced effects are found of decreasing amplitudes with respect to the lower temperature case of 4.2 K seen before. The significant role played by the black flow from the resonant state to the emitter (r1 contribution in Fig. 10 (alp)) limits the suppression value to a value of about 0.7. Again, the behavior of the Fano factor at voltages above kBT/q is essentially controlled by the behavior of a as reported in Fig. 10 (arp). In particular, enhanced shot noise remains associated with the negativity of v1 as, reported in Fig. 10 (brp). Here, Coulomb correlation is negative (i.e. of repulsion character) in the PDC region, while it provides positive feedback in the NDC region as evidenced by the negative values taken by v1 (see Fig. 10 (brp)). Similar results, which confirm the trend of different transport parameters, have been obtained forT= 300 K.
3. COHERENT TUNNELING APPROACH = rL + rR where For simplicity, we consider the case of coherent tunneling when there is only one resonant state with rL = rR = r /2 and we take unit square contacts. The double barrier transparency D(cz) is written in the standard form: The structure here investigated is the same double well reported in Fig. 8. We denote by
r
r L,R are the partial widths due to the tunneling through left and right barrier respectively. r2 D(cz)
= (c.,- cr +4 qu) 2 + 4r2
(6)
where cz,.l are the energies for electron motion perpendicular and along barriers. For the current flowing from the emitter to the collector we found: qm I=- 2rr2h3
with 124
roo lo dc.zdc.LD(c..)[h (c)- !R (c)]
h,R the electron distribution in the emitter (L) and collector (R) contacts, respectively.
Proc. of SPIE Vol. 5115
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