IEEE TRANSACTIONS ON ELECTRON DEVICES. VOL. 38, NO. 2 , FEBRUARY 1991

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A Theory of Shot Noise in Quantum Wells and Applications in Resonant Tunneling Heterojunction Bipolar Transistors

Jun Han, Student Member, IEEE, and Frank S. Barnes, Fellow, IEEE

Abstract-Theoretical calculations of shot noise are carried out on a double-barrier, one-well structure. The results show that the noise power is a function of the energy bandwidth of the transmitted elec- trons and that the shot noise may be significantly reduced by the quan- tum wells. These results can be applied to heterojunction bipolar transistors which contain a quantum well, and it is shown that these resonant tunneling heterojunction bipolar transistors (RTHBT’s) should have a lower noise figure than homojunction transistors.

I . INTRODUCTION OISE characteristics of devices are important in determin- N ing the limiting characteristics of circuits which can be built

from them. In semiconductor devices there are multiple sources of noise which contribute to the fluctuation in the current. They include Johnson noise, generation-recombination noise, parti- tion noise, l/f noise, and shot noise [l]. The shot noise results from the discreteness of the change on electrons and the fact that the current flow consists of individual pulses where the number and velocity of the carriers vary randomly. Quantum wells restrict the energy distribution of the electrons to narrow bands and thus reduce the uncertainty. A reduction in the shot noise has been observed at low frequencies for both the case of an asymmetric, double-barrier quantum well 121 and for the case of current flow through a quantum point [3]. We believe that this reduction in randomness can in tum be used to improve quantum-well device characteristics, just as the bandgap differ- ence in heterojunction has allowed us to improve the character- istics of bipolar transistors. In this paper we will present some semiclassical arguments to show that the noise associated with the current through a resonant tunnel (RT) diode is reduced by the feedback through the modulation of the barriers by the space charge in the well and at high frequencies by the reduction in the velocity spread. As an application, heterojunction-bipolar transistors containing an RT structure are proposed as lower noise amplifiers than ordinary homojunction bipolar transistors.

11. ANALYTICAL DISCUSSION We begin by observing the current-density expression in a

one-type-camer-dominated semiconductor

J = enu (1)

Manuscript received March 14, 1990; revised September 20, 1990. The work on this project supported by the National Science Foundation under Grant 1536929. The review of this paper was arranged by the Editor R. P. Jindal.

J . Han was with the Electrical and Computer Engineering Department, University of Colorado, Boulder, CO 80309-0425. She is now at 3471 Tnpp Drive, no. 6, Reno, NV 89512.

F. S. Barnes is with the Electrical and Computer Engineering Depart- ment, University of Colorado, Boulder, CO 80309-0425.

IEEE Log Number 9040642.

where e is the charge on an electron, n is the density of the carriers, and u is the velocity of the carrier in the direction of the current flow, Therefore, the reduction in the fluctuations of both n and U affects the overall current fluctuation. Shot noise is usually analyzed using a one-dimensional diffusion model. However, this approach is not appropriate for quantum-well structures. Our approach will be to assume distributions for both the velocity and the density of carriers incident on the quantum- well structure and then to use the results of quantum mechanics for describing the transport through the well.

The variation in the density of carriers is interpreted as the emission fluctuation of the source for the carriers incident on the quantum-well structure. The velocity fluctuations are as fundamental as the emission fluctuation in contributing to the shot noise even though they do not contribute quantitatively in the same way and are ignored in most analyses at low fre- quency. For example, the velocity fluctuation can be ignored in a p-n junction when the maximum frequency of interest is much less than the reciprocal of the transit time for the carriers. The analysis can be approached in either the time or frequency do- main with each approach giving somewhat different physical insight.

111. A TIME DOMAIN APPROACH

A simple analysis in the time domain will give a straightfor- ward physical picture of the contribution to shot noise from the velocity distribution. As illustrated in Fig. l(a), if a pulse of length T is emitted from a cathode, the number of electrons NT in the pulse obeys a Poisson distribution [4]

NT

N T ! .

If NT is the_average number of electrons emitted in time in- terval T , X = N T / T is the average emission rate. p ( t ) is defined as the time distribution of electrons arriving at the anode. The number of electrons which arrive in the time interval T about to at the anode is given by

From Fig. I(b), we can see that the wider the distribution p ( f ) , the fewer the number of electrons which arrive in the in- terval T centered about to. There are two sources of fluctuations. The first is in the number of electrons emitted and the second in the number arriving within the interval 7.

N ( 7) can also be expressed in terms of velocity distribution

0018-9383/91/0200-0237$01.00 0 1991 IEEE

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(a) (b) Fig. 1 . Electron numbers versus time. (a) At the cathode. (b) At the an

ode.

P ( V )

1% + A v / 2

N ( T ) = 5 N T p ( v ) d V (4) vg- A v / 2

where A V = v O ~ / t O + T and uO = d / t o for the average velocity. In the case with no accelerating voltage, the velocity distri-

bution is

PO(V) d v = D ( E ) f ( E ) du

where c is a normalization constant and E , is the Fermi energy level; v is related to E by E = $mu2.

In quantum-well structures, the number of electrons contrib- uting to the current corresponds to a narrow-band transmission function prt ( U ) because of resonant tunneling. Qualitatively, only electrons in certain energy ranges, and thus in certain ve- locity ranges, can tunnel through the barriers and contribute to the current

If p , ( uo) is maximized, which can be realized by adjusting the resonant level to match the emitter's most probable velocity, p , ( U,,) approaches 1 at U,, and decays very rapidly when v is detuned from U,,. In other words, p ( U ) is strongly concentrated around vO in this case.

If the emission process has a Poisson distribution and the ve- locity filter selects a subset of those which are emitted in the velocity range A U with a probability P (A U )

( 7 )

where v I and U* are the limits of the velocity filter. The result- ing probability distribution that electrons will be emitted within A U is also a Poisson process such that

-XP(Au)T] ( X P ( A V ) T ) ~ ( ~ )

The average number of carriers crossing the double barrier is N ( T ) = h P ( A v ) T a n d the variance U' = h P ( A v ) T . If we - take the signal power in the received pulse as proportional to N ( T ) ~ and the noise power proportional to U', the signal-to-

'RANSACTIONS ON ELECTRON DEVICES, VOL. 38, NO. 2, FEBRUARY 1991

noise ratio is given by

( 9 ) S - = XP(AV)T. N

Under normal operating conditions the signal-to-noise require- ments are fixed by the bit-error rate required for system opera- tion. Thus the reduction in the number of carriers received either by reducing the velocity pass-band or decreasing the pulse can be compensated by increasing the emission rate h so that the signal-to-noise ratio stays constant.

To get an estimate of when the velocity fluctuation becomes important, we define an average velocity. If the velocity is as- sumed to follow a one-sided, one-dimensional Maxwell (1SlDM) distribution by providing that a negative velocity can- not form a current pulse, the average velocity is the square root of the second moment. The transit time t,, is the distance di- vided by the average velocity at which an average electron passes through quantum wells or a transistor base. If we ignore the effects of mass variation in quantum structures and take the effective mass as 0.067 m, for the electrons in the conduction band, where mo is the free-electron mass, then t,, is estimated to be 4.0 X s for a 1.0-pm-wide structure at 300 K. A longer transit time for a quantum-well structure may be calcu- lated by taking into account the partial reflection of the electron wave function at each potential discontinuity, as is shown in [SI. Even if the transit time is modified for reasonable reflectiv- ity, it is still short compared to variations of the signal field at low frequencies. Therefore, the transit time can still be treated as a 6 function and the velocity fluctuation can be ignored.

IV. ANALYSIS OF SHOT NOISE IN THE FREQUENCY DOMAIN

Assume that there are N electrons emitted in a time interval T. Denoting the velocity distribution of these electrons a s p ( U ) ,

the number of electrons with noise velocity v will be

Consider first the randomness caused by fluctuations in the number of electrons at a given time t . The current pulses which consist of electrons at velocity v are [6]

N,.

i,, = C L,( t - t i ) . (11) i = I

Let f, ( t ) be the current pulse of an electron emitted at time t = 0, then

where t,, is the transit time corresponding to U . The Fourier transform F, ( w ) off, ( t ) is as follows:

F,,(w) = 'jm L , ( t ) exp ( - i w t ) dr 2~ - m

ei -- - [exp (-iwt,,) - 11 2 ~ w t , ,

HAN AND BARNES: A THEORY OF SHOT NOISE IN QUANTUM WELLS 239

and so (19)

S, ( f ) TR is negligible when T, becomes very small. A simple approximation to evaluate the effect of Tg on S, is to use the half-energy-width A E , which is defined by +mu; = A E , and T,( U , ) = iT,,, , . By assuming that the tunneling probability is zero outside 2 A E and unity inside 2AE, the fluctuation is reduced from a range 0 to 03 to 2v , about vo. vo is the velocity which corresponds to a resonant energy level.

uO+U“I

S ( f ) = S ” ( f ) dv ( 2 0 ) U0 - um

A resonant level in a quantum well works like a velocity fil- ter. If the resonant energy is located significantly above the band edge, and is large compared to the energy spread, then the fil- tered velocity distribution can be treated as a 6 function. The shot noise is given by

Shot noise can be interpreted as an equivalent current gen- erator in equivalent circuits

7: = Su( f ) dv A f (15)

where w = 27rf, and the noise power spectrum density SI,( f ) is [6]

I,, denotes the current pulses whose velocities are v and have the relationship I z , = e N I , / T .

This value for S,, ( f ) dv assumes that each of the f ( t ) are the same and are independent of each other. This will not be true in the case where the space charge in the quantum well modifies the transit time through the well.

Now consider the randomness due to different initial veloci- ties. The total current consists of the electron pulses at all pos- sible initial velocities. Therefore, the total power-spectrum density is the sum of SI,( f ) over all the possible U .

For a potential profile, such as in a p-n junction, there is no restriction on the maximum positive velocity of the carriers. All of these carriers contribute to the current, and the difference among them is only their transit time. The relevant p ( U ) is the product of the Fermi distribution and the density of state. The expression of p ( U ) is shown in (5). Thus

sin‘

The average transit time tav is much less than the field variation at low frequencies.

sin2 ( wtav /2 )

( Wtav / 2 >’ approaches to unity, because ft << 1, or wt << 1. Thus we get

= 2el. (18) This coincides with the well-known shot-noise expression of a p-n junction for frequency much less than the transit time.

The velocity fluctuations of electrons contributing to the cur- rent through an RT structure are, however, limited by the tun- neling probability Tg ( U ) , which decays very quickly when it is away from an eigenstate. Therefore, the noise power generated by electrons at each velocity is weighted by the tunneling coef- ficient, i.e.,

where t , is the transit time corresponding to the tunneling en- ergy level. The numerical results of noise power versus fre- quency are plotted in Fig. 2 . The transit time for the resonant tunneling electrons have been carried out by several research groups [7], [8]. In our calculations, we use the data from the paper of Anwar et al. [8] for a symmetric well with barrier thickness L and well width d.

When the transit time becomes larger, the reduced noise fre- quency becomes lower. This is reasonable since the reciprocal of the frequency approaches the larger transit time at relatively low frequencies.

The practical merit of reduced shot noise at high frequencies is also dependent on the rate at which the current falls off at high frequencies. Payne [9] predicted that the resonant current is proportional to

1 / 1 \

where 7’ is a characteristic time, and that there will not be a fall-off in the current until the frequencies are well above the reciprocal of 7. This is consistent with the experimental results done by Sollner’s [IO] group of measuring tunneling nonlinear- ities in terahertz region, but contrary to our intuition, where we would expect the signal and noise to decrease at the same rate.

Thus far we have treated the double-well barrier simply as a velocity filter which introduces a delay, with no improvement in the low-frequency signal-to-noise ratio. To realize a low-fre- quency reduction in the shot noise below

we need to modify the Poisson process either by reducing the effective size of the charge e or by providing negative feedback so that the emissions of charges from the quantum well are not independent events.

Negative feedback to reduce the shot noise may be obtained through the space charge in the quantum well. The space charge in the quantum well modifies the potential drop across each of the barriers and thus their transmission coefficient. Consider the case of the transmission of an electron through the barriers shown in Fig. 3 (from [5]). The transmission probability for an electron across the two barriers is given by

240 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38. NO. 2. FEBRUARY 1991

2 0 4 /i Frequency L o g ( f )

Fig. 2. Shot noise spectral density versus frequency for different transit times. f, -= 11749.0 ps when L = 100 A, d = 60 A; r2 = 716.0 ps when L = 50 A , d = 60 A; r3 = 17.93 ps when L = 50 p\, d = 40 A .

t ENERGY

across the first barrier into the well reduces the potential drop across it and increases the potential drop across the second bar- rier. Since the phase-electron wavefunctions in the wells are incoherently scattered, the overall transmission through the bar- riers can be treated as a sequential process, which is given in 11 11.

If the barriers are symmetric, the changes in the two resistors cancel. However, if the barriers are asymmetric, then the changes in the two barrier heights are different with

where AQ is the change in the well space charge, and the ca- pacitances C, and C2 are given by

Then

( 2 8 ) AE, - A E , - dl d2 .

If we take the case where d , > d2, then most of the potential change due to the emission of a change from the well across the second barrier will occur across that barrier and will decrease A E, and thus T,.. Only a much smaller change will occur in T, . To the first order, the change in transmission coefficient is lin- ear in the electric field, as shown experimentally in [ 1 2 ] . This means that the probability for emission goes from a Poisson distribution to a Markov process, where the emission of a charge is dependent on the number of charges in the well. This is sim- ilar to the smoothing of the noise in a space-charge-limited vac- uum diode where the depth of the retarding potential is controlled by the number of charges emitted by the cathode. This problem has been solved in [13 ] with the net result that

t ENERGY

- ii = 2eIAfc

(b) Fig. 3 . (a) Symmetrical double barrier (no field applied). (b) Double bar-

rier with applied field (no longer symmetrical).

where h is Planck's constant divided by 2a, m is the effective mass of the electron, the ''s are the heights of the potential as shown in Fig. 3 , e is the charge on the electron, and Fb is the electrical field across the barrier. The transport of an electron

where c is less than one and a complex function of the height of the barrier and the thermal energy of the electrons emitted from the cathode. We expect a similar result for the exponential variation in the tunneling probability with the potential across each barrier.

V. NOISE ANALYSIS OF AN RTHBT The proposed low-noise device is an HBT with a one-well,

two-barrier resonant structure inserted in the base region (RTHBT). The energy-band diagram is illustrated in Fig. 4. The noise figure F is used here to evaluate the noise perfor- mance of the device. F is the ratio signal-to-noise gain at output to input. The equivalent circuit of the RTHBT is illustrated in Fig. 5. The equivalent noise-voltage generators are as follows [ 1 4 ] :

Z: = 2kTr,Af (30)

Z i = 4kTrbAf (31 1 (32) Z: = 2 e c ( T , f ) l , ( l - CY) 1ZCl2Af.

Therefore

4kTR,a2 ( 3 3 )

HAN AND BARNES: A THEORY OF SHOT NOISE IN QUANTUM WELLS 24 I

A E B AE-zEzl

(b) Fig. 4. (a) Energy distribution of conduction-band carriers in the emitter.

(b) Energy-band diagram of RTHBT’s.

Fig. 5 . Equivalent circuit of RTHBT’s.

where a is the ratio of Z, to I, , a. is the value at low frequency. re, r,, and R, are the resistance of the emitter, base, and source load, respectively. Z , is the impedance andf, is the cutoff fre- quency.

There are three terms of the RTHBT with F less than that of a homojunction transistor. First, c ( T , f ) is less than 1 in the fourth term because the collector current has only a reduced shot noise instead of full shot noise. Second, the emitter effi- ciency and corresponding a are higher in HBT’s than in ho- mojunction devices. Third, the wide-bandgap emitter makes it possible to reduce the base resistance by increasing the base doping. Consequently, the second term of the noise figure is reduced. This is possible because the increased base doping will not decrease the injection coefficient significantly in an HBT. tn addition, the speed of a transistor can be enhanced with a heavily doped base by reducing the RC time constants [15]. Measurements showed that 1 /f noise is the dominating low- frequency noise [ 161.

Further improvement of HBT performance can be expected by inserting a quantum-well structure between the emitter and base to form a prewell. The prewell will confine the emitter electrons to a narrow energy range and increase the density of current transmitting through the double-barrier quantum well. A very sharp turn-on and saturation I-V curve is expected as the emitter-base voltage is varied, which should result in the reduction of re . A detailed theoretical analysis of the structure is in progress and some devices are being fabricated.

ACKNOWLEDGMENT The authors wish to express their appreciation to E. E. Gard-

ner, W. Eidelloth, H. D. Wu, L. Scharf, and S. Kwok for help- ful discussions. They also wish to thank R. P. Jindal for his interest and helpful suggestions.

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structures below I O kHz,” Phys. Rev. B , vol. 41, pp. 8388-8391, 1990.

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[4] W. Bennett, “Methods of solving noise problems,” Proc. IRE,

[5] B. Ricco and M. Azbel, “Physics of resonant tunneling. The one- dimensional double-barrier case,” Phys. Rev. B , vol. 29, pp.

[6] A. Yariv, Optical Electronics, 3rd ed. New York, NY: Holt. Rinehart and Winston, 1985, ch. 10, pp. 306-341.

[7] K. Jensen and F. Buot, “Numerical calculation of partical trajec- tories and tunneling times for resonant tunneling barrier struc- tures,” Appl. Phys. Lett . , vol. 5 5 , pp. 669-671, 1989.

[8] A. Anwar, A. Khondker, and M. Khan, “Calculation of the transversal time in resonant tunneling devices,” J . Appl. Phys., vol. 65, no. 7, pp. 2761-2765, 1989.

[9] M. Payne, “Transfer hamiltonian description of resonant tunnel- ing,” J . Phys. C: Solid State Phys., vol. 19, pp. 1145-1155, 1986.

[IO] T. Sollner, W. Goodhue, P. Tannenwald, C . Parker, andD. Peck, “Resonant tunneling through quantum wells at frequencies up to 2.5 TH,,” Appl. Phys. Lett . , vol. 43, pp. 588-590, 1983.

[ I l l M. Jonson and A. Grincwajg, “Effect of inelastic scattering on resonant and sequential tunneling in double barrier heterostruc- tures,” Appl. Phys. Lett . , vol. 51, pp. 1729-1731, 1987.

1121 T. Norris, X . Song, W. Schaff, L. Eastman, G. Wicks, and G . Mourou, “Tunneling escape time of electrons from a quantum well under the influence of an electric field,” Appl. Phys. Lett . ,

[13] D. North, “Fluctuations in space-charge-limited currents at mod- erately high frequencies; Part 11-Diodes and negative-grid triodes,” RCA Rev. , vol. 4 , pp. 441-472, 1940.

[14] E. Nielsen, “Behavior of noise figure in junction transistors,” Proc. IRE, vol. 45, pp. 957-963, 1957.

[15] G. Ladd and D. Feucht, “Performance potential of high fre- quency heterojunction transistor,” IEEE Trans. Electron Device,

1161 S. Jue, D. Day, A. Margittai, and M. Svilans, “Transport and nosie in GaAs/AlGaAs heterojunction bipolar transistors,’ ’ IEEE Trans. Electron Devices, vol. 36, pp. 1020-1024, 1989.

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* Jun Han (S’89) was born in Lanzhou, China, in June 1963. She received the B.S.E.E. de- gree from Zhejiang University at Hangzhou, China, in July 1985, and the M.S.E.E. degree from the University of Colorado at Boulder in December 1988.

She has been working on designing and pro- cessing heterostructure bipolar transistors, as well as researching on bipolar transistors’ noise.

* Frank S. Barnes (S’54-M’58-F’70) was born in Pasadena, CA, on July 31, 1932. He re- ceived the B.S. degree in electrical engineering from Princeton University, Princeton, NJ, in 1954, and the M.S., Engineer, and Ph.D. de- grees from Stanford University, Stanford, CA, in 1955, 1956, and 1958, respectively.

He is presently a Professor of Electrical and Computer Engineering at the University of Colorado, Boulder. His research interests in- clude lasers, semiconductor devices, optics,

and the application of lasers and electromagnetic fields to biological materials.

Dr. Barnes is a member of the American Physical Society and the Bioelectromagnetics Society, and is a Fellow of the AAAS and the American Society for Lasers in Medicine and Surgery. He is the Editor of the IEEE TRANSACTIONS ON EDUCATION, and is a recipient of the Curtis McGraw Research Award of the ASEE and the IEEE Centennial Medal.