Tunneling phenomena from non-equilibrium carrier distributions in semiconductor heterostructures

ELSEVIER

Physica B 225 (1996) 76-88

Tunneling phenomena from non-equilibrium carrier distributions in semiconductor heterostructures L. Meza-Montes, J.L. Carrillo, M.A.

Rodriguez*

lnstituto de Fisica, Universidad Aut6noma de Puebla, A.P. J-48 Puebla Pue. C.P. 72570, Mexico

Received 14 July 1995; revised 5 December 1995

Abstract

We study tunneling phenomena in semiconductor heterostructures taking into account nonequilibrium carrier distributions. Due to the strong dependence of tunneling probabilities upon the carrier energy, some of the characteristics of tunneling current depend critically upon the carrier distribution function. Thus, to describe these phenomena a correct shape of the carrier distribution function is required. We analyze those effects which can be associated with the nonequilibrium character of the carrier distribution, both in steady-state resonant tunneling transport and in timedependent tunneling phenomena. In time-dependent processes, we analyze the resonant and nonresonant escape time of extremely out-of-equilibrium carrier populations excited in a mesoscopic well. We found that these effects may have strong manifestations in resonant tunneling transport.

1. Introduction Resonant tunneling (RT) in semiconductor heterostructures has received considerable attention during the last few years due to its technological impact [1]. The strong dependence of resonant tunneling upon the energy of carriers makes this physical phenomenon critically dependent on the shape of the carrier distribution function (CDF). The shape of the C D F is governed by the main interaction mechanisms of the carrier system. Therefore, RT provides an interesting tool to study subtle aspects of the scattering mechanisms in a carrier population which is far from the equilibrium condition in many cases. A physical situation in which the out-of-equilibrium nature of the CDF is determinant is the escape of a far from equilibrium carrier population

* Corresponding author.

injected in a well via tunneling events. To determine the escape time of this carrier population, it is necessary to know the time evolution of the CDF, since the rate of escape of the carriers is influenced by the shape of the CDF. There have been many studies, both experimental and theoretical, on the tunneling transport [1]. The calculation and interpretation of the experimental results were carried out using a Fermi-Dirac CDF. However, in the case of extreme nonequilibrium of the system, this approach is mostly unsuitable, provided it yields significant deviations from the measured values [2]. Manybody techniques have been used to describe the out-of-equilibrium CDFs since it became evident that the one-particle scheme is not enough to take into account important features which affect the tunneling transport in real semiconductor systems [3]. As a result, a formulation based on the Wigner function model has been proposed to study a spatially variable mass system. This method is better

0921-4526/96/$15.00 ~) 1996 Elsevier Science B.V. All rights reserved PII S 0 9 2 1-45 2 6 ( 9 6 ) 0 0 0 0 7 - 5

L. Meza-Montes et al. /Physica B 225 (1996) 76-88

for the calculation of the tunneling transport than the one based on the transmission probability, even for the static case [4]. Other procedures have been proposed, some more fundamental than others. However, one of the central points in all of them, is the determination of the CDF for real conditions. Self-consistent calculations, taking into account spatial charge effects on tunneling transport, have been reported [5, 6]. These approaches are based on two different hypotheses: the first one (Ohnishi et al. approach [5]) assumes an equilibrium carrier distribution at the contacts region and ballistic transport through the barrier system. In the second one (Cahay et al. and Brennan approach [6]) the other extreme is assumed, i.e. a quantum ballistic transport through the whole region between the contacts. The agreement of these approaches with experiment has been analyzed for Si/SiGe and A1As/GaAs double-barrier tunneling diodes [7]. In contrast to these works, we study the effects of the nonequilibrium character of the CDF (under some conditions this means the hot-electron effects) on tunneling phenomena. We perform an analysis in two different semiconductor systems: the first is a double-barrier system (DBS) and the second is a mesoscopic well. In the former case we study the transport properties while in the latter case, the escape time of a carrier population excited by a laser. We have found that the proper determination of the CDF is an important aspect that one should consider in the description of tunneling transport. In this sense, we might consider the problem to be composed of two parts. First, the transmission coefficient related to the geometrical aspects of the problem. Second, the CDF related to the carrier-dynamical aspects. Of course, both parts are closely related, since the tunneling current is an important factor which contributes in the determination of the shape of the CDF. We shall discuss in detail this point below. The out-of-equilibrium CDFs that we use in this work are obtained from a kinetic nonlinear model which we have developed and used previously to analyze mainly bulk semiconductor systems [8]. With appropriate modifications, this model enables us to describe the CDFs in the electrode regions even for far from equilibrium conditions. We include in our kinetic model the main scattering

77

mechanisms which determine the behavior of the CDF. It is convenient to remark that our theoretical procedure resembles the Ohnishi's approach; however, we do not assume an equilibrium distribution in the electrode regions. Another important difference with the Ohnishi self-consistent procedure is that here we are neglecting the spatial charge effects induced by the spatial inhomogeneities. Thus, we focus our attention in the description of the CDF in the energy space. With the above-mentioned approximations, we explore the manifestations of the actual out-of-equilibrium condition of the carrier population in the tunneling transport. The paper is organized as follows. We describe in Section 2 our kinetic model with the modifications and extensions necessary to describe the CDF behavior in the electrode regions of a semiconductor heterostructure to obtain the tunneling rates. In Section 2 we compare our predictions with experimental J - V characteristics. After that, we extend our analysis to new and interesting situations for which, as far as we know, experiments have not yet been done.

2. Theory In this work we consider a GaAs-AlxGal_xAs heterostructure, where x represents the Al content in the alloy. One of the most important quantities used to characterize the tunneling transport in this kind of semiconductor systems is the integrated current across the DBS. This quantity is a function of the applied voltage. The expression which incorporates the two components of the current in a transport process through a heterostructure may be written [9] in the form

J = 4 -e~ fd ezd2k~lU(e)-f(e+eV)]T(e~ ),

(1)

wheref(e) and T(e) are the CDF, and the transmission coefficient, respectively. The tunneling direction is parallel to the growth direction of the heterostructure which is labeled as z-axis. A density of states of the form predicted by the effective mass approximation is usually assumed. We wish to recall here that the band structure effects can be

78

L. Meza-Mantes et al. IPhysica B 225 (1996) 76-88

strong enough to compete with those associated with the out-of-equilibrium nature of the CDF in hot-electron physics. In order to distinguish between these contributions, we also use here the effective mass approximation for the density of states. The transmission coefficient, which we obtain by means of the transfer matrix technique, is given [9] by

mtk, - 1

T(E) = -

mnh PII?

where mi and ki are, respectively, the effective mass and the wave vector in the ith region of the heterostructure, and RI I is the (1,l) element of the transfer matrix. To obtain the CDFs, we shall use our kinetic model conveniently extended, as explained below. The description is based on a set of nonlinear rate equations for the carrier density. We arrive to that set of nonlinear rate equations by working in a discretized energy space. The basic ideas of this approach were proposed and applied to hot-carrier problems by Kurosawa [lo] and Levinson [l 11. This scheme was extended to hot-phonon problems by Ferry [12]. The set of rate equations on which our model is based (without modifications to deal with tunneling yet) is the following [S]: dX”o dt=

- ox”0 + v(l)xT + v( - 1)x; - Ml)

111 x”o

+ A -

+f&;- 1,xT + ST+ 1& - ~“0) + 9o(t)T (34 o = - oxp+ v(1) ($+I - xl) -dxP+ dt

+ v( - 1) (x7+1 - x9

+

zNx(xP+

+

ZNXO(XY

-

I -

-

x3

+

28

I4

+

-

XL

1)(x:-

1 -

s E,+AE

Xi(&)

=

1x7 = (T

d&fb)D(4

EC

is the carrier population in the ith energy level, i=o,1,2, . CJis + ( - ) for electrons with ... >b?l, positive (negative) momentum, 6 = - 0, x = CXi is the total carrier population, which, for the sake of numerical calculation we consider normalized to N, the total carrier concentration. In Eq. (4), D(E)is the density of energy states calculated in the effective mass approximation. The quantities o, v, 11, 2, and fE are effective collision frequencies (ECF) for various mechanisms [8], namely, recombination, phonon emission, phonon absorption, electronelectron scattering, and carrier-electric field interaction, respectively. We calculate these ECF as the mean value of the corresponding transition probabilities per unit time, using standard time-dependent perturbation theory. The effective frequency associated with the electron-field interaction fE is calculated by evaluating the power gained by an electron in an electric field. In our theoretical procedure it is an easy task to include any other important interactions, for instance, intervalley scattering or interaction with impurities [S]. However, in order to not introduce more elements which can make the analysis less clear we neglect such other interactions. The following expression is the general definition of an effective collision frequency, say 0: @=’

+ ZNx(6 - $3 + ZN xox;io

+ P(1) (XP-1

Here,

E,,X de 0 (E)D(E), Emaxs 0

where O(E) is the transition probability per unit time due to a given collision mechanism and F,,, is an upper bound for the energy which depends on the material. Eqs. (3) must be solved coupled to the rate equation for the time evolution of the LO phonon population which can be written as follows:

XP)

dN$J -= dt

11

- iN% + tC(x - xo)(v(l) + v( - 1))

xP-11

+

+

fE

#-

gitt).

1) XT+ 1 +

q+

1) x4-

1 -

(5)

xf)

(W

x(/4) + A - 1))l. (6)

Here NY; is the total phonon population in excess with respect to the average number of phonons in

L. Meza-Montes et al./Physica B 225 (1996) 76-88

equilibrium at the same lattice temperature. The coefficient ~ is the phonon damping constant and U is the number of accessible phonon modes per unit volume [8]. The structure of the rate equations (3) allows us to incorporate the information concerning the screening by free carriers in the electron-electron and electron-phonon interactions, and the phonon-heating effects. This is done through the ECF which have to be evaluated at each step of the process. Eqs. (3) and (6) can be solved to describe time-dependent and steady-state processes. For given physical conditions of carrier excitation, lattice temperature, etc., the self-consistent solution Z~ also provides the values of the ECFs. These values tell us how the energy exchange through the collision mechanisms is carried out in that particular steady state of the system. Notice that Eq. (6) takes into account the changes in the total phonon population along the process. The value of N[~ is one of the factors which define the ECF associated with the emission and absorption of LO phonons, v and p, respectively. We incorporate in our description the phonon population effects by solving Eqs. (3) coupled to Eq. (6), see Ref. I-8]. A detailed description and discussion about the fundamentals of our theoretical model and its applications can be seen in Refs. [8, 13]. To deal with tunneling resonant and nonresonant) we need to modify our kinetic equations to take into account the presence of the heterostructure system. We include additional effective collision frequencies associated with the scattering by the barrier system and with the tunneling events. Thereby, we have to modify Eqs. (3) by including a term which describes the change of the carrier concentration at a given energy level due to the scattering with the DBS and a term which describes the concentration loss due to tunneling. The terms which we add to Eqs. (3) are the following. For the scattering with DBS, the change of the carrier population at the ith interval of energy is given by

(d q = dt f l

-- ri Z°~ q'- rl)('i'

(7)

79

for the tunneling frequency we use the expression

(dxq

=

dt lit

--JTi"

(8)

The respective effective collision frequencies, ri and J~i, are defined by 1 l '~ +At:

ri = A--~e,J~,/

r(~)de

(9)

and 1 /'~' +a~

fTi -- ~ J,,l

r(e)T(e)Z~(e) de.,

(10)

where r(e) = v(O/le and v(0 is the carrier velocity. The quantity le is the electrode length in the stationary case and the mesoscopic well width in the time-dependent processes. The effective frequency ri, associated with the scattering by the DBS, is calculated assuming that the electrons move with approximately constant speed between collisions [13]. For convenience and to have a reference quantity we have taken Ae = hOLo, the longitudinal optical (LO) phonon energy. A brief explanation of the hypotheses and physical concepts behind these expressions is the following. For zero bias voltage, we assume that in the contact regions the carrier system dynamics is governed by Eqs. (3) and (6), and the solution at a given time provides us the corresponding CDF. In these conditions, the dispersion of carriers by the barrier system is the result of a thermal diffusion. If a carrier is backscattered by the barriers, the other scattering mechanisms may make this carrier collide again with the barriers. For an applied voltage, an additional mechanism affects the frequency of the collision with the barrier system; this is the effect of local electric field. The effect must be taken into account in the equations through the frequency associated with the carrier-field interaction fE. The effect in the CDF of the collisions with the barriers is taken into account by the backscattering frequency ri. Notice that, due to the sensitivity of the tunneling probabilities on the carrier energy, the average involved in Eq. (5) is unsuitable in this case. As a result, the evaluation of the corresponding ECF has to be done locally, Eqs. (9) and (10). Special care is required in those energy regions

80

L. Meza-Montes et al./Physica B 225 (1996) 76-88

close to the resonances. Thus, unlike the other ECFs, ri and j~i have different values at different energy intervals. These frequencies comparatively change notoriously at those intervals of energy in which the resonances are included. By these considerations one can properly include tunneling events in the rate equations. We mentioned above the relationship between the geometric and the dynamic part of the tunneling transport. Physically, the linking between these parts is performed by the scattering mechanisms, since the scattering rates depend upon the carrier population at the energy levels, but these carrier populations are affected by the geometry of the DBS, i.e., by the transmission probability; furthermore, the tunneling events affect the shape of the CDF. There are second-order effects of the carrier dynamics on the transmission probability caused by screening and by the heating of the phonon system. This relationship between the carrier dynamics and the tunneling current introduces an important difficulty in the theoretical treatment of these systems. In our case, due to this relationship, a strong nonlinear dependence appears on the carrier population in the rate equations (3). This nonlinearity is due not only to the electron-electron interaction but also to the tunneling events and to the scattering by the barriers. It is important to remark that the ECFs, which are calculated as the mean value of the corresponding transition probabilities per unit time, depend on physical constants and material parameters. However, some of these parameters are not well known. Such is the case of LO phonon time decay and the electric field in the electrode regions. We have assumed a LO phonon time decay value of the order of 1012S 1-8] and we have used the electric field in the electrode regions as a fitting parameter. Thus, in order to describe tunneling phenomena in semiconductor heterostructures, we integrate Eqs. (3) including the terms given by Eqs. (7) and (8). This procedure provides us the CDF corresponding to a steady state, or the CDF at certain time t for time-dependent processes. We have assumed in our treatment that, in real tunneling devices, the interaction mechanisms can be intense enough to induce a shape of the CDF significantly different of a Fermi-Dirac distribu-

tion. On the other hand, Eq. (1) for the tunneling current is a noninteracting expression. For the sake of congruency, in our theoretical approach, it is more convenient to evaluate the tunneling current by counting directly the tunneling events. We use to this end, instead Eq. (1), the expression d = Ne~j~T,

(11)

i,a

although the adoption of a noninteracting scheme to evaluate the tunneling current still remains here. We proceed now to compare our results with experimental data for typical DBSs. This comparison is possible only for the case of steady-state tunneling transport, since there are not yet systematic experimental studies for time-dependent processes. We firstly present our general results of current-voltage characteristics. Secondly, we present (and compare with other results when possible) our predictions about the changes in some features of the tunneling current induced by variations in the DBS geometric parameters and variations in the carrier injection. Finally, we study some more involved out-of-equilibrium tunneling transport processes in semiconductor heterostructures.

3. General results 3.1. Tunneling in steady-state processes

In order to have a reference system for comparison purposes, we have calculated that J - V characteristics for some DBS for which there exist published experimental results [14-16]. The J - V characteristics of RT transport as predicted by our equations and the experimental data (dotted line) reported in Ref. [17] are presented in Fig. l(a). The heterostructure corresponds to x = 1, namely GaAs-A1As. We have taken the AlAs barrier height [14], Vo = 1.05eV and the barrier's effective mass factor [18], m * = 0.15. Other geometrical parameters, the barrier width b and the well width W, are 2.26 and 7 nm, respectively. The parameter set is completed with the lattice temperature T = 80 K and the carrier concentration N = 5x 101Scm -3.

L. Meza-Montes et al. /Physica B 225 (1996) 76-88

(a) 3

,

(b) I0~9

'

......... Experimental -This theory This theory ~ F~mi-Dirac: .'"

t~2

~&1 0 l0

Tkis theory ~

.f"

1

z Fermi-Dirac

0 L-0.0

1

0.5

v (v)

1.0

0.0

0.3

0.6

e (eV)

Fig. 1. (a) Current-voltage characteristics. Comparison of our results with experimental ones and with those obtained by the use of a Fermi-Dirac distribution; see text. (b) Transmission coefficient, T(e) x 1019, in the region close to the resonances and carrier concentration as a function of the energy: (dashed line) our theory; (dotted line) calculated with a Fermi-Dirac distribution.

In the figure, the thickest curve corresponds to the results obtained by using a Fermi-Dirac distribution 1-19]. The thinnest line corresponds to our results. The other continuous line shows the shift caused by the inclusion of load-charge effects. As it is well known, some series and contact resistances cause additional voltage drops in the circuit. When these voltage drops are artificially included in the theoretical description, strong hysteresis effects appear in the current behavior [19]. One can observe that our results predict more accurately the main trends shown by the measured values; this is true for the complete form of the curve for the tunneling current as a function of the applied voltage as well as for the peak and valley values of the current. In Fig. l(b) are depicted the transmission function T(e) scaled by a factor 1019, the carrier concentration, NX(e), for the voltage corresponding to the resonance as predicted by our theory, and the one corresponding to the Fermi-Dirac distribution. This figure explains the behavior of the J - V curves. Notice that the equilibrium C D F predicts a lower concentration in the resonance regions which, of course, modifies notoriously the magnitude of the tunneling current.

81

This happens even if more elaborated procedures are used to calculate the transmission function [5]. Therefore, we assert that the proper determination of the C D F might be as important as is the calculation of the transmission coefficient. We stress that in our theory we do not take into account spatial charge effects. In the limit of small electric fields our theory gives a carrier distribution which is essentially a Fermi-Dirac distribution. Therefore, we choose to compare our results with those obtained using a Fermi-Dirac distribution. A direct comparison, in this limit, with other theories like Ohnishi's, in which spatial charge effects on tunneling transport plays an important role is in progress and will be published elsewhere [20].

3.2. Geometry of DBS and injection effects As we shall show below, the details of the current curve, like the peak value Jp, the valley value Jr, and the ratio of the current at the peak to the current at the valley Jp/Jv, are also very sensitive to the shape of the CDF. For steady-state processes we discuss how the geometry of the DBS and the excitation conditions affect the C D F of system and consequently the tunneling transport. First, we calculate the tunneling current as a function of the applied voltage for several values of the parameters which characterize the DBS (for fixed carrier injection N = 5 x 1018cm-3). In Fig. 2, we exhibit the behavior of J p and Jv as functions of the barrier width, b. The well-known behavior of the transmission coefficient is shown in the inset [9, 153. Notice that the transmission function becomes sharper as b increases, namely, the resonance range becomes shorter. As we should expect, this fact has a clear effect on the resonant tunneling current. As it can be observed in the curves, when b increases both J p and Jv decrease but not in the same proportion. As a result, it is important to determine with the maximum accuracy the carrier concentration around the resonance energy levels to predict correctly the order of magnitude of these quantities. The width of the well plays also an important role in the tunneling transport [14]. This is due to its direct effect on the resonant levels of energy which affects directly the tunneling rate of carriers.

82

L. M e z a - M o n t e s et al. / P h y s i c a B 2 2 5 ( 1 9 9 6 ) 7 6 , 8 8

0

Jp

"

•

-lo

Jv

80 296 K

:I ..... /

,t

// ]L

-20

104

\

t.

i i

[]

.

;J'

A

,,

--

b = 1.42 a m

--

2.26 3.11

10 4

[]

Exp.

m Theory

-30

0.0

0.J

1.0

E (or)

103

80 296 K A

~k

~

\

3

Exp.

[]

Theory

\

10 3

[]

\\

10 2

10 2 i

1.0

I

1.5

i

I

2.0

i

L

2.5

i

I

3.0

i

b

3.5

1.0

i

1.5

2.0

2.5

3.0

3.5

b (nm) Fig. 2. Peak (dp) and valley (Jr) current values for various values of the barrier width b. Inset: logarithmic plot of the transmission function (see text).

In Fig. 3 we present, for the same excitation condition as above and for a barrier thickness b = 2.26 nm, the behavior of Jp and Jv for various values of the well width, W. For comparison, we have also plotted the experimental results [-14]. It is important to remark that there is no artificial shift in the curves. Notice that qualitatively the measured trends are reproduced by our results. Our predictions fit better the Jp measurements than those corresponding to Jv. We found that this particular calculation has a great sensitivity to changes in the CDF. This is why we present separately the behavior of Jp and Jv instead of the ratio Jp/Jv in which some details might be masked. Based on this theoretical approach, we analyze in the following discussion some other important factors which affect the RT transport. The carrier injection conditions have a direct effect upon the shape of the CDF. To some extent, the excitation conditions determine several features of important

kinetic quantities in the carrier system (the carrier temperature, for instance). Other important quantities also depend in an indirect way on the injection or excitation conditions, for example, dynamical screening depends on both carrier temperature and carrier concentration [21]. Thus, it is important to take into account the injection conditions to determine correctly the shape of the CDF, and then to calculate properly the tunneling current in this kind of systems [3, 22]. In Eqs. (3), the generation of carriers is taken into account by the term gi(t). Notice that several kinds of excitations can be modeled by this procedure: from thermal injection to monochromatic generation, from low to high rates of generation and from a constant rate of generation to generation by pulses [21]. The obtention of the CDF in these very involved physical conditions from first principles theories is practically impossible due to the mathematical complexity. In those fundamental proced-

83

L. Meza-Montes et al./ Physica B 225 (1996) 76-88

Jp

Jv

104

104

103

103

0

80 296 K

[]

Exp.

m

Theory

80

296K

~..

"--~

[]

Exp.

w

Theory

[ 02

102 1

4

I

6

I

I

8

I

i

10

4

I

6

i

I

10

8

W (nm) Fig. 3. Peak (Jp) and valley (Jr) current values for various values of the well width W.

ures, this fact forces the introduction of some oversimplifications, or makes necessary the inclusion of phenomenological information like collision terms or relaxation times. In our treatment we only need as phenomenological information, the damping constant for the LO modes which appears in the rate equation (6). The carrier concentration is one of the physical parameters which strongly affects the tunneling current due to its direct influence on the carrier dynamics 1-13]. Unfortunately, the experimental characterization of this quantity is not an easy task. As far as we know, a systematic study on this subject is still unexplored. Our results on this aspect appear in Fig. 4, where the ratio Jp/Jv is plotted for carrier concentrations N ranging from 1016 to 1 0 1 9 c m -3, for the same heterostructure as in Fig. 1. We also show in this figure the effect of the intensity of an applied electric field. We present results for two intensities (El) and (E2 = 10 x El).

15

10

5

• E1

"'',, IL.. . . . . . . . . "k-,--,k-7,7-'---- ! .... 1017

1018

1019

N ( c r n -3) Fig. 4. Jp/Jv as a function of the carrier concentration N.

84

L. Meza-Montes et al./Physica B 225 (1996) 76-88

Notice that the ratio Jp/Jv is larger in all the concentration interval for the lower value of the applied electric field. This apparently contradictory result can be explained as follows. The applied electric field lifts up the carrier population from the lowest resonance energy, diminishing the efficiency of this tunneling channel and increasing the contribution of the second one. In this way, the peak value of the current decreases and the valley value increases producing the trend shown in the figure. Charge accumulation in the resonant well might modify tunneling transport by producing changes in several geometric and dynamical properties of the system. Builtup charge changes the potential profiles producing an effective decrement in the height of the potential barriers [23]. One would expect that this fact propiciates an increment in tunneling current. The areal charge density in the well, Y~~ zJ, is proportional to the lifetime r of the carriers in the well and inversely proportional to the mean width of the resonance peak of the transmission function [-24,25], A F = h/z. This later quantity is very sensitive to the barrier height. Thus, the charge buildup effects should be considered mainly in the case of high and/or wide barrier systems. In Fig. 5, we present a comparison of the tunneling current as a function of the applied voltage considering (squares) and neglecting (continuous line) charge buildup effects. We estimate the charge buildup effects according to the Goldman's procedure [24]. The DBS parameters are the same as in Fig. 1, except for the well width which is in this case 5 nm. We observe a clear manifestation of the charge buildup. In our analysis we found that in more extreme conditions charge buildup effects can destroy the resonant features of the J - V characteristics [26]. In Fig. 6, we present our results for the peak and valley values of the current as functions of the lattice temperature. We compare our results with the measured values [16]. We consider here a short-barrier system to make the temperature effects easily observable. The values of the parameters are: Vo = 0.255 eV, b = W = 5 nm. In the absence of an applied field, there is a resonance at ~;~1= 0.0766 eV, and other at er2 0.2727 eV. We =

6.50

m

3.25

n

[]

0.00 0.(

0.1

0.2

v (v) Fig. 5. Charge buildup effects on the J V characteristics, see text.

40

[]

Ip

30

m mA

-mm mmmnmmmm

A A

<

E

A

20 ,It

I0

0

i

0

i

1 O0

J

i

n

200

I

300

i

i

400

T (K) Fig. 6. Ip and Iv as function of the lattice temperature.

observe in the figure that our results reproduce roughly the experimental trends. We attribute the discrepancies to band-bending effects and inelastic tunneling processes not included in our theory [16,27-].

85

L. Meza-Montes et al./Physica B 225 (1996) 76-88

3.3. Tunneling in time-dependent processes Now we discuss results for some more complex physical situations. We wish to remark again that the nonequilibrium nature of the CDF determines to great extent of features and, in some cases, the order of magnitude of several of the physical quantities by which resonant tunneling transport is characterized. It is known that a steady-state process is not an equilibrium one. However, a steady-state process is fundamentally driven by the slow interactions of the system. In these conditions the role of the fast interactions becomes masked [8]. On the other hand, since some of the interactions depend on quantities which change dynamically with the relaxation process, these interactions change too. Thus, at different stages, the process might be dominated by different mechanisms of energy exchange in the system. To get some insight into these physical aspects we approach this problem by the study of the relaxation process of a carrier population excited inside a mesoscopic well by a laser pulse. We analyze the time evolution of the tunneling current and we compare the efficiency of the different relaxation channels of the system at different stages of the relaxation process. First, we present our results for the nonresonant escape time. The geometric parameters of the system are the following: The barriers have an effective mass m* = 0.096, a height of Vo = 0.026187 eV and a width b = 1.5 nm. The lattice temperature is T = 8 0 K and the mesoscopic well width is Wm = 100 nm. In Fig. 7, we display results for the time evolution of the normalized total carrier population excited in the mesoscopic well. Notice that Z does not reach the unit value because the pulse has a finite duration and in this time interval some particles escape from the well by tunneling or by recombining. We are considering three carrier concentrations as it is indicated in the figure, and three pulse durations tp = 180, 195 and 243 fs, respectively. Here we are assuming a monochromatic injection at the energy e~ = 0.01 eV. Notice that the existence of tunneling as a channel of relaxation produces noticeable changes in the total carrier population in times of the order of picoseconds. In the absence of this channel of relaxation, these changes would take place in the order

1.0 0.8 0.6 "'--......

0.4 I "

N = 1 016 cm-3 1017

0.2

1018

.........

0.0 0.(

I

i

0.5

I

1.0

r

1.5

t (ps) Fig. 7. Time evolution of the total carrier population for three values of the carrier injection and pulse duration.

of nanoseconds. We can also observe that for regimes of high carrier concentrations, in which the dominant interaction is the electron-electron scattering, the carrier population escapes faster than in the other cases. This is because the electron-electron interaction quickly thermalizes the carrier system. Therefore, an important fraction of the total carrier population reaches energy values for which tunneling is more probable 1-13]. On the contrary, for low carrier concentration regimes, the dominant interaction is electron-phonon interaction. This scattering mechanism shifts the CDF to lower energies, so that fewer electrons reach the energies for which they have high tunneling probabilities. Obviously, the time taken by the carriers to escape from the mesoscopic well depends on the geometry of the heterostructure; in particular, it depends on the barrier width b. In Fig. 8, we show results for the escape time ~, as a function of b. This time is defined as that time at which the total carrier population in the well decreases by a factor e. The triangles in the figure represent the calculated results and the continuous line is a fitting curve given by re = 0.352e a'46b where b is given in nm and re in fs. The inset shows the time evolution (in characteristic time, to, units)

86

L. Meza-Montes et al./Physica B 225 (1996) 76-88

50

r

,

i

,

i

4000

,

i

N = 1016 cm-3 ,

1017

19 e

3000

! ' " : ' ' " £iiiiiiii,............

r~

25

t

80K

2000

1000

~ 0

i

0

i

1

N=1018cm-3 i

I

i

2

I

3

i

i

i

4

0

Fig. 8. Escape time (see definition in text) for various values of the barrier width b. Inset: Electronic temperature as a function of time; nonresonant case.

of the carrier temperature for three widths of the barriers; b = 1.5, 2.0, and 2.5 nm. The characteristic time of the system is defined by the expression 1 og + v + /a + Z N '

h

1

I

2

i

3

t (ps)

b (nm)

tc =

L

(12)

where o9, v, #, and ZN, are the effective collision frequencies associated with the main interaction mechanisms (see Eq. (5)). Not for all the physical conditions the electronic system reaches the thermalized condition showed in Fig. 8. In this case we might speak properly of hot carrier tunneling. Tunneling events contribute importantly to the escape in the time interval immediately following the beginning of the pulse. At intermediate stages the electron-phonon interaction will take out the energy from the carrier system and the recombination interaction will control the final stages of the relaxation process. In Fig. 9, we depict the competition among the collision mechanisms by means of the parameter 6, defined as the ratio of the escape by tunneling to the escape by recombination. It is interesting to compare the results shown in Figs. 7, 8, and 9 to those obtained for resonant

Fig. 9. Ratio 6 (see text) as a function of time. Nonresonant case.

escape. By resonant escape we mean that process in which the carriers escape from the mesoscopic well by tunneling resonantly through the barriers. We are assuming that the mesoscopic well is limited by DBSs. From our analysis we find that the escape time is drastically reduced if escape from the well occurs via resonant tunneling events. The contribution of resonant tunneling to the escape is more appreciable at the first stages after the starting of the laser pulse. The time evolution of the CDF and, consequently, the change in the associated carrier populations of the energy levels of the conduction band in the mesoscopic well region produces a change in time of the rate of carrier escape. This change is more easily seen in the resonant than in the nonresonant case because of the great sensitivity of the resonant tunneling probabilities upon the carriers energy. However, from a global point of view, the resonant tunneling phenomenon is a very selective process since the transmission probability falls drastically at energies slightly out of resonances. To get some insight on the relaxation process of a carrier population in these conditions, we display in Fig. 10 our results for the carrier population as a function of the energy and time. The continuous line in the e - t plane indicates the top of the barriers

L. Meza-Montes et al./ Physica B 225 (1996) 7 6 8 8

and the dotted line indicates the position of the resonance. The parameters are those used in the nonresonant calculation, and the well width is W = 6nm. To compare resonant with nonresonant escapes, we use the 6 parameter. In Fig. 11, we present the time behavior of 6 for various values of the well width W. We note that before the C D F reaches the

Z

'~[

W = 6 nm

(eV)

f

i

i\

....

i i W=Onm 3

.........

4000

\

5

6 "...

9

2000

0 0

1

2

thermalization condition, the 6 ratio suffers drastic changes. These early changes can have an important influence in the determination of the escape time. This influence obviously would be more noticeable if the energy of the carriers injected into the conduction band is close to the resonance energy. This is the case of the dotted curve with W = 4 nm. For this well width the resonance energy is very close to the energy of carrier injection ep, it propitiates that at early stages of the relaxation process, resonant tunneling events can carry out from the mesoscopic well a much greater number of carriers compared to the number of tunneling events which occur for other W values or in the nonresonant tunneling case. The inverse process, namely, the resonant electron capture in a quantum well, has been recently investigated. It is interesting to compare our results for escape times with the measured results of photoluminescence intensity as a function of time reported by Fujiwara et al. [28].

4. Summary and comments

Fig. 10. Time evolution of the carrier population in the mesoscopic well as function of the energy.

6000

87

3

4

t (ps) Fig. 11. Time evolution of the parameter 6 for the resonant case; W is the well width.

We have explored theoretically the effects of outof-equilibrium carrier distributions in tunneling phenomena in semiconductor heterostructure. To describe these phenomena, we have obtained the out-of-equilibrium CDFs, corresponding to the electrode regions, using a kinetic nonlinear model. Although this model is not derived from first principles, it does incorporate all the main interaction mechanisms of the carrier system, including tunneling effects. To calculate the transmission function we have used the transfer matrix technique. The influence of the C D F on the transmission coefficient is included in our approach only in the case in which buildup charge effects are considered. We found that in the steady-state and time-dependent processes of tunneling transport and, most noticeable in the case of resonant tunneling timedependent processes, a very important factor which must be considered is the out-of-equilibrium nature of the CDF. The shape of this C D F resembles an equilibrium distribution only in some particular conditions. Thus, due to the great sensitivity of resonant tunneling probabilities upon the energy of

88

L. Meza-Montes et al./ Physica B 225 (1996) 76-88

carriers, the shape of the CDF has a fundamental role in the determination of the tunneling current. Since there is a nonlinear dependence of space charge buildup upon the tunneling current, in steady and transient processes, this charge buildup might be too important to be neglected. Here, we have discussed these effects only in the steady-state processes.

Acknowledgements This work (M~xico).

was supported

by CONACyT

References [1] Resonant Tunneling in Semiconductors, NATO ASI Series B: Physics. Vol. 27 (Plenum, New York, 1991); Proc. 7th. Int. Conf. on Hot Carrier in Semiconductors. Sem. Sci. Tech. 7 (1992). [2] L. Meza-Montes, M.A. Rodriguez and J.L. Carrillo, Solid State Commun. 80 (1991) 37. [3] R. Lake and S. Datta, Phys. Rev. B 45 (1992) 6670. [4] H. Tsuchiya, M. Ogawa and T. Miyoshi, IEEE Trans. Electron. Dev. 38 (1991) 1246; H. Tsuchiya, M. Ogawa and T. Miyoshi, Jpn. J. Appl. Phys. 31 (1992) 745. [5] H. Ohnishi, T. Inata, S. Muto, N. Yokoyama and A. Shibatomi, Appl. Phys. Lett. 49 (1986) 1248; Y.W. Choi and C.R. Wie, J. Appl. Phys. 71 (1992) 1853. [6] M. Cahay, M. McLennan, S. Datta and M.S. Lundstrom, Appl. Phys. Lett. 50 (1987) 612; K.F. Brennan, J. Appl. Phys. 62 (1987) 2392. [7] D. Landheer and G.C. Aers, Superlattices Microstructures 7 (1990) 17. [8] J.L. Carrillo and M.A. Rodriguez, Phys. Rev. B 44 (1991) 2934.

[9] E.E. Mendez, in: Physics and Applications of Quantum Wells and Superlattices, eds. E.E. Mendez and K. von Klitzing (Plenum, New York, 1988) p. 159. [10] T. Kurozawa, J. Phys. Soc. Japan 20 (1965) 937. [11] I.B. Levinson, Fiz. Tverd. Tela. 6 (1964) 2113. [12] D.K. Ferry, Phys. Rev. B8 (1973) 1544. [13] L. Meza-Montes, M.A. Rodriguez and J.L. Carrillo, Sem. Sci. Tech. 7 (1992) B450. [14] M. Tsuchiya and H. Sakaki, Appl. Phys. Lett. 49 (1986) 88. [15] M. Tsuchiya and H. Sakaki, Jpn. J. Appl. Phys. 25 (1986) L185. [16] O. Vanb6sien et al., in: Resonant Tunneling in Semiconductors, op cite. p. 107. [17] E.E. Mendez, W.I. Wang, E. Calleja and C.E.T. Goncalves da Silva, Appl. Phys. Lett. 50 (1987) 1263. [18] S. Adachi, J. Appl. Phys. 58 (1985) R1. [19] S. Collins, D. Lowe and J.R. Barker, J. Appl. Phys. 63 (1988) 142; J. Chen et al., J. Appl. Phys. 70 (1991) 3131. [20] Yu.G. Gurevich, M.A. Rodriguez, J.L. Carrillo, L. MezaMontes and I.N. Volovichev, in: Proc. 9th Vilnius Symp. on Ultrafast Phenomena in Semiconductors, 5-7 September (1995), in press. [21] M.A. Rodriguez, J.L. Carrillo and J. Reyes, Phys. Rev. B 35 (1987) 6318. [22] W.R. Frensley, J. Vac. Sci. Technol. B 3 (1984) 1261; Rev. Mod. Phys. 62 (1990) 745. [23] B. Ricco and M.Ya. Azbel, Phys. Rev. B 29 (1984) 1970; V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. Lett. 58 (1987) 1256. [24] V.J. Goldman, D.C. Tsui and J.E. Cunningham, Phys. Rev. B 36 (1987) 7635; L. Meza-Montes, M.A. Rodriguez and J.L. Carrillo, to be published. [25] H. Sakaki et al. in: Resonant Tunneling in Semiconductors. op cite. p. 307; F.W. Sheard and G.A. Toombs, Appl. Phys. Lett. 52 (1988) 1228. [26] L. Meza-Montes, Ph.D. dissertation. Universidad Aut6noma de Puebla, M6xico (1993). [27] P.J. Turley and S.W. Teitsworth, Phys. Rev. B 44 (1991) 3199; Phys. Rev. B 44 (1991) 8181. [28] A. Fujiwara, Y. Takahashi, S. Fukatsu, Y. Shiraki and R. Ito, Phys. Rev. B 51 (1995) 2291.