Theory for nonequilibrium behavior of anisotype graded heterojunctions

Solid State Electronics Vol. 26. No. I. pp. 59-64, 1983 0038=I 101/83/010059-06503.00/0 Printed in Great Britain. Pergamon Press Ltd.

THEORY FOR NONEQUILIBRIUM BEHAVIOR OF ANISOTYPE GRADED HETEROJUNCTIONS

AMITAVA CHATFERJEE Electrical, Computer and Systems Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12181,

U,S.A.

and

ALAN H. MARSHAK

Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803, U.S,A.

(Received 13 March 1982; in revised form 12 July 1982)

Abstract--An analysis for the terminal behavior and capacitance of anisotype graded heterojunctions is presented. A closed form expression for the I-V characteristics is derived using the depletion approximation and the rigid band model. The expression is valid for uniform nondegenerate doping and low-level injection. The transition region is modeled to extend beyond the space-charge region with a constant bandgap gradient. Variations of the dielectric constant, carrier lifetime and mobility are ignored. The nonsaturating nature of the reverse current is demonstrated. The effects of variations of bandgap and electron affinity on the I-V characteristics and junction capacitance are discussed. Numerical results for a graded nGe-pGaAs device are given.

1. INTRODUCTION

The potential usefulness of special electrical and optical properties of a heterojunction for new or improved device designs is well recognized[l-6]. If the transition from one material to the other takes place over a distance large compared to interatomic distances, the junction is said to be graded. The graded heterojunction concept has been successfully applied for optimum designs of solar cells [7, 8] and in devices where tunneling and interface recombination actions are undesirable[9]. While earlier works[10--12] have examined the current- voltage relationship in a graded heterojunction, they do not provide a complete analytical closed form expression for the I - V characteristic. Consequently the effects of variation of some of the material properties on terminal behavior is not clearly recognizable. The major problems in predicting theoretically the electrical properties of a heterojunction is the lack of a proper characterization of the transition region, the diversity of materials used as heteroiunction pairs and variations in methods of fabri- cation. Various models for the behavior of abrupt heterojunctions have been proposed; a review of these is given in [2, 3, 13]. It is conspicuous, however, that graded heterojunction theory is not well developed despite the increasing number of possible applications[14].

In this paper a closed form expression for the current- voltage relationship is derived treating the diode as a device with a linear varying bandgap in the transition region. The transition region extends beyond the space- charge region. The model assumes uniform nondegenerate doping in both n and p regions. Effects of stress, interface states, recombination near the junction, variations of dielectric constant, effective mass, carrier lifetime and mobility are ignored. For an inhomogeneous

This work was supported in part by the National Science Foundation under grant ENG 78-09886.

59

mixed crystal where we have a slow macroscopic variation of composition with position, the concept of position-dependent band structure can be justified[15-18]. The basic equations for transport and carrier densities in semiconductors with nonuniform band structure are taken from [17-19]. Since the current equations are derived from the Boltzmann transport equation, this formalism does not account for quantum tunneling and reflection. The extra terms in the transport equations in addition to the conventional drift and diffusion arise from the variation of the energy band edges and from the spatial dependence of the electron and hole density of states. These two effects are referred to as the "rigid band effect" and the "density of states effect". The present analysis assumes the rigid band model. We have also made the fixed-field approximation which is valid for low-level injection and the notorious depletion approximation to facilitate the solution of Poisson's equation.

Despite the simplifying approximations, the derivation provides a framework for a more complete analysis. Our results clearly demonstrate the nonsaturating nature of the reverse current. Deviation from the classical diode eqtlation is marked for low doping and steep bandgap gradient (distance over which the bandgap changes' by 2 kT is small compared to the minority carrier diffusion lengths). An additional series term in the space-charge capacitance is found due to variations in electron affinity and bandgap. The model is applied to a graded nGe- pGaAs device; numerical results are consistent with experimental results of devices with varying bandgap.

2. DEVELOPME]~r

For a sufficiently graded junction between materials with a good lattice match, the change in band structure due to strain would be negligible. The electron and hole current densities, in the rigid band case, can then be

60

written as [ 1 7 ]

c~ c~n J. =-nt~. ~ ( e 6 + x)+eD. ~x

Op J~ -'- - Pg" ~x (e4, + X + E,~) - eD. ~x"

A. CHAYrERJEE and A. H. MARSHAK

(l)

(2)

Here n, p are electron and hole densities; g,, go are the carrier mobilities;/9,, Do are diffusion constants; and 6, X and E s denote the electrostatic potential, electron affinity and bandgap, respectively. The transition region is considered to be of finite width extending beyond the space-charge region near the interface. The doping changes from n-type to p-type at x = 0. The transition region extends from -x.~ to x~,g and the space-charge region is from - x . to xo, as shown in Fig. 1. We shall consider only completely singly ionized impurities and doping densities such that the semiconductor is nondegenerate and sufficientlyt extrinsic. Wriging j o = 0 for the quasi-neutral p-region (superscript 0 denotes thermal equilibrium), we have, for low-level injection,

, , i i i , t t i i k , , i I

-Xno-X n 0 Xp Xpg

q Fig. 1. Model of the n-p graded heterojunction; V/is the voltage across the space-charge region and L is the width of the tran-

sition region in which the composition varies with position.

d ~ d d x dEg (3) e ~ x - - k T (InNA) dx dx

where N^ is the ionized acceptor density. Thus the injected electron current is given by

(4) On dEs J, = kTg,, -&x + njz. --dx-x ' x >>- x,,

for uniform nondegenerate doping. Similarly, the injected hole current is given by

(5) 9 p dE, J, = - kTIz, ~ -p t t , --d-xx ' x <- x..

Using the SRH single-recombination level approximation, the continuity equation for injected electrons is, in steady state,

1 d J . = n - n ° (6) e d x 7 .

Here ~. and rp are the minority carrier lifetimes. For the boundary conditions that excess carrier densities are zero at -+~ and n, p and their gradients are continuous

~rSuflicient in thd sense that the doping densities ND and NA are much much greater than the effective intrinsic density n~,(x) in the region of interest.

everywhere we can solve the current and continuity equations for the minority carrier density; thus

I C, e k'* + C2 e k2.~ + ~ exp[-(x - Xr,~)/L~], I

X v < X < X p ,

n(x) = 1 xlLn It iF' X > Xpg.

(7)

Likewise. for holes we have

p ( x ) =

B, e <~ +/~2 e ~- + ND exp[-(x + x.s)/Lg],

- X n , < X < ~ Xn

n~ B 3 eXtLp + G , X < - - X n g

(8)

where Lj = ( D ~ ) 'n is the minority carrier diffusion length and

1 d E , 1 B E . llL, --- k--T d--~- kT L " (9)

In the transition region L. is assumed to be constant. Here ~Es = E.p - E.., L = x~, + x.,, and n,., n,. are the intrinsic carrier densities of the material of bafldgap E~., E,o, respectively. Other constants are defined by

I [I+ X/(I+(2LdL.)=)I (lOa)

k2 = - ~ L f i l -~/ (1 +(2L./L,,)2)] (10b)

C~ = h(xp)e k'-~p~ e -xP~/L"(k2+ 1 D .0c,

C3 = ~(xp) e <'"~ ek2*'*(k2 - kO/D (10e)

where ~ = n - n ° and

-ek~-X" e <:'P~(k, + ~7) ]. (lOf)

Also

a, = - 2-~[1 + V'(I + (2LdLp)2)] (1 la)

I [ l - V ( l + ( 2 L d L o f ) ] ( l ib)

¼)/ - , - A ( l id)

B3 =/~(-x.) e ...... e ":*"~(a2-a,)/A ( l ie)

where 1~ = P - pO and

Theory for nonequilibrium behavior of anisotype graded heterojunctions

A = e-".¢LO [e-~2~. e-a,x., (-~T- a, )

(110

The I - V equation Using these expressions for the minority carrier den-

sities (7) and (8) in the expressions for the minority currents (4) and (5), the electron and hole currents can be evaluated at the space-charge region boundaries. We obtain

J.(xo) = - eD.~t(Xo)A, (12)

and

Jp(-X.) = - eDpO(-x.)A2 (13)

where

(14)

and

X2 --- e -~'~" t~(-x.)

+ e-"2x" iO(_~.) ( ~ + ~-~ ) • (15)

Assuming no recombination in the space-charge region, the total forward current density is

J = eDpO(-x.)A2 + eDdi(xp)),,. (16)

The excess carrier densities at the space-charge region boundaries are given by the Law of the Junction which is derived for this special case. Integrating (1) from -x . to xp yields

'~P

eVj = dx + AX - kT In _ n(xp)- J . f ]

~. n/z. t n [ -x . / J (17)

where Vj =- dp(-x.)- dp(xp) and A X -~ X(Xp)- X(-x.). Making the usual assumption1" that the contribution of ./. in the above equation is negligible[20], and rearranging terms we obtain

61

where AE~ =-Eg(xp)-Es(-x.). The total junction voltage Vj in terms of the applied forward voltage V is given by

Vj ~- Vr,- V (20)

where Vo = Vj ° is the diffusion voltage. For low-level injection, the majority carrier densities are ap- proximately the equilibrium densities and when the changes in the space-charge region boundaries under bias are negligible, we can rewrite (18) and (19) as

n(xp) = n °(Xo) exp (eV/kT) (21)

p(-x . ) = p °(-x.) exp (eV/kT). (22)

The thermal equilibrium carrier densities n°(xp) and p°(-x,) are obtained by setting J, = Jp = 0 in (4) and (5). Thus the diode current density in (16) can be expressed as

r n~ J = LeDpA2-~o exp {(x. - x.~)/Lg}

+ eD.A, nN~: exp ((xpg- xp)/Lg} ]

× (exp (eV/kT)- 1). (23)

The space-charge boundaries To obtain x, and Xp explicitly as functions of the

applied voltage we need to solve Poisson's equation

d2~b e , dtb d In E (24) -d-~=-~ ~ ' n - p - N ° + N A ) dx dx"

Here E denotes the dielectric constant. Accurate numerical solution for an n+-p structure with a Gaussian doping profile [21] indicates that even a substantial variation of E has little effect on the solution for 4~(x). We simplify (24) by assuming ~ to be constant in the region of interest and by making the depletion approximation. From (3) and the corresponding equation for the quasi- neutral n region we obtain the boundary conditions

d~b(-x.) = _ 1 dx(-x.) (25) dx e dx

d~b(Xp) 1 [dX(Xp) , kT] dx = - e L ~ - r ~-~g]. (26)

In the absence of dipole layers tk is continuous. Solving Poissons equation under these conditions we obtain the voltage drop across the space-charge region

e AX n(Xp)=n(-x,) exp[~-~ (--~-- V,)]. (18)

Similarly,

P(-x") = p(xp) exp [FT (A)c + AE" - vs ) (19)

= e (NDX. 2 + NAXp 2) + 1 r dx(-x.) Vj 2 \ E. ep / e [ x" - ~

Evaluating (17) in thermal equilibrium we also have

(27)

tAs observed by Hall[10], this assumption may produce errors when electron affinity and bandgap vary. However, it is neces- sary in order to obtain a closed form solution.

V D _ I [ A O x _ k T l n ( n__~, ~ ~E~, o,3

(28)

62 A. CHA'I~EP, JEE and A. H. MARSHAK

The total charge per unit area in the space-charge region (SCR) is QscR = e N o x . - eNaxp. Since IQscRt'~eNDX. or eNaxp, we use

Using (27) and (29) to evaluate the derivative, we obtain

1 C - (37)

Nox.=Nax~. (29) ~ % C~

Thus (27) and (29) can be solved simultaneously for x, and xp in terms of Vj given a model for X. Often a hyperbolic tangent is a good approximation[22], i.e. X(x) = (~Xl2)tanh(mxlL)+b. Here m and b are constants, and 8X = Xp - X-. If the width of the space-charge region is small compared to that of the transition region, X(x) can be approximated by its Taylor series expansion up to the linear term. Then

d_.xX __- 8X m dx 2 L

and

AX = (x~ + xp) 8X m 2 L - '

The series term C~ is due to the boundary conditions imposed on the electric field at the edges of the space- charge region. Using (32) and (33) in (37) we have

C = ( ~ + V o ? V ) ,Je (38)

Equation (38) indicates that experimentally determined C -V characteristics cannot be extrapolated back to the voltage axis to yield V~,, as has been done [23]. From (38)

(30) a plot of 1/C 2 vs V gives a straight line with an intercept of Vo = VD + ~/C~ 2. The extra term ~/C~ 2, which to first order is temperature independent, will be significant for low doping.

For an abrupt heterojunction I/C~ = 0 and (38) reduces (31)

to the result first given by Anderson[24]

Solving (27) and (29) simultaneously and using (20) and (30) we obtain

C=(V_~_V)"~ [ abrupt \heterojunction/

(39)

_ 2~ [ 1 + V o - V ' x'/2]

Na x. = N--~ x~

with e V o = f x + k T l n ( N o N a / n . , ) . For this case a (32)

measure of the intercept V, will allow the change in electron affinity to be determined. For a homojunction,

(33) 8X = 0, n~, = n~ = n~, e, = e~, = e and (39) reduces to the classical result.

where

eepe.NaNo = 2(~Na + ~.ND) (34a)

is,-, m~x 1 1 6E ,~ , = 2e--~ (~--~a + ~DD)+ ~ . (34b)

A simplified version of (32) can be derived. Solving (20) and (29) simultaneously and using (30) and (31) and when the changes in the space-charge region boundaries under bias are negligible, we have

2 f f k T / NANo x~,g\ }]1/2 xo-mA[qv ln-- d-- )-V (35)

3. SUMMARY AND DISCUSSION

The I - V characteristic for the graded heterojunction p-n diode is of the form

where

J = Jo(V)[exp (eV/kT) - I]

eDpn~, Jo(V) = ~ exp [-(x,~ - x,)/L,])t2(x,)

eO, n ~p + ~ exp [(xpg - xo)fLg]h,(xp).

In the IlL = l/Lg = 0 limit L,(V) reduces to

(40)

(41)

This is an exact expression in equilibrium and can be used with (28) to evaluate Vr~ Equation (23) along with the auxiliary relations (32) and (33) give the terminal behavior of the diode in terms of material and doping parameters.

Space-charge capacitance

The capacitance per unit area is given by C = IdQ/dVI, where Q represents the charge per unit area in half the junction region. Because of the depletion approximation, we can write

. dxp C = er~/, ~-~/ (36)

eD~n~ 2 eD.ni: Js = ~ 4 NALn" (42)

which is the reverse saturation current for a conventional homojunction diode. The bandgap gradient affects the transport of the injected minority carriers by influencing the carrier distribution in the quasi-neutral regions: The current, therefore, depends on the width of the transition region outside the space-charge region. Since this width is determined by the width of the space-charge region on either side, Jo depends on the applied voltage only through x. and xp. In the expression for Jo (41), the first term is a function of x, via the factor h2(x,)exp(xdL~) and the second term is a function of x, via the factor

Theory for nonequilibrium behavior of anisotype graded heterojunctions

A:(x~) exp(-xo/Ls). The two terms are additive since A~ and ,~: are always positive. We observe that ,Xl and ~.2 are of the form

)~1 a l e klxp - - b l e k2xp

= Cl e km~p - d l e k2xp (43)

/12 e-CtlXn - b2 e-~2x"

~2 = c : e-~ '~" - d2 e -~2~" (44)

where the as, bs, cs, ds are all independent of x, and x,. Since the numerator and denominator are of the same order, A~ and A2 are only weak functions of applied voltage and we can focus attention on the factors exp (x,/Ls) and exp (-xp/Lg). If Ls is positive, that is, the n-side has a smaller bandgap, increasing reverse bias would cause the hole current to increase and the electron current to decay. If the p-side has a smaller bandgap increasing reverse bias would have the opposite effect. But, in both cases, large reverse bias would cause more reverse current. If the space-charge region extends beyond the transition region, i.e. if x. + xp > L, then the second term in (4) and (5) reduce to zero in the region of interest. The structure of the problem then becomes identical to that of a classical p-n homojunction.

As we have seen, Jo ceases to depend on x, and xp when the bandgap gradient tends to zero; it depends on the bandgap gradient through x,(Ls), x p ( L g ) , ( x . s -

x,)/L,, (xpg-xp)/Lg, kt, k2, al, and a2. A reexamination of the derivation of the diode equation shows that the first four members of this list have entered the I -V equation as parameters of the expressions for n°(x) and p°(x). Thus dependence of the current on bandgap gradient via these parameters is essentially due to the fact that the thermal equilibrium pn-product has to be modified for a semiconductor with position-dependent band structure. That the reverse current in p-n junctions as well as the common-emitter gain of bipolar transistors are directly affected by the modified pn-product was demonstrated in [18]. Note that kb k2, a t and az are parameters in the expressions for the excess carrier distributions and the dependence of Jo on the bandgap gradient through them represent the effect of the bandgap gradient as an additional force on the injected carriers. From the definitions of k~, k2, al, a2 (10) and (11), we see that they would differ appreciably from their limiting values (lim 1/L~ ~0) of - l /L, , 1/L,, -l/Lp and l/Lp, respectively, when the distance over which the bandgap changes by 2kT is small enough to be compar- able to the diffusion lengths. Osipov et a/.[12] have also demonstrated the change in the distribution of injected carriers in the quasi-neutral transition region for different thickness of this region and different ratios of the diffusion length and bandgap gradient.

While the expressions f o r the minority currents are independent of X(x), the diode equation depends on X via boundary conditions imposed for the solution of Poisson's equation. This is a direct consequence of the approximation that in the quasi-neutral region the applied field is negligible, which is a good approximation for low-level injection.

Figure 2 shows the numerical results from (40) for a

o_

o_

.-j

o

63

E

E ?

'o I I I I 0 . 0 5 0.10 0.15 0 . 2 0

v (v ) (a)

m

_o

2 0

6 0

f

I I I I 4 0 8 0 120 180

- v ( v )

(b)

Fig. 2. Terminal behavior of nGe-pGaAs device shown in Fig. 1 with x,g = xpg = U2; (a) forward bias, (b) reverse bias.

graded nGe-pGaAs heterojunction for v.arious values of L. The material parameters used are given in Table 1. We have assumed ND = NA = 10 ~8 cm -3, T, = Tp = 0.01/~s and m = 5.3.

We observe that the reverse current Jo(V) is larger for smaller L and that the nonsaturating behavior of Jo is more prominent for small L. Analysis of the effects of doping level indicates that the nonsaturating behavior is more marked for symmetric doping than for asymmetric doping and that lower doping produces higher reverse current, as in the conventional diode. Additionally, the nonsaturating behavior is more dramatic. This is prob- ably due to the greater movement of the SCR boundaries

64

Table l. Properties of Ge and GaAs at 300 K[25]

Ge

A. CHATTERJEE and A. H. MARSHAK

where ~(x) represents either ti or ~. Here f~ and f2 are known functions once models for the material

GaAs parameters are specified, This equation cannot be analy- tically solved in the most general case. However, computer simulation is possible as in, for example, [8]. Finally we note that as the grading becomes more pronounced other transport mechanisms like tunnelling and interface state recombination become important and must be included in the analysis [28, 29].

Eg(eV) 0.68 1.~3

)~(eV) 4.13 4.07

~ / [ 16 1 0 , 9 o

ni(cm -3) 2.5 x 1013 107

;~n(Cm2/V-s) 3900 8600

;: (cm2/V-s) 1900 250 P

under bias for lower doping. These results are insensitive to the value of m. The I - V characteristics are typical of a graded heterojunction and are consistent with experimental results for reverse current reported in [26, 27].

For an abrupt heterojunction (41) reduces to

eD~,ni~ eD, n~p lim Jo(V)=T-cr- . . + c o l , r~ L,/Q,~" (45)

Using (28) with the affinity rule BE,, = - ~(, we have

NANo eVD + BE, = kT In n~,, (46)

Integrating (2) from -x , , to x,, and using BE,,= - 8X - fiEf, we also have

eVD + BE, = kT In Na~N~. (47) n?,

By using (46) and (47) in (45), the diode eqn (40) can be written as

J = [ L~[eDpNA e ~Vo+SE,)/kT + eD.NDe-~vo*~E~);kT

( abrupt × (exp (eV/kT) - 1) \heterojunction]' (48)

Because of the discontinuities in the energy band edges at the interface, the barriers to electrons and holes have different magnitudes. The dominant term will depend on the doping and the relative values of 8E~ and ~E,. and, as observed by Anderson[24], in most cases the current will consist almost entirely of electrons or holes. When there are spikes in the conduction or valence band edges, (48) differs from Anderson's result, which was derived by considering thermal injection of carriers over barriers.

The analysis of graded compositon heterojunctions can be extended to include variations in mobility, dielectric constant, carrier lifetime, density of states and im- purity doping. When the general form of the electron and hole current equations[17-19] are used in the continuity equations, the excess carrier densities can be described by an equation of the form

d2s ~ , , d~ + - ~ - r f , ( x J - ~ x f2(x)l~ = 0 (49)

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Theory for nonequilibrium behavior of anisotype graded heterojunctions