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Modeling semiconductor heterojunctions in equilibrium
Solid-State Electronics Vol. 25, No. 8, pp. 683Jo91, 1982 Printed in Great Britain.
0038-1101/82/080683~9503.0010 Pergamon Press Ltd.
MODELING SEMICONDUCTOR HETEROJUNCTIONS IN EQUILIBRIUM M. S. LUNDSTROMand R. J. SCHUELKE Purdue University, School of Electrical Engineering,West Lafayette, IN 47907, U.S.A. (Received 30 November 1981; in revised form 4 February 1982)
Abstract--Computation of the equilibrium electrostatic potential and energy band diagram of semiconductor devices with nonuniform composition is considered. We first establish the relationship between the electrostatic potential and the energy band edges and then derive a Poisson-Boltzmannequation for the electrostatic potential. Although the equation is more general, we consider only the simple example of an abrupt heterojunction diode in detail. An analytical solution for doubly-intrinsic, abrupt heterojunctions is also presented. The modeling approach is then generalizedto include Fermi-Diracstatistics for free carriers. We also discuss the applicationof the formulation to a variety of heterojunction devices (e.g. heavily doped silicon and gate-modulateddevices).
l. INTRODUCTION
Although many semiconductor devices consist of compositionally nonuniform material, the term, "heterojunction", is usually restricted to abrupt (or nearly so) junctions between two dissimilar materials. We shall, however, refer to any compositionally nonuniform device as a heterojunction. Our definition includes not only conventional heterojunction devices, such as semiconductor lasers[I] and solar cells[2], but also metalinsulator-semiconductor (MIS) and silicon p-n junction devices. In the MIS device, a semiconductor heterojunction occurs at the insulator (wide-gap semiconductor)--semiconductor interface. Because heavy impurity doping modifies the band structure of silicon[3], a heterojunction occurs between the lightly and heavily regions of a p-n junction. In this paper we restrict our attention to semiconductor heterojunctions and do not consider the metal-semiconductor heterojunction. Our goal is to present a unified formulation for modeling a variety of semiconductor devices. Several different techniques have been employed to analyze various compositionally nonuniform semiconductor devices. Since 1962 when Anderson [4] applied the depletion approximation to analyze abrupt heterojunctions, the subject has received much attention (see, for example,J5] and references cited therein). However, because it is based on the depletion approximation and restricted to abrupt junctions, the Anderson model does not apply to many devices. By resorting to numerical solutions and assuming abrupt heterojunctions, Cserveny[13] and Adams and Nussbaum[14] removed the depletion approximation. Approximate techniques for modeling graded heterojunctions h~.ve also been presented [7-9]. Adams and Nussbaum, and recently van Roos [31], have, however, raised some fundamental questions concerning heterojunction energy band diagrams. These questions will be considered in this paper. Because they significantly influence silicon device performance, so-called bandgap narrowing effects must be considered[3, 10]. The nonuniform band structure (to
first order, a bandgap shrinkage) associated with heavy impurity doping lowers the gain of bipolar transistors [10] and the open-circuit voltage of solar cells [3]. To model these heavy doping effects, transport equations for semiconductors with nonuniform band structure have been developed[ll, 12], but it is difficult to apply them because our understanding of heavily doped semiconductors is limited. For this reason, approximate methods, quite different from conventional heterojunction theory, have been developed to model heavily doped silicon devices [12, 25]. In this paper we consider the equilibrium theory of ideal (i.e. no interface charge) semiconductor heterojunctions. To permit calculation of the energy band diagram of a heterojunction, we first relate the electrostatic potential to the band edges and to the intrinsic energy level and then derive a simple, yet general, Poisson-Boltzmann equation for the electrostatic potential. We then use this equation, which includes free carrier charge and applies to both abrupt and graded structures, to analyze abrupt heterojunctions and compare the results to recently proposed alternative models[14,31]. Since carrier degeneracy is important in many heterojunctions, we also generalize the theory to include Fermi-Dirac statistics. Finally, we discuss the application of this equation to MIS and silicon p-n junction devices and show that these devices can be modeled as heterojunctions.
2. ENERGY BANDS IN NONUNIFORM SEMICONDUCTORS
Figure 1 shows the energy band diagram of a semiconductor with nonuniform composition[7, l l]. The validity of this energy band model (the underlying assumption of our paper) has been discussed by Marshak and van Vliet[l 1]. From Fig. l we observe that
683 SSE VoL 25, No. ~-A
E~(x) =Eo-qV(x),
(1)
E~(x) =Eo-X(X)- qV(x),
(2)
684
M.S. LUNDSTROMand R.J. SCHUELKE and I
no = N c ( x ) e ¢~F ~ u k ~ .
~/
qV(x)
[EL(X)----t------"
(5)
Using (2)-(5) we obtain the intrinsic energy level, kr
x(x) -
x(x) = E o -
log r/ N,,(x)]Nc(x)J
+
EG(X)
POSITION (x)
-
Fig. 1. Energy vs position for a semiconductor with nonuniform composition.
qv(x),
which is not, in general, parallel to V(x). Consequently, for a heterojunction one cannot identify ( - E t / q ) with electrostatic potential as is commonly done for homo junctions [17]. Although the energy band model outlined above applies only to materials in which X and Eo vary slowly, we will follow conventional practice and apply it to abrupt heterojunctions. Consider, for example, the energy band diagram for an abrupt heterojunction presented in Fig. 2. To evaluate the discontinuity in the conduction and valence band edges at the junction (x~), we use (2) and (3) to write
and
AEc = Ec(xj-) - Ec(xj +) = Ix(x() - X(xj )] Eo(x) = E c ( x ) - E ~ ( x ) ,
(3)
where Eo is a reference level, EL the local vacuum level, V is the electrostatic potential, X the electron affinity, and q is the magnitude of the electronic charge. It is important to note that the electrostatic potential is not, in general, parallel to either band edge or to the intrinsic level[15, 16] as it is in a uniform semiconductor. To relate the intrinsic energy level to Ec and Ev we assume Boltzmann statistics and write the equilibrium carrier densities as po = Nv(x) e ~v- EF)/~T
AEv = Ev(xj ) - Ev(xj ÷) = [x(xj +) - x(xj-)] + [E~(xj +) - Eo(xj )],
a E , -- Ex(xj ) - Er(xj +) 1
kT, rN,,(x~ ) N c ( x / ) ] -~-log [Nv(x+) Nc(xi_)j.
(4)
n-GoAs
---v E
0.0
.
V
4.13
, E¢
0"12
i
.07
i
.14
I
t i
.21
+
= [x(xj*) - x(x~-)] + ~[E~(x~ ) - E~(x~ )]
.
T
(8)
which are known as affinity rules. Similarly, for the discontinuity in the intrinsic level, we use (6) to obtain
.52
E
(7)
and
p -Ge
EL
(6)
i
.29
i
.36
x(/=m)---~
Fig. 2. Ge--GaAsenergy band diagram.
i
,43
i
.50
(9)
Modeling semiconductor heterojunctions in equilibrium 3. THE POISSON-BOLTZMANN EQUATION
Since the built-in potential of the junction is v~, =
v( vo-
v(o),
(lO)
To obtain the electrostatic potential within a heterojunction, we must solve Poisson's equation, dD dx
where x = 0 and x = W are the locations of the contacts, we can use (2), (3) and (6) to express Vbi as qVb, = - [ E c ( W ) - Ec(O)] - [X(W) - X(0)], qVb, = -
[E~(W)-
E,,(0)] - [x(W)-
(11)
x(0)]
- [E~(W)-
E~(0)],
685
(12)
q(P
(19)
n + N+),
where D is the electric flux density and N + = N o + - N a is the ionized dopant density. In equilibrium, the carrier concentrations, p and n, are related to the electrostatic potential by (14), (15) and (6) as
(X(x) - x(O)) + k T log l Nc(O) J + q(V(x) - V~) kT
n = ni(O) exp
(20)
and - (X(X) - x(O)) - (Eo(x) - Ea(O)) + k r log ~v(O)
p = ni(0) exp
- q(V(x) - V~)
kT
(21)
'
where
or
qVb, = - [ E , ( w )
- E,(O)] - Ix(W)-
-{[EG(W)-
..... Eo(O) kr, fNv(0)l qV, : - EF * r,o - X(O)- ~ + T log [ N---~-~J"
x(O)]
[ N v ( W)Nc(O) ]
EG(O)]+-~ log L ~ J "
(22) (13)
Each of the above three equations is just the difference of the field-free work functions between the two contacts. To relate Vb~ to the carrier densities, we first express the carrier densities in terms of the intrinsic level
To simplify these equations, we define two new variables,
[ m(x)N (x) ]
AG = - [Ea(x) - Ea(O)] + k T log L
~
J
(23)
as
po(x) = n~(x) e ~EI EV~/kT
(14)
and
and
(X(x) - X(0)) + k T log [ N c ( x ) ] ] LNc(O)JJ
no(x) = ni(x) e tE~-~1)/kr,
Y
(15)
Ac
(24)
'
which enable us to write
where hi(X) = x / ( N v ( x ) N c ( x ) ) e--EO(x)/2kT
(16)
is the position-dependent intrinsic carrier concentration. Using (14) and (15) we obtain kT ,
f no( W)po(O)]
- [ E l ( W ) - El (0)] = -~- log [no(0)po( W)J'
(17)
which we then use with (13) to express Vbi as q Vb, = ~ - log [no( W)po(O)]
t ~ J
-
Ix(W)
- x(O)]
o(o)1 + ,og [ N o ( W)Nv(0)J" (18)
no = n,(O) exp [ qV' ; ~ Aa ]
(25)
po = n~(0)exp r / - q V ' - (y - I)A6/,] kT J L
(26)
and
where V' = V - V,~ Note that our choice of reference for the potential makes V' zero at x = 0 if the material there is intrinsic. From (25) and (26) we obtain the equilibrium np product as noPo = n,2(x) = ni2(O) e ('~kr).
(27)
686
M.S. LUNDSTROMand R.J. SCHUELKE
Since Aa affects noPo like a bandgap shrinkage, we call it an effective bandgap shrinkage[12]. The second parameter, 3' (called the effective asymmetry factor [12]), measures the change in the conduction band edge and density of states relative to the change in both bands. Since (25) and (26) relate the equilibrium carrier densities to the electrostatic potential, we can use them with (19) to express Poisson's equation in terms of V' alone as
which is the intrinsic Debye length in the reference material. The resulting, normalized Poisson-Boltzmann equation is
d2U~-4 RI dRdUdrdr
R-~[Cuc92)sinh[U+(y-l/2)U~] -
D].
(35)
4. THE IDEAL, ABRUPT ItETEROJUNCTION
d2V '
1 dKdV'
d ~ - + K dx dx
-q[ni(O)e, Ke~
~v, ~ ,a~)/kT~
t
- ni(O) e ""v'+~a~)/kr) + N+],
(28)
where K is the relative dielectric constant. Equation (28) describes the equilibrium electrostatic potential in a material with nonuniform band structure (assuming Boltzmann statistics). The solution, V'(x), can be used with (2), (3) and (6) to determine the energy band diagram. Since the terms, yAG and (1 - y)AG are zero in homogeneous materials, (28) reduces to the conventional Poisson-Boltzmann equation[6] when the energy band structure is uniform. In many cases we can assume that the dopants are completely ionized (i.e. N ~ = N = Nn - NA). For typical lightly doped semiconductors, this assumption is valid near room temperature, and, because the overlap of wavefunctions precludes the existence of localized states, it is also a good approximation for heavily doped semiconductors[12]. However, for lightly doped semiconductors at low temperature, or even at room temperature when accumulation layers exist [29], the dopants may be only partially ionized. In such cases, N + must be evaluated using Fermi-Dirac statistics. When Fermi-Dirac statistics also apply to the free carriers, (23) and (24) must be modified as discussed in Section 6. To simplify the ensuing algebra, we make the following change of variables: V¢ U = -
kT q
AG
u~ = ~-~ N ~
D=- 2ni(0) r=
X
Lo(O)
(29)
In this section, we apply (35) to an ideal heterojunction consisting of an abrupt junction (located at x = xj) between two materials (surface states at the interface are not considered). The contacts of the device are located at x = 0 and x = W, and the subscripts L and R are used to denote the materials to the left and right of xj respectively. For x < xj we obtain the potential by solving d2U
-~
• I
= stun (U) - DL
d2U d P = R ~[e~ud~ sinh [U + ( y - I/2)U~] - DR]. r > rj
U(0) = sgn (DL) log [rDLI + ~/(DL2 + 1)l
U(W) = - (Y - 1/2) U~ + sgn (DR)log [IDR[ e(-Uc;:' + x/(DR2 e~- u ° + I)], (39) where sgn(D) is the sign of D. The normalized built-in potential is simply the difference in the above two quantities. We integrate (36) and (37) once to obtain the normalized electric field as d--U-U= -+ [2(cosh(U) - DLU + CL)I ~/2 dr r < r~
(32)
d---r-= -+
(40)
(e (Ud2~ cosh [U + (7 - 1/2) UG] -- DRU -ll/Z + CR)J •
(34)
(38)
and
and
[ eoK (O)k T ] 1/2 Lo(O) = [ ~ j ,
(37)
To evaluate the potentials at the two ohmic contacts we assume space charge neutrality at the contacts and solve (36) and (37) for
(31)
(33)
(36)
and for x > xj by solving
(3O)
K R= K(0)
r < rj
r > rj
(41)
The ambiguity in the sign of the electric field is removed by considering the potential drop between contacts; the proper sign is the sign of U ( W ) - U(0). If we assume dU/dr = 0 at the ohmic contacts, then
where
CL = DL U(0) - cosh [ U(0)]
(42)
Modeling semiconductor heterojunctions in equilibrium and CR = DR U(W) - e' uc/2) cosh [ U(W) + ( y - 1/2) Uo ] (43) are the two constants of integration in (40) and (41). To evaluate the electrostatic potential at the metallurgical junction we apply the boundary condition on the normal displacement field,
687
the band parameters as Ao = -0.83 eV and Y = 0.17. The electrostatic potential at the contacts is then evaluated from (38) and (39) as V'(0)= 0.0 and V ' ( W ) = - 0 . 2 7 volts and the two constants of integration from (42) and (43) as CL = --1 and CR = - 9 . 7 X 10-s. For the doublyintrinsic device we solve (45) for the potential at the junction to obtain
U(rj) = log [a2 -+ x/(a2 z - a3)]
(46)
1 - Re t uc/2) a2 = 1 - Re
(47)
["1 - Re "-v)uG ] a3= [. ~_~e~UG j,
(48)
where r=rj
and obtain cosh [ U(rj)] - Re Iu°/2) cosh [ U(rj) + (3' - I/2) Uol + ( R D , , - D , 3 U ( r j ) + C , - R C R =0.
and (45)
Except for the special cases of homojunctions and doubly-intrinsic heterojunctions, closed-form solutions to (45) do not exist. Nevertheless, (45) is readily solved for U(rj) by iterative methods. Since we can proceed no further analytically, we resort to numerical methods[6] to integrate (40) and (41) and thereby obtain U(r). With U(r) known, the equilibrium energy band diagram is computed from (2), (3) and (20). Figure 2 shows such an energy band diagram for a room temperature Ge-GaAs heterojunction assuming complete ionization of dopants with NL = - 3 x 10'6 and NR = 1 x 10 '6 cm -3. Table 1 lists the other parameters used for the computations. For this example, our solution (which includes free carriers) agrees well with the simple Anderson model based on the depletion approximation. Our formulation, however, also applies to heterojunctions with inversion or accumulation layers and to graded structures. The doubly-intrinsic heterojunction we consider in the following section is an example for which depletion layer theory cannot be applied.
which, for the Ge--GaAs example, gives V'(xj)= -1.4 x 10-3 volts. Since the doping density is zero we can integrate the expressions for the electric field, (40) and (41), and write
(r-rj)=log[
tanh(U/4) ] [tanh [ U(rj)/4]J
r
(49)
and l" t a n h [ ( U - U(W))/4I ] ( r - rj) = - R '/: e ~-ua4) log [tanh [(U(rj) - U( W))/4]J" r > r~
(50)
Figure 3 shows a computed energy band diagram for this doubly-intrinsic device. The solution agrees with the result recently obtained by Chatterjee and Marshak[30] but not with that of Adams and Nussbaum [14].
5. T H E DOUBLY-INTRINSIC A B R U P T I I E T E R O J U N C T I O N
6. GENEIO.LIZArlONTO l~l~fl-DnOtC STATtS'TlCS
Adams and Nussbaum[14] noted that an analytical solution for the electrostatic potential can be obtained for abrupt, doubly-intrinsic heterojunctions. The theory outlined in the previous section can be directly applied to this case by setting DL = DR = 0. In this section we consider a specific example, a germanium-gallium arsenide doubley-intrinsic structure. We begin by using the paramters in Table 1 to compute
When Boltzmann statistics cannot be assumed, the preceding formulation must be extended. Fermi-Dirac statistics may be important, for example, in MOS or n-N heterojunctions with strong accumulation (or inversion) layers. In this section we extend the formulation to include carrier degeneracy while retaining a simple, Boltzmann-like form. When Fermi-Dirac statistics apply, (4) and (5) become
Table 1. Material parameters for Ge--GaAsheterojunction
po = Nv(x)F,~2(no)
(51)
no = Nc(x)F,/2(nc),
(52)
Material
Ge
GaAs
4.13
4.07
and
Parameter X Eo
0.7
1.45
Nc
1 . 0 4 x I 0 TM c m -a
4 . 7 x 10 t? crrt -3
Nv
6.0 x l O l a crn -3
7.0 x l 0 t S e m -a
K
16
11.5
where F,2 is the Fermi-Dirac integral of order one-half with
nc =
(EF - E~) kT
(53)
688
M.S. LUNDSTROMand R. J. SCHUELKE Ge
27
GaAs
/
i
4.07
4.13 E
(ev)
1.45
E I =E F
0.7
0.69
I
6
,o
ioo'oo x (/~m)
-
Fig. 3. Doubly-intrinsicGe--GaAsenergy band diagram (for illustrative purposes, the band-bending and discontinuities have been exaggerated).
and ( E v - E~) kT
*Iv =
(54)
These parameters are related to the normalized potential, U, by
yet incorporate Fermi-Dirac statistics, by including the influence of carrier degeneracy in the two parameters, Ac and 3,[12, 18]. To accomplish this, we generalize the definitions of 3' and Ao as •
[Nv(x)Nc(x)3
EG(0)I + k T log [
Ao = - [ Ea(x ) -
~
j
[F,;2(,o)F,;:(,c) ] e~ t, e , c j
+ kT log [ ~¢ = U ( x ) 4 X ( X ) - x(O) + !/2 log J " V'V'~C(O ) kT
] -
kT
[ F,;z(r/~(0))F,;2(r/c (0))1 log [ e~o~O~e.~O > j
(57)
(55) and
(x(x)- x(o)) + k r
fNc(x)Fv2(nc)]
log [ ~
?
AG
and fly = - U ( x ) -
X ( x ) - x(O) kT
.j -
E a ( x ) - Ea(O) kT
fm(0)e<-~o
- 1/2 log L
(56)
When deriving (55) and (56), we made the assumption that Boltzmann statistics could be applied for intrinsic semiconductors. We can retain the Boltzmann-like formulation, (28),
.~. ~.
fF, n(nAO))]
,og [ e,-~-~--J
(58)
If we attempt to analyze abrupt heterojunctions including carrier degeneracy, then the Boltzmann-like formulation only complicates the integrations. As we discuss in the following section, however, for many devices the complexities introduced by graded transitions or nonuniform doping make it necessary to solve (28) directly by numerical methods. When direct numerical techniques are employed to solve the nonlinear equation, (28), little additional complexity is introduced by the weak nonlinearities in AG and 3'. The
Modeling semiconductorheterojunctions in equilibrium Boltzmann-like formulation, therefore, is a simple, convenient way to include carrier degeneracy in a numerical model.
7. APPLICATION TO OTHER HETEROJUNCTION STRUCTUR~
Although the formulation is general (it applies to semiconductor devices with nonuniform composition) we discussed only the abrupt heterojunction because it was possible to make some progress analytically for that case. The mathematical difficulties introduced by graded composition and nonuniform doping may preclude analytical solutions, but (28) remains valid. We now discuss the application of this equation to several heterojunction devices. For heterojunctions with compositional grading, we must either make simplifying approximations to achieve analytical solutions or resort to numerical methods. Although solutions for graded heterojunctions have been reported[7-9], the assumptions involved limit the applicability of these models. Oldham and Milnes[7], for example, calculated the electrostatic potential in a graded heterojunction by assuming uniform doping, neglecting minority carriers, and dropping the term in (28) that involves the gradient of the relative dielectric constant. Pearson et a1.[8,9] modeled graded heterojunctions by computing the electrostatic potential for the corresponding abrupt structure and then using (3) and (4) to compute Ec and Ev. As they noted, their procedure is valid if the graded region is narrow. Since (28) includes majority and minority carrier charge, the variation of the dielectric constant, and Fermi-Dirac statistics, a direct numerical solution can remove many commonly used approximations. Numerical simulations of graded bandgap solar cells (but neglecting the effects of carrier degeneracy) using a formulation similar to ours have already been reported[24]. Since the numerical solution to (28) is readily obtained[20], its use may be preferred in many cases. Within the last decade, it has been demonstrated that the heavy impurity doping used to fabricate pn junction devices modifies the energy band structure of silicon[3, 10]. Because of these so-called bandgap narrowing effects, silicon devices that contains heavy impurity doping must be regarded as heterojunctions. The formulation we have presented should, therefore, be equally applicable to modeling heavily doped silicon devices. Indeed, we choose our notation to correspond to that commonly used in modeling heavily doped silicon devices [3, 12]. When modeling heavily doped silicon devices, we find it convenient to choose the reference material as lightly doped silicon (with unperturbed energy band structure) rather than the material located at x = 0. If we denote the intrinsic carrier concentration in the lightly doped semiconductor (with unperturbed energy bands) as n,o and the equilibrium np product in a heavily doped semiconductor as n~e,then (27) becomes ni,2 =
nio 2
e (aOlkT).
(59)
689
Since the effective intrinsic carrier concentration in a heavily doped semiconductor, n~e, can be inferred from electrical measurements[27], the parameter, A~, can be determined. Most authors choose 3' arbitrarily when modeling heavily doped semiconductors because no experimental measurements for it have been reported. Fortunately, this expedient is justified because many properties of the device are insensitive to T[12, 25]. Simulations of heavily doped silicon devices based on this formulation have been reported[18,26] and provide confirmation that terminal I-V characteristics are not sensitive to 3' under many operating conditions. The metal-oxide-semiconductor (MOS) capacitor is a heterojunction device controlled by a gate voltage. Similarly, the properties of a heterojunction between GaAs and AIGaAs can also be controlled by a voltage applied to a metal gate[21,22]. Such gated heterojunction structures show promise as high-speed MESFETS [21,22]. The modeling approach we presented is directly applicable to these gated heterojunction structures; we simply define the boundary conditions (by setting the gate voltage), then solve (28) numerically. Computations performed by the authors show that (28) is readily solved by numerical techniques, even for MOS structures in which an abrupt change in E~ of about 7 eV occurs.
8. DISCUSSION
Although the abrupt heterojunction is only a special case described by (28), it deserves further discussion. Cserveny, who also included the effects of mobile charge in the abrupt heterojunction[13], obtained results that agreed with the Anderson model (for structures in which both semiconductors were depleted). The same problem, however, has recently been reconsidered by Adams and Nussbaum[14] and by yon Roos[31]. These authors proposed alternative models which differ from each other and from Anderson's. The preceding discussion has shown that our model agrees with the Anderson model of an abrupt heterojunction. In this section we compare the underlying assumptions of Adams and Nussbaum and those of yon Roos with ours. As Adams and Nussbaum noted, they used the intrinsic level to measure the electrostatic potential within a heterojunction[14]. When this assumption is made, the intrinsic level is forced to be continuous at the junction in order to insure continuity of the electrostatic potential. As a result, the discontinuities in the conduction and valence bands differ substantially from those predicted by the Anderson model, and, in addition, a discontinuity in the local vacuum level must be hypothesized[14]. Although using the intrinsic level to measure the electrostatic potential follows long-standing practice in semiconductor device physics[17], it cannot be done for a material with nonuniform composition because the electrochemical potentials contain position-dependent terms[15, 16,28]. For compositionally nonuniform materials, the chemical potential for electrons (under
M. S. LUNDSTROMand R. J. SCHUELKE
690 nondegenerate conditions) is given by[28] #c n = k T log (n(x)) + f(x, T),
where f(x, T) represents the contribution to the internal energy arising from the interaction of the electron with the crystal lattice. As a result of the position-dependent term, /', the quantity ( E p - E t ) is not the chemical potential and, consequently, the intrinsic level is not parallel to the electrostatic potential. Figure I illustrates the relationship between the energy band edges and the electrostatic potential, and, as (2), (3) and (6) show, neither the conduction band edge, nor the valence band edge, nor the intrinsic level may serve as a reference for the potential in a material with nonuniform composition. Another approach has recently been taken by von Roos who assumed that the conduction band was parallel to the electrostatic potential[31]. Because of his assumption, the energy band diagram he obtained has a continuous conduction band edge but a discontinuous valence band edge and intrinsic level. As we discussed in Section 2, however, the conduction band (nor for that matter the valence band or intrinsic level) is not parallel to the electrostatic potential. The validity of the energy band model we assumed for compositionally nonuniform material (Fig. 1) has been discussed by Marshak and van Vliet[ll]. For an abrupt heterojunction, (7)-(9) show that discontinuities in the conduction band, valence band, and the intrinsic level exist at the junction, however, no discontinuity in the vacuum level exists. Our computations for the doped and undoped Ge-GaAs abrupt heterojunction show that the difficulties which led Adams and Nussbaum to propose their alternative model do not arise if the electrostatic potential is defined as shown in Fig. 1.
9. SUMMARY AND CONCLUSIONS
In this paper we presented a Poisson-Boltzmann equation that describes the electrostatic potential within equilibrium semiconductor heterojunctions. The equation includes the effects of free carriers, is valid for both Boltzmann and Fermi-Dirac statistics, and applies to graded or abrupt structures. We used this equation to analyze both doped and undoped, ideal (no interface charge) abrupt heterojunctions. The importance of not using the intrinsic energy level to measure the electrostatic potential within a heterojunction has been stressed. Several different techniques for modeling semiconductor devices with nonuniform composition exist, but the generality of our formulation is an important advantage. When the device being modeled has a uniform band structure (and when Boltzmann statistics apply) (28) reduces to the conventional Poisson-Boltzmann equation. Consequently, the homojunction is a special case of this formulation. Our modeling approach has previously been used to analyze heavily doped silicon devices [12, 18, 26], but with the proper choice for Aa and 7, heterojunction or MIS devices can also be modeled. Another advantage is that Fermi-Dirac statistics are
included, but the simple Boltzmann-like formulation is preserved. The formulation presented in this paper should be particularly valuable in numerical simulation of semiconductor devices since a wide variety of devices (all viewed as heterojunctions) can be modeled using a single formulation. We note that the extension to non-equilibrium devices can be readily made[12]. Previously reported simulations using this[18,26] and a similar[24] formulation, demonstrate the viability of this approach. Acknowledgement--This material is based upon work supported
by the National Science Foundation under Grant No. ECS8105956 and, in part, by the 3M Company. We also thank F. Sanii and R. J. Schwartz for critical reviews that improved the manuscript. REFERENCES
1. M. B. Parish, Proc. IEEE. 64, 1512-1540(1976). 2. H. J. Hovel, Solar cells. In Semiconductors and Semimetals, Vol. 11. Academic Press, New York (1975). 3. F. A. Lindholm and C. T. Sah, IEEE Trans. Electron Dev. ED-24, 299-304 (1977). 4. R. L. Anderson, Solid-St. Electron. 5, 341-351 (1962). 5. A. G. Milnes and D. L. Feuct, Heterojunctions and Metal Semiconductor Junctions. Academic Press, New York (1972). 6. A. Nussbaum, Semiconductor Device Physics. Prentice-Hall, New Jersey (1962). 7. W. G. Oldham and A. G. Milnes, Solid-St. Electron. 6, 121-132 (1963). 8. D. T. Cheung, S. Y. Chiang and G. L. Pearson, Solid-St. Electron. lg, 263-266 (1975). 9. S. C. Lee and G. L. Pearson, Solid-St. Electron. 24, 563-568 (1981). 10. H. J. J. DeMann, IEEE Trans. Electron Dev. ED-Ig, 833-835 (1971). 11. A. H. Marshak and K. M. van Vliet, Solid-St. Electron. 21, 417-427 (1978). 12. M. S. Lundstrom, R. J. Schwartz and J. L. Gray, Solid-St. Electron. 24, 195-202(1981). 13. S. I. Cserveny, Int. J. Electronics. 25, 65-80 (1968). 14. M J. Adams and A. Nussbaum, Solid-St. Electron. 22,783-791 ( 1979). 15. C. T. Sah and F. A. Lindholm, Carrier generation, recombination, trapping, and transport in semiconductors with position-dependent composition. IEEE Trans. ED, ED-24, 358-362 (1977). 16. K. M. van Vliet and A. H. Marshak, Solid-St. Electron. 23, 49-53 (1980). 17. W. Shockley, Electrons and Holes in Semiconductors. van Nostrand, Princeton, New Jersey (1950). 18. M. S. Lundstrom, R. J. Schwartz and J. L. Gray, Modeling solar cells containing heavily doped regions, Conf. Rec., 15th IEEE Photovoltaic Spec. Con[., 400--405(1981). 19. H. Kroemer, J. AppL Phys. 52, 873-878 (1981). 20. R. W. Klopfenstein and C. P. Wu, IEEE Trans. Electron Dev. ED-22, 329-333 (1975). 21. H. L. Stormer, A. C. Gossard and W. Wiegmann,Appl. Phys. Lett. 39, 493-495 (1981). 22. D. Delagebeaudeufand N. T. Linh, IEEE Trans. Electron Dev. ED-28, 790-795 (1981). 23. H. Kroemer, Problems in the theory of heterojunction discontinuities. CRC Critical Reviews in Solid-St. Sci. 5, 555564 (1975). 24. J. E. Sutherland and J. R. Hauser, IEEE Trans. ED-24,363-372 (1977). 25. M. S. Adler, A method of achieving and choosing variable density grids in finite difference formulations and the importan~:e of degeneracy and band gap narrowing in device modeling. Proc. of the NASECODE Conf. pp. 3-30, Trinity College, Dublin, Ireland (1978).
Modeling semiconductor heterojunctions in equilibrium 26. M. S. Lundstrom, Numerical simulation of silicon solar cells Purdue University, School of Electrical Engineering Tech. Rept., TR-EE 80-27 (1980). 27. J. W. Siotboom and H. C. de Graaff, Solid-St. Electron. 19, 857-862 (1976). 28. A. C. Smith, J. F. Janak and R B. Adler, Electronic Conduction in Solids. McGraw-Hill, New York (1967).
691
29. R. C. Jaeger and F. H. Gaensslen, IEEE Trans. Electron Dev. ED-27, 914-920 (1980). 30. A. Chatterjee and A. H. Marshak, Solid-St. Electron. 24, 1111-1115 (1981). 31. O. von Roos, Solid-St. Electron. 23, 1069-1075 (1980).
0038-1101/82/080683~9503.0010 Pergamon Press Ltd.
MODELING SEMICONDUCTOR HETEROJUNCTIONS IN EQUILIBRIUM M. S. LUNDSTROMand R. J. SCHUELKE Purdue University, School of Electrical Engineering,West Lafayette, IN 47907, U.S.A. (Received 30 November 1981; in revised form 4 February 1982)
Abstract--Computation of the equilibrium electrostatic potential and energy band diagram of semiconductor devices with nonuniform composition is considered. We first establish the relationship between the electrostatic potential and the energy band edges and then derive a Poisson-Boltzmannequation for the electrostatic potential. Although the equation is more general, we consider only the simple example of an abrupt heterojunction diode in detail. An analytical solution for doubly-intrinsic, abrupt heterojunctions is also presented. The modeling approach is then generalizedto include Fermi-Diracstatistics for free carriers. We also discuss the applicationof the formulation to a variety of heterojunction devices (e.g. heavily doped silicon and gate-modulateddevices).
l. INTRODUCTION
Although many semiconductor devices consist of compositionally nonuniform material, the term, "heterojunction", is usually restricted to abrupt (or nearly so) junctions between two dissimilar materials. We shall, however, refer to any compositionally nonuniform device as a heterojunction. Our definition includes not only conventional heterojunction devices, such as semiconductor lasers[I] and solar cells[2], but also metalinsulator-semiconductor (MIS) and silicon p-n junction devices. In the MIS device, a semiconductor heterojunction occurs at the insulator (wide-gap semiconductor)--semiconductor interface. Because heavy impurity doping modifies the band structure of silicon[3], a heterojunction occurs between the lightly and heavily regions of a p-n junction. In this paper we restrict our attention to semiconductor heterojunctions and do not consider the metal-semiconductor heterojunction. Our goal is to present a unified formulation for modeling a variety of semiconductor devices. Several different techniques have been employed to analyze various compositionally nonuniform semiconductor devices. Since 1962 when Anderson [4] applied the depletion approximation to analyze abrupt heterojunctions, the subject has received much attention (see, for example,J5] and references cited therein). However, because it is based on the depletion approximation and restricted to abrupt junctions, the Anderson model does not apply to many devices. By resorting to numerical solutions and assuming abrupt heterojunctions, Cserveny[13] and Adams and Nussbaum[14] removed the depletion approximation. Approximate techniques for modeling graded heterojunctions h~.ve also been presented [7-9]. Adams and Nussbaum, and recently van Roos [31], have, however, raised some fundamental questions concerning heterojunction energy band diagrams. These questions will be considered in this paper. Because they significantly influence silicon device performance, so-called bandgap narrowing effects must be considered[3, 10]. The nonuniform band structure (to
first order, a bandgap shrinkage) associated with heavy impurity doping lowers the gain of bipolar transistors [10] and the open-circuit voltage of solar cells [3]. To model these heavy doping effects, transport equations for semiconductors with nonuniform band structure have been developed[ll, 12], but it is difficult to apply them because our understanding of heavily doped semiconductors is limited. For this reason, approximate methods, quite different from conventional heterojunction theory, have been developed to model heavily doped silicon devices [12, 25]. In this paper we consider the equilibrium theory of ideal (i.e. no interface charge) semiconductor heterojunctions. To permit calculation of the energy band diagram of a heterojunction, we first relate the electrostatic potential to the band edges and to the intrinsic energy level and then derive a simple, yet general, Poisson-Boltzmann equation for the electrostatic potential. We then use this equation, which includes free carrier charge and applies to both abrupt and graded structures, to analyze abrupt heterojunctions and compare the results to recently proposed alternative models[14,31]. Since carrier degeneracy is important in many heterojunctions, we also generalize the theory to include Fermi-Dirac statistics. Finally, we discuss the application of this equation to MIS and silicon p-n junction devices and show that these devices can be modeled as heterojunctions.
2. ENERGY BANDS IN NONUNIFORM SEMICONDUCTORS
Figure 1 shows the energy band diagram of a semiconductor with nonuniform composition[7, l l]. The validity of this energy band model (the underlying assumption of our paper) has been discussed by Marshak and van Vliet[l 1]. From Fig. l we observe that
683 SSE VoL 25, No. ~-A
E~(x) =Eo-qV(x),
(1)
E~(x) =Eo-X(X)- qV(x),
(2)
684
M.S. LUNDSTROMand R.J. SCHUELKE and I
no = N c ( x ) e ¢~F ~ u k ~ .
~/
qV(x)
[EL(X)----t------"
(5)
Using (2)-(5) we obtain the intrinsic energy level, kr
x(x) -
x(x) = E o -
log r/ N,,(x)]Nc(x)J
+
EG(X)
POSITION (x)
-
Fig. 1. Energy vs position for a semiconductor with nonuniform composition.
qv(x),
which is not, in general, parallel to V(x). Consequently, for a heterojunction one cannot identify ( - E t / q ) with electrostatic potential as is commonly done for homo junctions [17]. Although the energy band model outlined above applies only to materials in which X and Eo vary slowly, we will follow conventional practice and apply it to abrupt heterojunctions. Consider, for example, the energy band diagram for an abrupt heterojunction presented in Fig. 2. To evaluate the discontinuity in the conduction and valence band edges at the junction (x~), we use (2) and (3) to write
and
AEc = Ec(xj-) - Ec(xj +) = Ix(x() - X(xj )] Eo(x) = E c ( x ) - E ~ ( x ) ,
(3)
where Eo is a reference level, EL the local vacuum level, V is the electrostatic potential, X the electron affinity, and q is the magnitude of the electronic charge. It is important to note that the electrostatic potential is not, in general, parallel to either band edge or to the intrinsic level[15, 16] as it is in a uniform semiconductor. To relate the intrinsic energy level to Ec and Ev we assume Boltzmann statistics and write the equilibrium carrier densities as po = Nv(x) e ~v- EF)/~T
AEv = Ev(xj ) - Ev(xj ÷) = [x(xj +) - x(xj-)] + [E~(xj +) - Eo(xj )],
a E , -- Ex(xj ) - Er(xj +) 1
kT, rN,,(x~ ) N c ( x / ) ] -~-log [Nv(x+) Nc(xi_)j.
(4)
n-GoAs
---v E
0.0
.
V
4.13
, E¢
0"12
i
.07
i
.14
I
t i
.21
+
= [x(xj*) - x(x~-)] + ~[E~(x~ ) - E~(x~ )]
.
T
(8)
which are known as affinity rules. Similarly, for the discontinuity in the intrinsic level, we use (6) to obtain
.52
E
(7)
and
p -Ge
EL
(6)
i
.29
i
.36
x(/=m)---~
Fig. 2. Ge--GaAsenergy band diagram.
i
,43
i
.50
(9)
Modeling semiconductor heterojunctions in equilibrium 3. THE POISSON-BOLTZMANN EQUATION
Since the built-in potential of the junction is v~, =
v( vo-
v(o),
(lO)
To obtain the electrostatic potential within a heterojunction, we must solve Poisson's equation, dD dx
where x = 0 and x = W are the locations of the contacts, we can use (2), (3) and (6) to express Vbi as qVb, = - [ E c ( W ) - Ec(O)] - [X(W) - X(0)], qVb, = -
[E~(W)-
E,,(0)] - [x(W)-
(11)
x(0)]
- [E~(W)-
E~(0)],
685
(12)
q(P
(19)
n + N+),
where D is the electric flux density and N + = N o + - N a is the ionized dopant density. In equilibrium, the carrier concentrations, p and n, are related to the electrostatic potential by (14), (15) and (6) as
(X(x) - x(O)) + k T log l Nc(O) J + q(V(x) - V~) kT
n = ni(O) exp
(20)
and - (X(X) - x(O)) - (Eo(x) - Ea(O)) + k r log ~v(O)
p = ni(0) exp
- q(V(x) - V~)
kT
(21)
'
where
or
qVb, = - [ E , ( w )
- E,(O)] - Ix(W)-
-{[EG(W)-
..... Eo(O) kr, fNv(0)l qV, : - EF * r,o - X(O)- ~ + T log [ N---~-~J"
x(O)]
[ N v ( W)Nc(O) ]
EG(O)]+-~ log L ~ J "
(22) (13)
Each of the above three equations is just the difference of the field-free work functions between the two contacts. To relate Vb~ to the carrier densities, we first express the carrier densities in terms of the intrinsic level
To simplify these equations, we define two new variables,
[ m(x)N (x) ]
AG = - [Ea(x) - Ea(O)] + k T log L
~
J
(23)
as
po(x) = n~(x) e ~EI EV~/kT
(14)
and
and
(X(x) - X(0)) + k T log [ N c ( x ) ] ] LNc(O)JJ
no(x) = ni(x) e tE~-~1)/kr,
Y
(15)
Ac
(24)
'
which enable us to write
where hi(X) = x / ( N v ( x ) N c ( x ) ) e--EO(x)/2kT
(16)
is the position-dependent intrinsic carrier concentration. Using (14) and (15) we obtain kT ,
f no( W)po(O)]
- [ E l ( W ) - El (0)] = -~- log [no(0)po( W)J'
(17)
which we then use with (13) to express Vbi as q Vb, = ~ - log [no( W)po(O)]
t ~ J
-
Ix(W)
- x(O)]
o(o)1 + ,og [ N o ( W)Nv(0)J" (18)
no = n,(O) exp [ qV' ; ~ Aa ]
(25)
po = n~(0)exp r / - q V ' - (y - I)A6/,] kT J L
(26)
and
where V' = V - V,~ Note that our choice of reference for the potential makes V' zero at x = 0 if the material there is intrinsic. From (25) and (26) we obtain the equilibrium np product as noPo = n,2(x) = ni2(O) e ('~kr).
(27)
686
M.S. LUNDSTROMand R.J. SCHUELKE
Since Aa affects noPo like a bandgap shrinkage, we call it an effective bandgap shrinkage[12]. The second parameter, 3' (called the effective asymmetry factor [12]), measures the change in the conduction band edge and density of states relative to the change in both bands. Since (25) and (26) relate the equilibrium carrier densities to the electrostatic potential, we can use them with (19) to express Poisson's equation in terms of V' alone as
which is the intrinsic Debye length in the reference material. The resulting, normalized Poisson-Boltzmann equation is
d2U~-4 RI dRdUdrdr
R-~[Cuc92)sinh[U+(y-l/2)U~] -
D].
(35)
4. THE IDEAL, ABRUPT ItETEROJUNCTION
d2V '
1 dKdV'
d ~ - + K dx dx
-q[ni(O)e, Ke~
~v, ~ ,a~)/kT~
t
- ni(O) e ""v'+~a~)/kr) + N+],
(28)
where K is the relative dielectric constant. Equation (28) describes the equilibrium electrostatic potential in a material with nonuniform band structure (assuming Boltzmann statistics). The solution, V'(x), can be used with (2), (3) and (6) to determine the energy band diagram. Since the terms, yAG and (1 - y)AG are zero in homogeneous materials, (28) reduces to the conventional Poisson-Boltzmann equation[6] when the energy band structure is uniform. In many cases we can assume that the dopants are completely ionized (i.e. N ~ = N = Nn - NA). For typical lightly doped semiconductors, this assumption is valid near room temperature, and, because the overlap of wavefunctions precludes the existence of localized states, it is also a good approximation for heavily doped semiconductors[12]. However, for lightly doped semiconductors at low temperature, or even at room temperature when accumulation layers exist [29], the dopants may be only partially ionized. In such cases, N + must be evaluated using Fermi-Dirac statistics. When Fermi-Dirac statistics also apply to the free carriers, (23) and (24) must be modified as discussed in Section 6. To simplify the ensuing algebra, we make the following change of variables: V¢ U = -
kT q
AG
u~ = ~-~ N ~
D=- 2ni(0) r=
X
Lo(O)
(29)
In this section, we apply (35) to an ideal heterojunction consisting of an abrupt junction (located at x = xj) between two materials (surface states at the interface are not considered). The contacts of the device are located at x = 0 and x = W, and the subscripts L and R are used to denote the materials to the left and right of xj respectively. For x < xj we obtain the potential by solving d2U
-~
• I
= stun (U) - DL
d2U d P = R ~[e~ud~ sinh [U + ( y - I/2)U~] - DR]. r > rj
U(0) = sgn (DL) log [rDLI + ~/(DL2 + 1)l
U(W) = - (Y - 1/2) U~ + sgn (DR)log [IDR[ e(-Uc;:' + x/(DR2 e~- u ° + I)], (39) where sgn(D) is the sign of D. The normalized built-in potential is simply the difference in the above two quantities. We integrate (36) and (37) once to obtain the normalized electric field as d--U-U= -+ [2(cosh(U) - DLU + CL)I ~/2 dr r < r~
(32)
d---r-= -+
(40)
(e (Ud2~ cosh [U + (7 - 1/2) UG] -- DRU -ll/Z + CR)J •
(34)
(38)
and
and
[ eoK (O)k T ] 1/2 Lo(O) = [ ~ j ,
(37)
To evaluate the potentials at the two ohmic contacts we assume space charge neutrality at the contacts and solve (36) and (37) for
(31)
(33)
(36)
and for x > xj by solving
(3O)
K R= K(0)
r < rj
r > rj
(41)
The ambiguity in the sign of the electric field is removed by considering the potential drop between contacts; the proper sign is the sign of U ( W ) - U(0). If we assume dU/dr = 0 at the ohmic contacts, then
where
CL = DL U(0) - cosh [ U(0)]
(42)
Modeling semiconductor heterojunctions in equilibrium and CR = DR U(W) - e' uc/2) cosh [ U(W) + ( y - 1/2) Uo ] (43) are the two constants of integration in (40) and (41). To evaluate the electrostatic potential at the metallurgical junction we apply the boundary condition on the normal displacement field,
687
the band parameters as Ao = -0.83 eV and Y = 0.17. The electrostatic potential at the contacts is then evaluated from (38) and (39) as V'(0)= 0.0 and V ' ( W ) = - 0 . 2 7 volts and the two constants of integration from (42) and (43) as CL = --1 and CR = - 9 . 7 X 10-s. For the doublyintrinsic device we solve (45) for the potential at the junction to obtain
U(rj) = log [a2 -+ x/(a2 z - a3)]
(46)
1 - Re t uc/2) a2 = 1 - Re
(47)
["1 - Re "-v)uG ] a3= [. ~_~e~UG j,
(48)
where r=rj
and obtain cosh [ U(rj)] - Re Iu°/2) cosh [ U(rj) + (3' - I/2) Uol + ( R D , , - D , 3 U ( r j ) + C , - R C R =0.
and (45)
Except for the special cases of homojunctions and doubly-intrinsic heterojunctions, closed-form solutions to (45) do not exist. Nevertheless, (45) is readily solved for U(rj) by iterative methods. Since we can proceed no further analytically, we resort to numerical methods[6] to integrate (40) and (41) and thereby obtain U(r). With U(r) known, the equilibrium energy band diagram is computed from (2), (3) and (20). Figure 2 shows such an energy band diagram for a room temperature Ge-GaAs heterojunction assuming complete ionization of dopants with NL = - 3 x 10'6 and NR = 1 x 10 '6 cm -3. Table 1 lists the other parameters used for the computations. For this example, our solution (which includes free carriers) agrees well with the simple Anderson model based on the depletion approximation. Our formulation, however, also applies to heterojunctions with inversion or accumulation layers and to graded structures. The doubly-intrinsic heterojunction we consider in the following section is an example for which depletion layer theory cannot be applied.
which, for the Ge--GaAs example, gives V'(xj)= -1.4 x 10-3 volts. Since the doping density is zero we can integrate the expressions for the electric field, (40) and (41), and write
(r-rj)=log[
tanh(U/4) ] [tanh [ U(rj)/4]J
r
(49)
and l" t a n h [ ( U - U(W))/4I ] ( r - rj) = - R '/: e ~-ua4) log [tanh [(U(rj) - U( W))/4]J" r > r~
(50)
Figure 3 shows a computed energy band diagram for this doubly-intrinsic device. The solution agrees with the result recently obtained by Chatterjee and Marshak[30] but not with that of Adams and Nussbaum [14].
5. T H E DOUBLY-INTRINSIC A B R U P T I I E T E R O J U N C T I O N
6. GENEIO.LIZArlONTO l~l~fl-DnOtC STATtS'TlCS
Adams and Nussbaum[14] noted that an analytical solution for the electrostatic potential can be obtained for abrupt, doubly-intrinsic heterojunctions. The theory outlined in the previous section can be directly applied to this case by setting DL = DR = 0. In this section we consider a specific example, a germanium-gallium arsenide doubley-intrinsic structure. We begin by using the paramters in Table 1 to compute
When Boltzmann statistics cannot be assumed, the preceding formulation must be extended. Fermi-Dirac statistics may be important, for example, in MOS or n-N heterojunctions with strong accumulation (or inversion) layers. In this section we extend the formulation to include carrier degeneracy while retaining a simple, Boltzmann-like form. When Fermi-Dirac statistics apply, (4) and (5) become
Table 1. Material parameters for Ge--GaAsheterojunction
po = Nv(x)F,~2(no)
(51)
no = Nc(x)F,/2(nc),
(52)
Material
Ge
GaAs
4.13
4.07
and
Parameter X Eo
0.7
1.45
Nc
1 . 0 4 x I 0 TM c m -a
4 . 7 x 10 t? crrt -3
Nv
6.0 x l O l a crn -3
7.0 x l 0 t S e m -a
K
16
11.5
where F,2 is the Fermi-Dirac integral of order one-half with
nc =
(EF - E~) kT
(53)
688
M.S. LUNDSTROMand R. J. SCHUELKE Ge
27
GaAs
/
i
4.07
4.13 E
(ev)
1.45
E I =E F
0.7
0.69
I
6
,o
ioo'oo x (/~m)
-
Fig. 3. Doubly-intrinsicGe--GaAsenergy band diagram (for illustrative purposes, the band-bending and discontinuities have been exaggerated).
and ( E v - E~) kT
*Iv =
(54)
These parameters are related to the normalized potential, U, by
yet incorporate Fermi-Dirac statistics, by including the influence of carrier degeneracy in the two parameters, Ac and 3,[12, 18]. To accomplish this, we generalize the definitions of 3' and Ao as •
[Nv(x)Nc(x)3
EG(0)I + k T log [
Ao = - [ Ea(x ) -
~
j
[F,;2(,o)F,;:(,c) ] e~ t, e , c j
+ kT log [ ~¢ = U ( x ) 4 X ( X ) - x(O) + !/2 log J " V'V'~C(O ) kT
] -
kT
[ F,;z(r/~(0))F,;2(r/c (0))1 log [ e~o~O~e.~O > j
(57)
(55) and
(x(x)- x(o)) + k r
fNc(x)Fv2(nc)]
log [ ~
?
AG
and fly = - U ( x ) -
X ( x ) - x(O) kT
.j -
E a ( x ) - Ea(O) kT
fm(0)e<-~o
- 1/2 log L
(56)
When deriving (55) and (56), we made the assumption that Boltzmann statistics could be applied for intrinsic semiconductors. We can retain the Boltzmann-like formulation, (28),
.~. ~.
fF, n(nAO))]
,og [ e,-~-~--J
(58)
If we attempt to analyze abrupt heterojunctions including carrier degeneracy, then the Boltzmann-like formulation only complicates the integrations. As we discuss in the following section, however, for many devices the complexities introduced by graded transitions or nonuniform doping make it necessary to solve (28) directly by numerical methods. When direct numerical techniques are employed to solve the nonlinear equation, (28), little additional complexity is introduced by the weak nonlinearities in AG and 3'. The
Modeling semiconductorheterojunctions in equilibrium Boltzmann-like formulation, therefore, is a simple, convenient way to include carrier degeneracy in a numerical model.
7. APPLICATION TO OTHER HETEROJUNCTION STRUCTUR~
Although the formulation is general (it applies to semiconductor devices with nonuniform composition) we discussed only the abrupt heterojunction because it was possible to make some progress analytically for that case. The mathematical difficulties introduced by graded composition and nonuniform doping may preclude analytical solutions, but (28) remains valid. We now discuss the application of this equation to several heterojunction devices. For heterojunctions with compositional grading, we must either make simplifying approximations to achieve analytical solutions or resort to numerical methods. Although solutions for graded heterojunctions have been reported[7-9], the assumptions involved limit the applicability of these models. Oldham and Milnes[7], for example, calculated the electrostatic potential in a graded heterojunction by assuming uniform doping, neglecting minority carriers, and dropping the term in (28) that involves the gradient of the relative dielectric constant. Pearson et a1.[8,9] modeled graded heterojunctions by computing the electrostatic potential for the corresponding abrupt structure and then using (3) and (4) to compute Ec and Ev. As they noted, their procedure is valid if the graded region is narrow. Since (28) includes majority and minority carrier charge, the variation of the dielectric constant, and Fermi-Dirac statistics, a direct numerical solution can remove many commonly used approximations. Numerical simulations of graded bandgap solar cells (but neglecting the effects of carrier degeneracy) using a formulation similar to ours have already been reported[24]. Since the numerical solution to (28) is readily obtained[20], its use may be preferred in many cases. Within the last decade, it has been demonstrated that the heavy impurity doping used to fabricate pn junction devices modifies the energy band structure of silicon[3, 10]. Because of these so-called bandgap narrowing effects, silicon devices that contains heavy impurity doping must be regarded as heterojunctions. The formulation we have presented should, therefore, be equally applicable to modeling heavily doped silicon devices. Indeed, we choose our notation to correspond to that commonly used in modeling heavily doped silicon devices [3, 12]. When modeling heavily doped silicon devices, we find it convenient to choose the reference material as lightly doped silicon (with unperturbed energy band structure) rather than the material located at x = 0. If we denote the intrinsic carrier concentration in the lightly doped semiconductor (with unperturbed energy bands) as n,o and the equilibrium np product in a heavily doped semiconductor as n~e,then (27) becomes ni,2 =
nio 2
e (aOlkT).
(59)
689
Since the effective intrinsic carrier concentration in a heavily doped semiconductor, n~e, can be inferred from electrical measurements[27], the parameter, A~, can be determined. Most authors choose 3' arbitrarily when modeling heavily doped semiconductors because no experimental measurements for it have been reported. Fortunately, this expedient is justified because many properties of the device are insensitive to T[12, 25]. Simulations of heavily doped silicon devices based on this formulation have been reported[18,26] and provide confirmation that terminal I-V characteristics are not sensitive to 3' under many operating conditions. The metal-oxide-semiconductor (MOS) capacitor is a heterojunction device controlled by a gate voltage. Similarly, the properties of a heterojunction between GaAs and AIGaAs can also be controlled by a voltage applied to a metal gate[21,22]. Such gated heterojunction structures show promise as high-speed MESFETS [21,22]. The modeling approach we presented is directly applicable to these gated heterojunction structures; we simply define the boundary conditions (by setting the gate voltage), then solve (28) numerically. Computations performed by the authors show that (28) is readily solved by numerical techniques, even for MOS structures in which an abrupt change in E~ of about 7 eV occurs.
8. DISCUSSION
Although the abrupt heterojunction is only a special case described by (28), it deserves further discussion. Cserveny, who also included the effects of mobile charge in the abrupt heterojunction[13], obtained results that agreed with the Anderson model (for structures in which both semiconductors were depleted). The same problem, however, has recently been reconsidered by Adams and Nussbaum[14] and by yon Roos[31]. These authors proposed alternative models which differ from each other and from Anderson's. The preceding discussion has shown that our model agrees with the Anderson model of an abrupt heterojunction. In this section we compare the underlying assumptions of Adams and Nussbaum and those of yon Roos with ours. As Adams and Nussbaum noted, they used the intrinsic level to measure the electrostatic potential within a heterojunction[14]. When this assumption is made, the intrinsic level is forced to be continuous at the junction in order to insure continuity of the electrostatic potential. As a result, the discontinuities in the conduction and valence bands differ substantially from those predicted by the Anderson model, and, in addition, a discontinuity in the local vacuum level must be hypothesized[14]. Although using the intrinsic level to measure the electrostatic potential follows long-standing practice in semiconductor device physics[17], it cannot be done for a material with nonuniform composition because the electrochemical potentials contain position-dependent terms[15, 16,28]. For compositionally nonuniform materials, the chemical potential for electrons (under
M. S. LUNDSTROMand R. J. SCHUELKE
690 nondegenerate conditions) is given by[28] #c n = k T log (n(x)) + f(x, T),
where f(x, T) represents the contribution to the internal energy arising from the interaction of the electron with the crystal lattice. As a result of the position-dependent term, /', the quantity ( E p - E t ) is not the chemical potential and, consequently, the intrinsic level is not parallel to the electrostatic potential. Figure I illustrates the relationship between the energy band edges and the electrostatic potential, and, as (2), (3) and (6) show, neither the conduction band edge, nor the valence band edge, nor the intrinsic level may serve as a reference for the potential in a material with nonuniform composition. Another approach has recently been taken by von Roos who assumed that the conduction band was parallel to the electrostatic potential[31]. Because of his assumption, the energy band diagram he obtained has a continuous conduction band edge but a discontinuous valence band edge and intrinsic level. As we discussed in Section 2, however, the conduction band (nor for that matter the valence band or intrinsic level) is not parallel to the electrostatic potential. The validity of the energy band model we assumed for compositionally nonuniform material (Fig. 1) has been discussed by Marshak and van Vliet[ll]. For an abrupt heterojunction, (7)-(9) show that discontinuities in the conduction band, valence band, and the intrinsic level exist at the junction, however, no discontinuity in the vacuum level exists. Our computations for the doped and undoped Ge-GaAs abrupt heterojunction show that the difficulties which led Adams and Nussbaum to propose their alternative model do not arise if the electrostatic potential is defined as shown in Fig. 1.
9. SUMMARY AND CONCLUSIONS
In this paper we presented a Poisson-Boltzmann equation that describes the electrostatic potential within equilibrium semiconductor heterojunctions. The equation includes the effects of free carriers, is valid for both Boltzmann and Fermi-Dirac statistics, and applies to graded or abrupt structures. We used this equation to analyze both doped and undoped, ideal (no interface charge) abrupt heterojunctions. The importance of not using the intrinsic energy level to measure the electrostatic potential within a heterojunction has been stressed. Several different techniques for modeling semiconductor devices with nonuniform composition exist, but the generality of our formulation is an important advantage. When the device being modeled has a uniform band structure (and when Boltzmann statistics apply) (28) reduces to the conventional Poisson-Boltzmann equation. Consequently, the homojunction is a special case of this formulation. Our modeling approach has previously been used to analyze heavily doped silicon devices [12, 18, 26], but with the proper choice for Aa and 7, heterojunction or MIS devices can also be modeled. Another advantage is that Fermi-Dirac statistics are
included, but the simple Boltzmann-like formulation is preserved. The formulation presented in this paper should be particularly valuable in numerical simulation of semiconductor devices since a wide variety of devices (all viewed as heterojunctions) can be modeled using a single formulation. We note that the extension to non-equilibrium devices can be readily made[12]. Previously reported simulations using this[18,26] and a similar[24] formulation, demonstrate the viability of this approach. Acknowledgement--This material is based upon work supported
by the National Science Foundation under Grant No. ECS8105956 and, in part, by the 3M Company. We also thank F. Sanii and R. J. Schwartz for critical reviews that improved the manuscript. REFERENCES
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