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Boundary value problems in Boltzmann modelling of hot electron transport
Solid-State
Electronics Vol. 30, No. 4, pp. 364374,
1987
0038-I IO]/87
BOUNDARY MODELLING
$3.00 + 0.00
Copyright 0 1987 Pergamon Journals Ltd
Printed in Great Britain. All rights reserved
VALUE PROBLEMS IN BOLTZMANN OF HOT ELECTRON TRANSPORT w. cox
Department
of Computer
Science and Mathematics, Aston Birmingham B4 7ET, U.K. (Received
University,
Aston
Triangle,
28 July 1986)
Abstract-We investigate the types of boundary value problem arising in modelling of hot electron transport by the nearly isotropic approximation to the Boltzmann equation. Models with a uniform electric field, and with specific collision term approximations, are considered for the cases of non-polar optical and acoustic phonon scattering and for piezoelectric scattering. In a number of interesting cases there exist models utilizing boundary value problems with relatively straightforward analytical solutions.
solutions satisfying realistic boundary conditions could be obtained in this case. Other@] have expressed interest in the extension of such simple analytical models to MHET’s in GaAs, for example. Here we will investigate quite generally the range of boundary value problems which can arise by application of the nearly isotropic approximation in nonpolar optical, acoustic phonon and peizoelectric scattering, with or without a field, with a spatially dependent distribution. We will discuss the availability of simple solutions and the sort of boundary conditions one can handle. The purpose at this stage is not to consider specific device models in detail, but to lay out the general approach and display the range of possibilities. Also, the approach is not confined to the scattering processes referred t-it will be suitable for modelling any device in which the electronic processes are adequately modelled by the nearly isotropic approximation to the 1-D BTE in a uniform electric field, in finite or infinite geometry, subject to mathematically well posed boundary conditions. The resulting equation may be analytically intractable in some cases, as for example in the case of polar optical phonon scattering, but often simple analytical solutions may be found. The approach is to take the first two terms in the Legendre polynomial expansion of the distribution function f(x, E) (E is the kinetic energy) about the field direction as axis of symmetry, splitting it into a symmetric part f0 and asymmetric part f, in energy space. Substitution in the BTE, with the usual appropriate approximations for the collision integral, yield two coupled equations for fu, f,. With a relaxation approximation for the asymmetric part of the collision integral, fi may be eliminated to yield a single equation forfo which one solves subject to approximate boundary conditions. In the spatially homogeneous case normally considered[3-51 the result is an ordinary differential equation for f0 in energy or velocity space. When f is spatially dependent the result is a linear partial differential equation for _&in
1. INTRODUCTION
For a wide range of applications, the Boltzmann transport equation (BTE) has proved a useful model of hot electron or high field effects in semiconductors. For extreme behaviour and submicron devices, there is little doubt that more sophisticated quantum transport equations are more appropriate[l,2]. However, even in such cases it is necessary to explore as widely as possible the implications of simpler BTE modelling applied to specific devices, so that comparisons with observations may be made and the limits of validity defined. For a given device structure the solution of the BTE under realistic boundary conditions may tax the largest computer, especially if one solves self consistently with the Poisson equation determining the electric field. Much work has been done on this, initially by semi-analytical methods such as moment methods, or Legendre polynomial expansion, and more recently by Monte-Carlo or iterative methods requiring large computational effort. This paper is motivated by the desire for relatively simple analytical or semi-analytical solutions which may provide models for particular device structures under hot electron behaviour. In particular we investigate the use of the “nearly isotropic” approximation (first two terms of the Legendre expansion) to the BTE for hot electron or high field transport, in formulating simple boundary value problems which may serve as useful models in specific applications. Semi-analytical treatments of such models for the case of homogeneous materials (i.e. spatially independent distribution function) have been available for some time[3-51, but little has been done on the spatially inhomogeneous case necessary for modelling devices. Ridley[6] has recently used such a model in studying high energy injection in monolithic hot electron transistors (MHET’s) in silicon, and for the calculation of the base-collector current-transfer ratio, in the zero field case. The author[7] extended this model to incorporate a uniform electric field and showed how 365
366
w. cox
x and E (or x and velocity) space. For example in the case treated by Ridley[6] of a first order approximation for non-polar optical phonon scattering and zero field the resulting equation for f0 is a simple parabolic equation which can be converted to a diffusion equation for the current density per unit energy interval, j(x, E). The problem therefore reduces to a boundary value problem (BVP) for the diffusion equation. In the presence of an electric field one still arrives at a parabolic equation[7], but this is no longer reducible to a simple equation for j(x, E). However, there exist analytical solutions for realistic boundary conditions for&(x, E) in this case too. In this paper we extend this type of modelling to other scattering mechanisms-acoustic and piezoelectric phonon-with or without a uniform field, and analyze the types of BVP one can envisage to model various physical situations. If the electric field is absent the resulting& equations can still be reduced to a BVP for the diffusion equation for j(x, E), similar to Ridley’s analysis. Naturally, the presence of the field complicates matters, as does the use of higher order approximations to the collision term, but in some cases there are still accessible analytical solutions. The modelling of the boundary conditions in a particular situation is delicate. Care must be taken to ensure that realistic physical situations are modelled by well posed BVP’s. Often of course, realistic boundary modelling is difficult and can preclude use of analytical solutions, but in some cases a useful compromise can be achieved. 2. LEGENDRE
so may
be regarded as functions of the energy instead of k, as indicated by the noEquation (2.2) may be written:
E = hZk/2M:
tation.
f
=h + k cosOf, =fo+ k,f,,
the asymmetric part off being characterized by k,. The conditions under which eqn (2.2) is a good analytical approximation are well documented in the literature[3,4]. Note however that discussion is usually confined to homogeneous transport in which there is no diffusion gradient term afiax, and in the present case it is necessary to ensure that combined diffusion and field drift effects remain sufficiently close to isotropy not to invalidate eqn (2.2). In the context of the Legendre expansion method, BaraffllO] has given an alternative truncation of the series, which allows the case of significant carrier streaming to be treated. It would be interesting to explore the formulation of the present approach in Baraff s treatment. The collision integral C(f) may also be expanded in the form eqn (2.2):
c(f)= Cop,,+ kC,P,. The forms of ing process below. Substituting orthogonality results in the
eqn (2.2) in eqn (2.1) and using the properties of the Legendre polynomials well known decomposition[9]:
(2.6)
We consider the problem of electron transport in a I-D material with a uniform electric field, [, and a I-D concentration gradient along the x-axis. Then the distribution function for the electron f(r, k, 1) taken in the quasi-classical effective mass approximation, in the steady state, depends only on x and k, the wave vector, and satisfies the BTE:
where k, = k cos 0 is the component of the wave vector in the direction of the electric field, with which it makes an angle 0; M: is the effective mass; C(f) is the collision integral[9]. Because of the axial symmetry in k-space, about the k, axis, we can expand f in the usual manner in a series of Legendre polynomials (in cos e), and adopting the standard nearly isotropic approximation one retains only the first two terrns[3,4,6,9]:
The general approximation
approach for C, :
c,=
0).
is to assume
--
Written in this manner both f0 and fi are functions only of x and the magnitude of the wave number, k,
’
a relaxation
(2.7)
where the relaxation time T,(E) depends on the particular collision process, and the approximation used to model it. With this assumption f, may be eliminated from eqn (2.5) giving an equation for fO:
where A = -e[, and EC, x, are characteristic energy and distance parameters used to define the dimensionless variables:
-xc
(2.2)
.A TI (~9
x=x
0) + kf,(x, E)P,(cos
(2.4)
C,, , C, depend on the particular scatterwe are modelling, and are discussed
EXPANSION OF THE BTE AND COLLISION TERM
f(x, k) =X,(x, E)P,(cos
(2.3)
w=z. kc
(2.9)
We will refrain from identifying x,, EC here, since the best choice will depend on the particular problem at hand.
Boundary value problems in hot electron transport Even without knowledge of T, it is clear that the left hand side of eqn (2.8) is a second order partial differential operator in (A’, W) space acting onh. The next stage is to adopt an approximation for the symmetric part of the collision integral C,, which results in a partial differential expression in W of at most second order. Then eqn (2.8) becomes a second order partial differential equation in (A’, W) for fO, which can be solved once we have defined suitable boundary conditions. Once& has been determinedf, is obtained from eqns (2.6) and (2.7) whence physically interesting quantities may be obtained in the usual way, averaging over J Note that there are two semi-analytical approximation steps here-the original near isotropy assumption eqn (2.2) and the assumption about the approximate form of the collision term. It will turn out that for the scattering processes and approximations we will consider the eqn (2.8) is parabolic if a first order (in phonon energy) approximation is used for C,,, as, for example, that used for non-polar optical scattering in Ref.[6]; while it becomes elliptic if a more accurate second order form is used, as for example in Ref.[4]. There are a range of standard boundary value problems (or strictly initial-boundary value problems, but we will ignore this distinction in terminology) for parabolic and elliptic equations for which existence and uniqueness theorems are well known, leading to a number of possible models of different physical situations. There are standard numerical routines for solving such BVP’s, but here we are specifically interested in investigating the possible semi-analytical approaches. There are a number of these available and in some cases these lead readily to simple analytical solutions describing quite realistic physical circumstances. In some cases however there are complications which lead to interesting mathematical problems, where the best coordinate system for analytical solution is not convenient for application of the boundary conditions. In the particular case of zero field it is possible to change the dependent variable fromf, to j(X, W), the current density per unit energy interval, as done by Ridley[6] for non-polar optical scattering, resulting in a BVP for the diffusion equation if a first order collision approximation is also used. The corresponding equations for acoustic and piezoelectric scattering are slightly more complicated, but still may be converted to a diffusion equation BVP by a change of variables. The approximation to be used for the collision term C(f) in particular cases is still a subject of debate, but we will simply use standard models which have already proved their worth. We will not concern ourselves with their range of validity in terms of the relevant parameters. Such questions are more readily dealt with in specific device applications. Further, we emphasize that the models given should be treated as examples of the general procedure. So long as one is
361
modelling by a near isotropic approximation, one is sure to arrive at a BVP for a parabolic or elliptic equation of the sort discussed here-its solution may be a different matter! Thus, it may be possible to find simple semi-analytical models for other scattering processes-polar optical phonon, intervalley; defects, impurities and so on. One may also consider combinations of these. The collision processes we will consider, and their Legendre expansions, are discussed fully in Nag[4], and we will simply review the results. For nondegenerate semiconductors the collision integral may be written in wave vector space as:
C(f)=
V -(27[)3
x
[P(k, k’)f(k) - P(k’, k)f(k’)] dk’,
(2.10)
s with P(k, k’) the usual transition probability from state k to state k’, V the volume. For purely lattice scattering modelled by electron-phonon coupling in first order perturbation theory P(k, k’) is given by the Golden Rule: P(k, k’) =;
IM(k, k’)(*6&
- Ek, k hw,),
(2.11)
where M(k, k’) is the transition matrix element, and wq the phonon energy ( + for absorption, - for emission), and k -k’ f q = 0 (no Umklapp processes). The genera1 form of M(k, k’) is: M(k, k’) = [A(q)]“‘,& = [A (q)]“‘Jn,+r
absorption emission
(2.12)
on the usual assumption of an infinite equilibrium phonon bath with occupation numbers {n,} (for range of validity of this assumption (see Ref.[llJ). Here A(q) depends on the type of scattering process. Substituting for P(k, k’), using the Legendre approximation eqn (2.3), and considering all emission and absorption processes there results:
C(f) ---
V (Zn)3
s
A(q)[6(E,-E,.+hw,)
x Nn,+ llf,(&)
-q&Q)
+ d(E, - 4’ - ~wJ{n,h(E,.) - (“9 + MdEdj + 66% - Er + Ro,) x {(n, + 1)k’ cos O’h(E,.) - n,k ~0s ofi( + 6(E, - Ek, - hw,){n,k’cos - (n4 + I)k cos Of,(E,J}] dk’
O’J(E,.) (2.13)
The terms involvingf, in this integral constitute the symmetric part of the collision integral. If, as is sometimes assumed,f, is the equilibrium distribution,
368
w. cox
then this part of the collision not so in the present case-it
integral is zero. This is in fact gives C,:
s
Y
co= A(q)[G(E, - &+ hw,) ml’ x ih, + IlAdE,,) - qwk)~
Piezoelectric
a’.
cosH
‘+
(2.15)
(For details and discussion see Ref.[4]), so eqn (2.7) is obtained for C,. For the specific processes we will consider the W dependences of 7, are as follows: Non-polar optical phonon scattering:
Acoustic
phonon
Piezoelectric
7"pt Iv
1.
(2.16)
scattering: 7, =
phonon 7, =
7a.cmw-"2.
(2.17)
scattering: tP w"?
7sp are all constants, independent of W, whose values do not concern us here-see Ref.[4]. Note that there are only essentially two distinct forms of 7, to deal with, W +I’*.
Suitable approximations for the symmetric part of the collision term C,, may be obtained by rewriting it utilizing the presence of the h-function terms as:
s
A(q)F’(& - Ek + hw,){(n, +
+ 6(E, -
EL. -
dk’
(2.19)
assuming Ek >>hw,, and expanding the f. (Ek & hw,) in a Taylor series. This procedure, and its range of validity is detailed in Ref.[4] and the results are: Non-polar optical phonon scattering:
co=$
w-“2
(;;+wg>I. [ w;;+pop
x fo+
phonon
woaf dW
B,,
(
^
2g;+w5&
11
(2.21)
’
’
scattering:
36 ’ + Bpzrw+w$
(
’
(2.22)
>I
Again, the T’S and ,0’s are constants, of no direct interest here, see Ref.[4]. The significant thing about the j?‘s is that they are of the order of the phonon energy and therefore represent corrections in the collision terms, which we may neglect in a first approximation, as done in Ref.[6]. In homogeneous high field transport such as that modelled in Nag[4] it is customary to retain these higher orders and we will allow ourselves this option here. Taking the /I’s non-zero changes the nature of the BVP, as we will see. The eqns (2.16H2.18) and (2.20)-(2.22) provide the necessary ingredients for use of the eqn (2.8) for determining fo. However, as shown by Ridley for non-polar optical scattering, in the field free case, the resulting& equation can be converted to one for the more physically relevant quantity j(X, W), the current density per unit energy interval. For this we will need the relation between j andf,. The current in the x-direction:
may be written, on substituting eqn (2.3), introducing polar coordinates, v2 = vt + v: + v’,, and using the symmetry properties off,, f,, in the form:
JxCC
WU W”% d W
(2.23)
s0
(the J,, Jz are zero) or
1)
hw,) x {n,f,(E, - ho,)
- hj + IXCWI
2fo,+
(2.18)
7opt3 t,,,,
co=
[
(2.14)
1
7, =
C&W”2
scattering:
+
The terms involvingf, constitute the asymmetric part of the collision integral. As mentioned earlier, we will now confine ourselves to collision processes for which there exists a simple relaxation approximation for this part. For example, this is appropriate to the randomizing collision model for collisions with optical and intervalley phonons and for elastic collision models for piezoelectric and impurity atom scattering. In either case the asymmetric part may be approximated by the form: -k
phonon
T?,ce
+ SC%- 4 - fq)In,h(E,) - (nq+ IM4N
Acoustic
The expression for the current density per unit energy interval can therefore be taken as, up to a multiplicative constant:
In the field free case (E, = 0) this becomes, multiplicative constant: j(X, W) = ;;
(2.20)
(7,
W”?fO).
up to a
(2.26)
Since eqn (2.8), and the T, and Co in eqns (2.16)-(2.18) and (2.20)-(2.22) only contain coefficients in Wit is
Boundary value problems in hot electron transport clear that in the free field case we can always convert the equation for & to one for j of the same order. Before treating the different scattering processes it is worth briefly reviewing the sort of BVP one encounters with parabolic and elliptic equations. Consider the general homogeneous parabolic equation in one spacelike variable x and one timelike variable t : a(x,r)~+b(x,t)~+~(~,r)~=~
(2.27)
a
t>O
[a(~, t) positive definite]. There are three classic (initial-) boundary value problems for this equation for which existence and uniqueness theorems are well known, and for which the diffusion equation in particular has readily obtainable analytical solutions[l2-141. Problem 1. Finite spatial domain q5 must be continuous and: Boundary
in and on boundary
region,
conditions
ti(a,t)=g,(t)
t 20
4(b,t)=g,(t)
r 20
Initial conditions $(x,O)=h(x)
a
gi and h given functions. The requirement of continuity of the boundary conditions is essential for guaranteeing uniqueness of the solution, and in particular it means that the functions gj, h are not arbitrary, but must satisfy g,(O) = h(a), gr(0) = h(b). Solutions violating such conditions are not unique. This does not of course invalidate them from representing real physical situations, but it means that this representation does not follow uniquely from the prescribed BVP. The solution given by Ridley for non-polar optical phonon scattering is precisely of this non-unique type-i.e. His BVP is not well posed. Problem 2. Infinite spatial domain 4(x, t) must be continuous and:
in and on its domain,
Initial condition 4(x,0)=/t(x) Boundary
--co
condition
14(x, t)l exponentially Problem 3. Semi-infinite 4(x, t) continuous
bounded
as lx ( + co.
Boundary
369
conditions
4(O,t)=IL(t)
t>O
(4 (x, t) 1exponentially
bounded
as 1x ) + co.
Again, the continuity of boundary conditions and the boundedness properties in Problems 2 and 3 are essential for uniqueness. In each of the above Problems the conditions have been applied to 4 itself, but of course there are equally important versions in which the conditions are applied to its derivatives. We will not consider such derivative BVP’s here, but technically they provide further useful models which may yield simple analytical solutions. Also, Problems l-3 merely scratch the surface of the parabolic BVP and we would not wish to discourage readers from exploring more complicated situations. The treatments of such problems in other areas such as heat conduction, chemical and nuclear transport, diffusion theory and so on has a vast literature where doubtless many useful results may be found. Now in fact, as we shall see, in the case of first order collision approximation one does not obtain, with the simplest choice of canonical coordinates, a parabolic equation of the form eqn (2.27). Instead, one obtains the adjoint form, with a(x, t) negative definite[l2,13]. This is easily seen in particular cases, and can be generally appreciated by consideration of the general form eqn (2.8). In general there are no solutions to the “initial” BVP’s of types 1-3, a fact most easily understood by considering the so called backward_ diffusion equation:
The type of problem one considers for the adjoint equation is typified by the “final” BVP of the form: 4(0, t) = h(r) lim 4(x, t) = 0. I-CC
(2.31)
Again there are many alternative types of such problems. In the following sections we will only consider a few of the simplest examples leading to analytical solutions. One can, with suitable interpretation of the boundary conditions applying in physical situations, convert the adjoint equation to the form eqn (2.27) with a(x, t) positive definite, by changing the timelike variable to say to - t. This was done by Ridley in Ref.[6]. However, we will tend to retain the adjoint form and use the BVP in the final value form eqn (2.31). For the general homogeneous elliptic equation:
spatial domain
in and on its domain:
$+2+a(x,y)g w
Initial condition 4(x,0)=/r(x)
o
+b(x,v)$+c(x.y)~
=o,
370
w.
the type of BVP is simpler, the standard possibilities being the prescription of either C$ or its normal derivative on segments of the boundary. We now turn to the specific types of scattering of interest to us.
3. NON-POLAR
OPTICAL
SCATTERING
Using eqns (2.16) and (2.20) in eqn simplifying yields the equation:
( >a2fo al+/?
a-22ay-
a’fo
5’fo
axaw +y $-@x 2
+(%+1)%+&o,
(2.8) and
cox
differentiating
eqn (3.2) with respect to X. We obtain: 2 ;+;;=0. (3.3)
Ridley solved this equation subject to boundary conditions purporting to represent high energy ( W,) injection of monoenergetic electrons at x = 0 into a semiconductor base, which attenuates the maximum energy within an infinitesimally small distance. Mathematically this corresponds to the BVP: j(0, W) = J,6( W - W,)
SY dfo
J, constant
lim j(x, W) = 0, w-w, (3.1)
where
(3.4)
for the eqn (3.3), which is not, as it stands, a parabolic equation of the form eqn (2.27); and this BVP does not correspond to any of Problems 1-3. Ridley remedies this by substituting s = W,, - Wand reformulating the BVP as
yz” /lx; ’
j(O,s)=J,6(3) j(x, s = 0) = 0
with
o
0 < x < co
(3.5)
for the resulting diffusion equation in (x, s). This corresponds to Problem 3, the semi-infinite spatial domain, but because of the d-function, continuity of the boundary conditions is not satisfied, so uniqueness is lost. This is not a serious problem-the 6(s) may be replaced by a sharply peaked Gaussian with the required continuity properties, the calculation repeated, and the (now unique) result of Ridley recovered as an approximation. Here we will determine j directly from equation (3.3) and the formulation of the BVP typified in equation (3.4), which is more representative of the physical situation. Equation (3.3) is in fact an example of the adjoint form of the parabolic equation[ 12,131. As discussed in Section 2, in general there is no solution to the “initial-BVP” Problem 3 to the adjoint equation, but solutions exist for the “final-BVP”:
The CK,/?, y are all dimensionless parameters whose numerical values depend on the scales x,, ECadopted to measure distance and energy. We are not interested in the precise values here, but it is important to note the general physical significance of the parameters. tl is proportional to the (uniform) electric field, and the zero field case corresponds to c( = 0. p essentially signals the presence of the collision term and will always be assumed non-zero. /I on the other hand corresponds to the second order approximation in the collision term and in those cases for which the first order approximation is adequate we can take p = 0; otherwise, note that /? > 0. For the full model including a field and second order collision approximation all the c(, /?, y are non-zero independent parameters. In this case, because b > 0, eqn (3.1) is elliptic, and j(0, W) = h(W) for a unique solution, boundary conditions have to be applied toy0 for some region in (X, W) space. On lim j(x, W) = 0. (3.6) the other hand, if we take /? = 0 then eqn (3.1) w- wrJ becomes a parabolic equation and unique solutions These are derived using superposition with the dewill be obtained to the sorts of well posed BVP’s rived source solution (Green’s function for the exemplified in Section 2. There are three distinct cases of eqn (3.1), with TV adjoint equation): or b or both taken to be zero. The simplest case, G*(x, W; 5, H’,,) r = /? = 0, corresponding to zero field and first order (x-r) collision approximation, is that considered by e-(x-;)~,‘4W-W w < w, Ridley[6], and with x = X/y it can be written as =Jm
a2(wJ)+ a(wxl) = ax2
aw
=o
w> w,.
(3.7)
o,
(3.2)
Ridley has already analysed this case, but it is worth briefly reviewing it. From eqns (2.26) and (2.16), we can replace a( Wfo)/ax, up to a factor, by j(x, W), by
The required
solution
f (& W) =
is:
= G*(x, W; 0, r)h(z) dr s0
(3.8)
Boundary value problems in hot electron transport so with h(T)=JO6(T
-
w,,)
i(x, W) J&e-x2i”+
H?6(r - w,) dr
(3.9) as obtained by Ridley[6]. Before leaving this simple case, it is worth remarking that BVP’s of the Problem 1 type are also very relevant to the sort of modelling we are discussing here, and may be treated by standard analytical methods (cf: heat conduction in a finite bar). Thus for example one could construct BVP’s on a finite domain as models of hot electron devices in a finite base region. The generalization of Ridley’s model to include the presence of a uniform field has been described by the author[7], and corresponds to the choice LY# 0, p = 0 in (3.1): 8% --
-
2
8%
1 ato +axt-TZ 8%
axaw
aw2
+ ;+ 1 g;+;fo=o, (3.10) ( > where, for convenience we have chosen EC, xc to make CL,y both unity. This equation is still parabolic and may be converted to canonical form by using the variables r = X + W, r~= W:
atro ah
ah
9=$+9~+(n+l)&-+fo=o.
In these equations ,Fl is the confluent hypergeometric function. Determination off,, and the evaluation of the current in this case is presented in Ref.[lS]. The generalization of Ridley’s analysis to include the second order collision approximation, but with zero field ;,:corresponds to tl = 0, p # 0 in eqn (3.1) resulting
-$+3+
;+ 1 ;;++=o, (3.15) ( > where we have replaced W, X by W/p, X/y respectively. Unlike the previous cases this equation is elliptic. As with the zero field first order case, equation (3.15) may be rewritten in terms of
wm
AX, W = ax in the form: W2g+
W’$+
W(W-l)Fi+j=O
(3.16) However, eqn (3.15) is easier to deal with in this case. In fact, apart from the X and < derivatives, eqn (3.15) is identical in form to eqn (3.11) and so separable solutions may be again obtained involving the confluent hypergeometric equation in W. We may use these to solve BVP’s withy, prescribed on a rectangle a
(3.11)
Again, this has the adjoint form to eqn (2.27). In Ref.[7] it has shown that solutions to the final BVP of the form:
371
4. ACOUSTIC
PHONON
SCATTERING
Equations (2.17) and (2.21) in eqn (2.8) and simplication gives:
MO, W) = h(W), lim fo(X, W) = 0,
(3.12)
W-U
exist, and in terms of X, W these can be written in the form
MK
/J = 3Md/2E
w cc
=
dtB(t)exp[-(t
+ l)2uT]
s
xUexp[--(r2+ t)X] t 1; (2t + 1)W (3.13) x IF, -’ 2t + 1’ ( > where B(t) is determined from the condition: h(W)=
(4.1) + $+2/l + w -&+2J,=o ( 1 a, 8, y are as in eqn (3.1), but with /I = /I, and
t
mdt B(t)exp(-(t s0
Since W 2 0, eqn (4.1) is elliptic if /I # 0 and is parabolic otherwise. Consider first the simplest case LY= p = 0, which generalizes Ridley’s analysis to the case of acoustic phonon scattering. With z =X/y the equation becomes: 8% g + w;;
+ 2fo = 0.
(4.2)
in this case, from eqns (2.17) and (2.26) we again have, up to a constant factor: Now
+ l)2 W]
x 16 -’2t+ 1’ 1;
c T a.mTace.
(2t + 1)W .
>
(3.14)
j(z,
W)=f!$i?.l
w. cox
312
for the current density per unit energy terms of j eqn (4.2) may be rewritten:
a*i
J(W) _.
s+ aw changing
interval.
the variables
In
(4.3)
’
to:
C$= Wj,
W = e’,
in turn reduces this to the adjoint
produces
(4.4)
diffusion
equation
^
s+gT=O T>O
Unfortunately the q-equation which results on separation of variables in this case does not appear to be readily reducible to a standard form and requires a systematic development before being of practical use. In the case of x = 0, b # 0 the equation (4.1) becomes, replacing W, X with W/p, X/y respectively
(4.5) w$$+
for C#J(Z,T). The BVP eqn (3.6) translates
again elliptic. This equation of the form:
#(O, T) = erh(er) JirnoemT+(z, T) = 0. We may again use the solution obtain
(4.6)
eqns (3.7) and (3.8) to
h(X
T > To
,= Jwxe’W-eT”)d7 the integral
with
W = e’
wj(z, w) = Jot e-Z2i4’n(waiw) (4.8)
J47r [ln( W. I WI’ so Joz =
,-z~:41n(Wo#w
WJ4n
(4.9)
[ln( W,/ W)13
This is the solution for acoustic phonon scattering which corresponds to Ridley’s result eqn (3.9) for non-polar optical scattering. Next consider the non-zero field but p = 0 case, again choosing E,, x, to make x, y both unity. There results:
3%
8%
+=-wax ax
axdw
+
;
1
constant. By substituting may be converted to the equation:
&!2+(2_ W)A!&*
=o,
dW
H(W)
_ W+’
(4.17)
and solutions well behaved for large W may be obtained if we restrict w < 2. These may be suitable for certain physical conditions.
PHONON
SCATTERING
$+
2f,=o.
2
(4.10) (cr*W*+aw,~-2,,W2~w+p2W:~
form with the variables:
<=x+w q=w
(4.16)
finite at W = 0. , F, denotes the confluent hypergeometric function. Therefore, we have solutions in analytical form and, subject to boundary conditions imposed on the rectangle a < X < b; c < W < d, we may have a reasonably tractable BVP. If this proves too complicated then we must resort to numerical methods. Also it is worth noting that, since the asymptotic behaviour of ,F, (a; b; X) for large X is -exXueb, we see that:
5. PIEZOELECTRIC
so
>
to canonical
(4.15)
In this case, eqns (2.18) and (2.22) in eqn (2.8) gives:
+ w
( Reduction
confluent
separation eqn (4.14) hypergeometric
II/ = ,F,(w; 2; W) = e’“H
with W, = e’O. Evaluating gives:
l?WZ
the
(4.14)
which has solution:
s
2
o
H = e-W$,
dW*
e-r2.‘4(r- n
32% --
WI,
w d*H ~+(W+2)~w+(2-+0 with
WI
w = woH(
solutions
where
(4.7)
If we use the boundary condition for high energy injection used by Ridley, eqn (3.4) with the qualification about uniqueness mentioned in Section 3, we have:
jk
has separable
$+mF=O = 0
r$(z, T) = J,z
(4.12)
w,~;+2fo4
w%+(z+
into:
-2ogW$+[(2&+
l)w+p1;;+,+0, (5.1)
Boundary value problems in hot electron transport where again G(,/?, y as in eqn (3.1) but with /3 = &
373
gives F(r) = e_OEand H satisfying:
and: ‘!2$+39~+(l--w~)H=0.
3M: ’ = ZE,r,r,,, The treatment is now familiar from Sections 3 and 4 and we only outline the results.
The substitution tion:
(5.11)
H = $1~ reduces this to the equa-
(5.12)
or=/?=0 The equation is
The solution of this which is finite at the origin, q = 0, is the zero order modified Bessel function Refs[l6,17].
w:gj+ wg+f+)
(5.2)
with x = X/y. In terms of the current density per unit energy interval. which in this case is
) = J0(2iJG
)
(5.13)
So the separable solution of eqn (5.9) finite at W = 0 can be written: rn “Xe~“‘w10(2&%) dw, (5.14)
(5.3)
i(X W)=-&(W%)
where we must take o > 0 to ensure that &(X, W) remains finite as X -+ co. Further, the asymptotic form of I,(z) is[18]:
eqn (5.2) can be rewritten: W2.?L!+W-!L_j=0
ax2
$ = &(2&G
aw
(5.4)
.
I,(z)
With the change of variables:
T=q,
j;
IzI+m,
this reduces to the adjoint diffusion equation: T>O
$$+FT=O
(5.6)
The solution corresponding to the Ridley high energy injection conditions eqn (3.4) is found using eqn (3.7) and (3.8) and the boundary conditions in the form:
’ .i(O,~)=~6($?JZT m
440, 7’) = -
lim @4(x, T- r0
T) = 0.
w,)
(5.7)
J,,xWfi
W) =
?r(Wi-
e-X*/2V4’- t+?
(5.8)
wy
With EC, x, chosen to make CC,y = 1 there results: 8%
aw2
2w2
<(2-c)n/2,t
-;),
= &-1
where 2Fo(a, b, x) is a generalized hypergeometric series which is finite at x = 0. It follows that for w > 0, e~“W1,(2~)-+0 as W + co. Thus the solution eqn (5.14) for f0 is finite as both X and W become infinitely large, and there is therefore a possibility of a solution to the final BVP of the type eqn (3.12) wherein A (co) would be determined from the integral equation: h(W) =
z A(w)eeUWZ,,(2m)dw.
(5.15)
s0
cr=O,/!I#O W$$+
azo,p=o
w2
-(2+t)rt/2
2Fo(;,;;
The possibility of solutions for other BVP’s should also not be excluded.
The result is: Ax,
+G
(5.5)
8th +w2a2fo
axaw
ax?
--2W~+3W~;+/,=O.
Using the canonical coordinates this becomes:
,a%
ah
{= X+
ah
q &.Y+3tl~+)l-@+fo=o.
Looking for separable solutions: fo(& ‘I) = E(r)H(rl),
(5.9) W, 9 = W
(5.10)
W’g+(W+
l)F;+fo=O,
(5.16)
on replacing Wand X by W/B and X/By respectively. This equation is elliptic. On separating the variables the W equation is, with separation constant w, w d2H d~‘+(W+l)g~+(l-mW’)H=O.
(5.17)
We have not been able to reduce this to any of the standard equations with well known solutions. 6. DISCUSSION
We have considered the sorts of BVP’s which arise on using the nearly isotropic approximation to the BTE for electron transport with or without a con-
314
w. cox
stant field, with specific collision integral approximations for the cases of non-polar optical, acoustic and piezoelectric scattering. In each case one obtains a second order partial differential equation for the symmetric part of the distribution function fO. If one uses only a first order (in phonon energy) approximation for the collision integral this equation is parabolic, whether or not a constant field is present, but otherwise it is elliptic. In the zero field case we can, following Ridley[6], obtain also an equation for the current density per unit energy interval; and in the first order collision integral approximation this may be converted to a diffusion equation (or its adjoint) and solved explicitly for simple boundary conditions. We have done this, extending Ridley’s results, for acoustic and piezoelectric scattering. The BVP arising when a field is present, or a second order collision integral approximation is used, is naturally more complicated, but in all but two cases (acoustic with field and first order collision approximation; and piezoelectric, zero field and second order collision approximation) one can obtain analytical separable solutions. We have outlined these solutions and the possible BVP’s which they can solve. We have not examined specific solutions in detail or considered their applications. These aspects are best treated in the context of specific applications and will be published elsewhere. What we have tried to do here is give an overview of the types of analytical BVP’s which are available for modelling nearly isotropic hot electron transport in the presence of a field and a diffusion gradient, under various conditions. Acknowledgement-The Bailey for many useful
author would like to thank E. J. discussions relating to this work.
REFERENCES 1. W. Paul (Ed.), Handbook of Semiconductors. Vol. 1. North Hohand, Amsterdam-(1982). 2. E. Gornik. G. Bauer and E. Vass (Eds). Proc. 4th Inf. Conf. Hot Electrons in Semiconductors: Physica. Vol. 1349. North Holland, Amsterdam (1985). 3. E. M. Conwell, High field transport in semiconductors. In: Solid State Physics, Suppl. 9. (Edited by F. Seitz and D. Turnbull). Academic Press. New York (1967). 4. 9. R. Nag. Theory of Electrical Transport in Semiconductors. &gamon Press, Oxford (1972). 5. P. J. Price. Solid-St. Electron. 21. 9 (1978). 6. 9. K. RidJey, Solid-St. Electron. 24, ‘147 (1981). J. Comput. Maths elect1 elec7. W. Cox, COMPEL-lnt. tron. Engng 4, 183 (1985). 8. J. M. Woodcock, J. J. Harris and J. M. Shannon, Proc. 4th Int. Conx Hot Electrons in Semiconductors: Physica, Vol. 1349, (Edited by E. Gomik, G. Bauer and E. Vass), p. I1 I. North Holland, Amsterdam (1985). 9. 9. R. Nag, Phys. Reo. 11B, 3031 (1975). 10. G. A. Baraff, Phys Rec. 133A, 26, (1964). II. J. G. Boulton. Ph.D. Thesis, University of Cambridge (1984). 12. A. Friedman, Partial Dtflerential Equations ofParabolic Type. Prentice-Hall, New Jersey (1964). 13. D. V. Widder, The Heat Equation. Academic Press, New York (1975). M. M. Smirnov and E. 9. Gliner, 14. N. S. Koshlyakov, Dtrerential Equations ofMathematical Physics. North Holland, Amsterdam (1964). 15. W. Cox and E. J. Bailey, in preparation. D@erentialgleichungen-Ltisungsmethoden 16. E. Kamke, and Losungen, Bande. I. Gewohnliche Differentialgleichungen. Chelsea, New York (1971). 17. 1. N. Sneddon, Special Functions of Mathematical Physics and Chemistry. Oliver and Boyd, Edinburgh (1961). 18. Y. J-. Luke, Integrals qf Bessel Functions. McGraw-Hilt, New York (1962).
Electronics Vol. 30, No. 4, pp. 364374,
1987
0038-I IO]/87
BOUNDARY MODELLING
$3.00 + 0.00
Copyright 0 1987 Pergamon Journals Ltd
Printed in Great Britain. All rights reserved
VALUE PROBLEMS IN BOLTZMANN OF HOT ELECTRON TRANSPORT w. cox
Department
of Computer
Science and Mathematics, Aston Birmingham B4 7ET, U.K. (Received
University,
Aston
Triangle,
28 July 1986)
Abstract-We investigate the types of boundary value problem arising in modelling of hot electron transport by the nearly isotropic approximation to the Boltzmann equation. Models with a uniform electric field, and with specific collision term approximations, are considered for the cases of non-polar optical and acoustic phonon scattering and for piezoelectric scattering. In a number of interesting cases there exist models utilizing boundary value problems with relatively straightforward analytical solutions.
solutions satisfying realistic boundary conditions could be obtained in this case. Other@] have expressed interest in the extension of such simple analytical models to MHET’s in GaAs, for example. Here we will investigate quite generally the range of boundary value problems which can arise by application of the nearly isotropic approximation in nonpolar optical, acoustic phonon and peizoelectric scattering, with or without a field, with a spatially dependent distribution. We will discuss the availability of simple solutions and the sort of boundary conditions one can handle. The purpose at this stage is not to consider specific device models in detail, but to lay out the general approach and display the range of possibilities. Also, the approach is not confined to the scattering processes referred t-it will be suitable for modelling any device in which the electronic processes are adequately modelled by the nearly isotropic approximation to the 1-D BTE in a uniform electric field, in finite or infinite geometry, subject to mathematically well posed boundary conditions. The resulting equation may be analytically intractable in some cases, as for example in the case of polar optical phonon scattering, but often simple analytical solutions may be found. The approach is to take the first two terms in the Legendre polynomial expansion of the distribution function f(x, E) (E is the kinetic energy) about the field direction as axis of symmetry, splitting it into a symmetric part f0 and asymmetric part f, in energy space. Substitution in the BTE, with the usual appropriate approximations for the collision integral, yield two coupled equations for fu, f,. With a relaxation approximation for the asymmetric part of the collision integral, fi may be eliminated to yield a single equation forfo which one solves subject to approximate boundary conditions. In the spatially homogeneous case normally considered[3-51 the result is an ordinary differential equation for f0 in energy or velocity space. When f is spatially dependent the result is a linear partial differential equation for _&in
1. INTRODUCTION
For a wide range of applications, the Boltzmann transport equation (BTE) has proved a useful model of hot electron or high field effects in semiconductors. For extreme behaviour and submicron devices, there is little doubt that more sophisticated quantum transport equations are more appropriate[l,2]. However, even in such cases it is necessary to explore as widely as possible the implications of simpler BTE modelling applied to specific devices, so that comparisons with observations may be made and the limits of validity defined. For a given device structure the solution of the BTE under realistic boundary conditions may tax the largest computer, especially if one solves self consistently with the Poisson equation determining the electric field. Much work has been done on this, initially by semi-analytical methods such as moment methods, or Legendre polynomial expansion, and more recently by Monte-Carlo or iterative methods requiring large computational effort. This paper is motivated by the desire for relatively simple analytical or semi-analytical solutions which may provide models for particular device structures under hot electron behaviour. In particular we investigate the use of the “nearly isotropic” approximation (first two terms of the Legendre expansion) to the BTE for hot electron or high field transport, in formulating simple boundary value problems which may serve as useful models in specific applications. Semi-analytical treatments of such models for the case of homogeneous materials (i.e. spatially independent distribution function) have been available for some time[3-51, but little has been done on the spatially inhomogeneous case necessary for modelling devices. Ridley[6] has recently used such a model in studying high energy injection in monolithic hot electron transistors (MHET’s) in silicon, and for the calculation of the base-collector current-transfer ratio, in the zero field case. The author[7] extended this model to incorporate a uniform electric field and showed how 365
366
w. cox
x and E (or x and velocity) space. For example in the case treated by Ridley[6] of a first order approximation for non-polar optical phonon scattering and zero field the resulting equation for f0 is a simple parabolic equation which can be converted to a diffusion equation for the current density per unit energy interval, j(x, E). The problem therefore reduces to a boundary value problem (BVP) for the diffusion equation. In the presence of an electric field one still arrives at a parabolic equation[7], but this is no longer reducible to a simple equation for j(x, E). However, there exist analytical solutions for realistic boundary conditions for&(x, E) in this case too. In this paper we extend this type of modelling to other scattering mechanisms-acoustic and piezoelectric phonon-with or without a uniform field, and analyze the types of BVP one can envisage to model various physical situations. If the electric field is absent the resulting& equations can still be reduced to a BVP for the diffusion equation for j(x, E), similar to Ridley’s analysis. Naturally, the presence of the field complicates matters, as does the use of higher order approximations to the collision term, but in some cases there are still accessible analytical solutions. The modelling of the boundary conditions in a particular situation is delicate. Care must be taken to ensure that realistic physical situations are modelled by well posed BVP’s. Often of course, realistic boundary modelling is difficult and can preclude use of analytical solutions, but in some cases a useful compromise can be achieved. 2. LEGENDRE
so may
be regarded as functions of the energy instead of k, as indicated by the noEquation (2.2) may be written:
E = hZk/2M:
tation.
f
=h + k cosOf, =fo+ k,f,,
the asymmetric part off being characterized by k,. The conditions under which eqn (2.2) is a good analytical approximation are well documented in the literature[3,4]. Note however that discussion is usually confined to homogeneous transport in which there is no diffusion gradient term afiax, and in the present case it is necessary to ensure that combined diffusion and field drift effects remain sufficiently close to isotropy not to invalidate eqn (2.2). In the context of the Legendre expansion method, BaraffllO] has given an alternative truncation of the series, which allows the case of significant carrier streaming to be treated. It would be interesting to explore the formulation of the present approach in Baraff s treatment. The collision integral C(f) may also be expanded in the form eqn (2.2):
c(f)= Cop,,+ kC,P,. The forms of ing process below. Substituting orthogonality results in the
eqn (2.2) in eqn (2.1) and using the properties of the Legendre polynomials well known decomposition[9]:
(2.6)
We consider the problem of electron transport in a I-D material with a uniform electric field, [, and a I-D concentration gradient along the x-axis. Then the distribution function for the electron f(r, k, 1) taken in the quasi-classical effective mass approximation, in the steady state, depends only on x and k, the wave vector, and satisfies the BTE:
where k, = k cos 0 is the component of the wave vector in the direction of the electric field, with which it makes an angle 0; M: is the effective mass; C(f) is the collision integral[9]. Because of the axial symmetry in k-space, about the k, axis, we can expand f in the usual manner in a series of Legendre polynomials (in cos e), and adopting the standard nearly isotropic approximation one retains only the first two terrns[3,4,6,9]:
The general approximation
approach for C, :
c,=
0).
is to assume
--
Written in this manner both f0 and fi are functions only of x and the magnitude of the wave number, k,
’
a relaxation
(2.7)
where the relaxation time T,(E) depends on the particular collision process, and the approximation used to model it. With this assumption f, may be eliminated from eqn (2.5) giving an equation for fO:
where A = -e[, and EC, x, are characteristic energy and distance parameters used to define the dimensionless variables:
-xc
(2.2)
.A TI (~9
x=x
0) + kf,(x, E)P,(cos
(2.4)
C,, , C, depend on the particular scatterwe are modelling, and are discussed
EXPANSION OF THE BTE AND COLLISION TERM
f(x, k) =X,(x, E)P,(cos
(2.3)
w=z. kc
(2.9)
We will refrain from identifying x,, EC here, since the best choice will depend on the particular problem at hand.
Boundary value problems in hot electron transport Even without knowledge of T, it is clear that the left hand side of eqn (2.8) is a second order partial differential operator in (A’, W) space acting onh. The next stage is to adopt an approximation for the symmetric part of the collision integral C,, which results in a partial differential expression in W of at most second order. Then eqn (2.8) becomes a second order partial differential equation in (A’, W) for fO, which can be solved once we have defined suitable boundary conditions. Once& has been determinedf, is obtained from eqns (2.6) and (2.7) whence physically interesting quantities may be obtained in the usual way, averaging over J Note that there are two semi-analytical approximation steps here-the original near isotropy assumption eqn (2.2) and the assumption about the approximate form of the collision term. It will turn out that for the scattering processes and approximations we will consider the eqn (2.8) is parabolic if a first order (in phonon energy) approximation is used for C,,, as, for example, that used for non-polar optical scattering in Ref.[6]; while it becomes elliptic if a more accurate second order form is used, as for example in Ref.[4]. There are a range of standard boundary value problems (or strictly initial-boundary value problems, but we will ignore this distinction in terminology) for parabolic and elliptic equations for which existence and uniqueness theorems are well known, leading to a number of possible models of different physical situations. There are standard numerical routines for solving such BVP’s, but here we are specifically interested in investigating the possible semi-analytical approaches. There are a number of these available and in some cases these lead readily to simple analytical solutions describing quite realistic physical circumstances. In some cases however there are complications which lead to interesting mathematical problems, where the best coordinate system for analytical solution is not convenient for application of the boundary conditions. In the particular case of zero field it is possible to change the dependent variable fromf, to j(X, W), the current density per unit energy interval, as done by Ridley[6] for non-polar optical scattering, resulting in a BVP for the diffusion equation if a first order collision approximation is also used. The corresponding equations for acoustic and piezoelectric scattering are slightly more complicated, but still may be converted to a diffusion equation BVP by a change of variables. The approximation to be used for the collision term C(f) in particular cases is still a subject of debate, but we will simply use standard models which have already proved their worth. We will not concern ourselves with their range of validity in terms of the relevant parameters. Such questions are more readily dealt with in specific device applications. Further, we emphasize that the models given should be treated as examples of the general procedure. So long as one is
361
modelling by a near isotropic approximation, one is sure to arrive at a BVP for a parabolic or elliptic equation of the sort discussed here-its solution may be a different matter! Thus, it may be possible to find simple semi-analytical models for other scattering processes-polar optical phonon, intervalley; defects, impurities and so on. One may also consider combinations of these. The collision processes we will consider, and their Legendre expansions, are discussed fully in Nag[4], and we will simply review the results. For nondegenerate semiconductors the collision integral may be written in wave vector space as:
C(f)=
V -(27[)3
x
[P(k, k’)f(k) - P(k’, k)f(k’)] dk’,
(2.10)
s with P(k, k’) the usual transition probability from state k to state k’, V the volume. For purely lattice scattering modelled by electron-phonon coupling in first order perturbation theory P(k, k’) is given by the Golden Rule: P(k, k’) =;
IM(k, k’)(*6&
- Ek, k hw,),
(2.11)
where M(k, k’) is the transition matrix element, and wq the phonon energy ( + for absorption, - for emission), and k -k’ f q = 0 (no Umklapp processes). The genera1 form of M(k, k’) is: M(k, k’) = [A(q)]“‘,& = [A (q)]“‘Jn,+r
absorption emission
(2.12)
on the usual assumption of an infinite equilibrium phonon bath with occupation numbers {n,} (for range of validity of this assumption (see Ref.[llJ). Here A(q) depends on the type of scattering process. Substituting for P(k, k’), using the Legendre approximation eqn (2.3), and considering all emission and absorption processes there results:
C(f) ---
V (Zn)3
s
A(q)[6(E,-E,.+hw,)
x Nn,+ llf,(&)
-q&Q)
+ d(E, - 4’ - ~wJ{n,h(E,.) - (“9 + MdEdj + 66% - Er + Ro,) x {(n, + 1)k’ cos O’h(E,.) - n,k ~0s ofi( + 6(E, - Ek, - hw,){n,k’cos - (n4 + I)k cos Of,(E,J}] dk’
O’J(E,.) (2.13)
The terms involvingf, in this integral constitute the symmetric part of the collision integral. If, as is sometimes assumed,f, is the equilibrium distribution,
368
w. cox
then this part of the collision not so in the present case-it
integral is zero. This is in fact gives C,:
s
Y
co= A(q)[G(E, - &+ hw,) ml’ x ih, + IlAdE,,) - qwk)~
Piezoelectric
a’.
cosH
‘+
(2.15)
(For details and discussion see Ref.[4]), so eqn (2.7) is obtained for C,. For the specific processes we will consider the W dependences of 7, are as follows: Non-polar optical phonon scattering:
Acoustic
phonon
Piezoelectric
7"pt Iv
1.
(2.16)
scattering: 7, =
phonon 7, =
7a.cmw-"2.
(2.17)
scattering: tP w"?
7sp are all constants, independent of W, whose values do not concern us here-see Ref.[4]. Note that there are only essentially two distinct forms of 7, to deal with, W +I’*.
Suitable approximations for the symmetric part of the collision term C,, may be obtained by rewriting it utilizing the presence of the h-function terms as:
s
A(q)F’(& - Ek + hw,){(n, +
+ 6(E, -
EL. -
dk’
(2.19)
assuming Ek >>hw,, and expanding the f. (Ek & hw,) in a Taylor series. This procedure, and its range of validity is detailed in Ref.[4] and the results are: Non-polar optical phonon scattering:
co=$
w-“2
(;;+wg>I. [ w;;+pop
x fo+
phonon
woaf dW
B,,
(
^
2g;+w5&
11
(2.21)
’
’
scattering:
36 ’ + Bpzrw+w$
(
’
(2.22)
>I
Again, the T’S and ,0’s are constants, of no direct interest here, see Ref.[4]. The significant thing about the j?‘s is that they are of the order of the phonon energy and therefore represent corrections in the collision terms, which we may neglect in a first approximation, as done in Ref.[6]. In homogeneous high field transport such as that modelled in Nag[4] it is customary to retain these higher orders and we will allow ourselves this option here. Taking the /I’s non-zero changes the nature of the BVP, as we will see. The eqns (2.16H2.18) and (2.20)-(2.22) provide the necessary ingredients for use of the eqn (2.8) for determining fo. However, as shown by Ridley for non-polar optical scattering, in the field free case, the resulting& equation can be converted to one for the more physically relevant quantity j(X, W), the current density per unit energy interval. For this we will need the relation between j andf,. The current in the x-direction:
may be written, on substituting eqn (2.3), introducing polar coordinates, v2 = vt + v: + v’,, and using the symmetry properties off,, f,, in the form:
JxCC
WU W”% d W
(2.23)
s0
(the J,, Jz are zero) or
1)
hw,) x {n,f,(E, - ho,)
- hj + IXCWI
2fo,+
(2.18)
7opt3 t,,,,
co=
[
(2.14)
1
7, =
C&W”2
scattering:
+
The terms involvingf, constitute the asymmetric part of the collision integral. As mentioned earlier, we will now confine ourselves to collision processes for which there exists a simple relaxation approximation for this part. For example, this is appropriate to the randomizing collision model for collisions with optical and intervalley phonons and for elastic collision models for piezoelectric and impurity atom scattering. In either case the asymmetric part may be approximated by the form: -k
phonon
T?,ce
+ SC%- 4 - fq)In,h(E,) - (nq+ IM4N
Acoustic
The expression for the current density per unit energy interval can therefore be taken as, up to a multiplicative constant:
In the field free case (E, = 0) this becomes, multiplicative constant: j(X, W) = ;;
(2.20)
(7,
W”?fO).
up to a
(2.26)
Since eqn (2.8), and the T, and Co in eqns (2.16)-(2.18) and (2.20)-(2.22) only contain coefficients in Wit is
Boundary value problems in hot electron transport clear that in the free field case we can always convert the equation for & to one for j of the same order. Before treating the different scattering processes it is worth briefly reviewing the sort of BVP one encounters with parabolic and elliptic equations. Consider the general homogeneous parabolic equation in one spacelike variable x and one timelike variable t : a(x,r)~+b(x,t)~+~(~,r)~=~
(2.27)
a
t>O
[a(~, t) positive definite]. There are three classic (initial-) boundary value problems for this equation for which existence and uniqueness theorems are well known, and for which the diffusion equation in particular has readily obtainable analytical solutions[l2-141. Problem 1. Finite spatial domain q5 must be continuous and: Boundary
in and on boundary
region,
conditions
ti(a,t)=g,(t)
t 20
4(b,t)=g,(t)
r 20
Initial conditions $(x,O)=h(x)
a
gi and h given functions. The requirement of continuity of the boundary conditions is essential for guaranteeing uniqueness of the solution, and in particular it means that the functions gj, h are not arbitrary, but must satisfy g,(O) = h(a), gr(0) = h(b). Solutions violating such conditions are not unique. This does not of course invalidate them from representing real physical situations, but it means that this representation does not follow uniquely from the prescribed BVP. The solution given by Ridley for non-polar optical phonon scattering is precisely of this non-unique type-i.e. His BVP is not well posed. Problem 2. Infinite spatial domain 4(x, t) must be continuous and:
in and on its domain,
Initial condition 4(x,0)=/t(x) Boundary
--co
condition
14(x, t)l exponentially Problem 3. Semi-infinite 4(x, t) continuous
bounded
as lx ( + co.
Boundary
369
conditions
4(O,t)=IL(t)
t>O
(4 (x, t) 1exponentially
bounded
as 1x ) + co.
Again, the continuity of boundary conditions and the boundedness properties in Problems 2 and 3 are essential for uniqueness. In each of the above Problems the conditions have been applied to 4 itself, but of course there are equally important versions in which the conditions are applied to its derivatives. We will not consider such derivative BVP’s here, but technically they provide further useful models which may yield simple analytical solutions. Also, Problems l-3 merely scratch the surface of the parabolic BVP and we would not wish to discourage readers from exploring more complicated situations. The treatments of such problems in other areas such as heat conduction, chemical and nuclear transport, diffusion theory and so on has a vast literature where doubtless many useful results may be found. Now in fact, as we shall see, in the case of first order collision approximation one does not obtain, with the simplest choice of canonical coordinates, a parabolic equation of the form eqn (2.27). Instead, one obtains the adjoint form, with a(x, t) negative definite[l2,13]. This is easily seen in particular cases, and can be generally appreciated by consideration of the general form eqn (2.8). In general there are no solutions to the “initial” BVP’s of types 1-3, a fact most easily understood by considering the so called backward_ diffusion equation:
The type of problem one considers for the adjoint equation is typified by the “final” BVP of the form: 4(0, t) = h(r) lim 4(x, t) = 0. I-CC
(2.31)
Again there are many alternative types of such problems. In the following sections we will only consider a few of the simplest examples leading to analytical solutions. One can, with suitable interpretation of the boundary conditions applying in physical situations, convert the adjoint equation to the form eqn (2.27) with a(x, t) positive definite, by changing the timelike variable to say to - t. This was done by Ridley in Ref.[6]. However, we will tend to retain the adjoint form and use the BVP in the final value form eqn (2.31). For the general homogeneous elliptic equation:
spatial domain
in and on its domain:
$+2+a(x,y)g w
Initial condition 4(x,0)=/r(x)
o
+b(x,v)$+c(x.y)~
=o,
370
w.
the type of BVP is simpler, the standard possibilities being the prescription of either C$ or its normal derivative on segments of the boundary. We now turn to the specific types of scattering of interest to us.
3. NON-POLAR
OPTICAL
SCATTERING
Using eqns (2.16) and (2.20) in eqn simplifying yields the equation:
( >a2fo al+/?
a-22ay-
a’fo
5’fo
axaw +y $-@x 2
+(%+1)%+&o,
(2.8) and
cox
differentiating
eqn (3.2) with respect to X. We obtain: 2 ;+;;=0. (3.3)
Ridley solved this equation subject to boundary conditions purporting to represent high energy ( W,) injection of monoenergetic electrons at x = 0 into a semiconductor base, which attenuates the maximum energy within an infinitesimally small distance. Mathematically this corresponds to the BVP: j(0, W) = J,6( W - W,)
SY dfo
J, constant
lim j(x, W) = 0, w-w, (3.1)
where
(3.4)
for the eqn (3.3), which is not, as it stands, a parabolic equation of the form eqn (2.27); and this BVP does not correspond to any of Problems 1-3. Ridley remedies this by substituting s = W,, - Wand reformulating the BVP as
yz” /lx; ’
j(O,s)=J,6(3) j(x, s = 0) = 0
with
o
0 < x < co
(3.5)
for the resulting diffusion equation in (x, s). This corresponds to Problem 3, the semi-infinite spatial domain, but because of the d-function, continuity of the boundary conditions is not satisfied, so uniqueness is lost. This is not a serious problem-the 6(s) may be replaced by a sharply peaked Gaussian with the required continuity properties, the calculation repeated, and the (now unique) result of Ridley recovered as an approximation. Here we will determine j directly from equation (3.3) and the formulation of the BVP typified in equation (3.4), which is more representative of the physical situation. Equation (3.3) is in fact an example of the adjoint form of the parabolic equation[ 12,131. As discussed in Section 2, in general there is no solution to the “initial-BVP” Problem 3 to the adjoint equation, but solutions exist for the “final-BVP”:
The CK,/?, y are all dimensionless parameters whose numerical values depend on the scales x,, ECadopted to measure distance and energy. We are not interested in the precise values here, but it is important to note the general physical significance of the parameters. tl is proportional to the (uniform) electric field, and the zero field case corresponds to c( = 0. p essentially signals the presence of the collision term and will always be assumed non-zero. /I on the other hand corresponds to the second order approximation in the collision term and in those cases for which the first order approximation is adequate we can take p = 0; otherwise, note that /? > 0. For the full model including a field and second order collision approximation all the c(, /?, y are non-zero independent parameters. In this case, because b > 0, eqn (3.1) is elliptic, and j(0, W) = h(W) for a unique solution, boundary conditions have to be applied toy0 for some region in (X, W) space. On lim j(x, W) = 0. (3.6) the other hand, if we take /? = 0 then eqn (3.1) w- wrJ becomes a parabolic equation and unique solutions These are derived using superposition with the dewill be obtained to the sorts of well posed BVP’s rived source solution (Green’s function for the exemplified in Section 2. There are three distinct cases of eqn (3.1), with TV adjoint equation): or b or both taken to be zero. The simplest case, G*(x, W; 5, H’,,) r = /? = 0, corresponding to zero field and first order (x-r) collision approximation, is that considered by e-(x-;)~,‘4W-W w < w, Ridley[6], and with x = X/y it can be written as =Jm
a2(wJ)+ a(wxl) = ax2
aw
=o
w> w,.
(3.7)
o,
(3.2)
Ridley has already analysed this case, but it is worth briefly reviewing it. From eqns (2.26) and (2.16), we can replace a( Wfo)/ax, up to a factor, by j(x, W), by
The required
solution
f (& W) =
is:
= G*(x, W; 0, r)h(z) dr s0
(3.8)
Boundary value problems in hot electron transport so with h(T)=JO6(T
-
w,,)
i(x, W) J&e-x2i”+
H?6(r - w,) dr
(3.9) as obtained by Ridley[6]. Before leaving this simple case, it is worth remarking that BVP’s of the Problem 1 type are also very relevant to the sort of modelling we are discussing here, and may be treated by standard analytical methods (cf: heat conduction in a finite bar). Thus for example one could construct BVP’s on a finite domain as models of hot electron devices in a finite base region. The generalization of Ridley’s model to include the presence of a uniform field has been described by the author[7], and corresponds to the choice LY# 0, p = 0 in (3.1): 8% --
-
2
8%
1 ato +axt-TZ 8%
axaw
aw2
+ ;+ 1 g;+;fo=o, (3.10) ( > where, for convenience we have chosen EC, xc to make CL,y both unity. This equation is still parabolic and may be converted to canonical form by using the variables r = X + W, r~= W:
atro ah
ah
9=$+9~+(n+l)&-+fo=o.
In these equations ,Fl is the confluent hypergeometric function. Determination off,, and the evaluation of the current in this case is presented in Ref.[lS]. The generalization of Ridley’s analysis to include the second order collision approximation, but with zero field ;,:corresponds to tl = 0, p # 0 in eqn (3.1) resulting
-$+3+
;+ 1 ;;++=o, (3.15) ( > where we have replaced W, X by W/p, X/y respectively. Unlike the previous cases this equation is elliptic. As with the zero field first order case, equation (3.15) may be rewritten in terms of
wm
AX, W = ax in the form: W2g+
W’$+
W(W-l)Fi+j=O
(3.16) However, eqn (3.15) is easier to deal with in this case. In fact, apart from the X and < derivatives, eqn (3.15) is identical in form to eqn (3.11) and so separable solutions may be again obtained involving the confluent hypergeometric equation in W. We may use these to solve BVP’s withy, prescribed on a rectangle a
(3.11)
Again, this has the adjoint form to eqn (2.27). In Ref.[7] it has shown that solutions to the final BVP of the form:
371
4. ACOUSTIC
PHONON
SCATTERING
Equations (2.17) and (2.21) in eqn (2.8) and simplication gives:
MO, W) = h(W), lim fo(X, W) = 0,
(3.12)
W-U
exist, and in terms of X, W these can be written in the form
MK
/J = 3Md/2E
w cc
=
dtB(t)exp[-(t
+ l)2uT]
s
xUexp[--(r2+ t)X] t 1; (2t + 1)W (3.13) x IF, -’ 2t + 1’ ( > where B(t) is determined from the condition: h(W)=
(4.1) + $+2/l + w -&+2J,=o ( 1 a, 8, y are as in eqn (3.1), but with /I = /I, and
t
mdt B(t)exp(-(t s0
Since W 2 0, eqn (4.1) is elliptic if /I # 0 and is parabolic otherwise. Consider first the simplest case LY= p = 0, which generalizes Ridley’s analysis to the case of acoustic phonon scattering. With z =X/y the equation becomes: 8% g + w;;
+ 2fo = 0.
(4.2)
in this case, from eqns (2.17) and (2.26) we again have, up to a constant factor: Now
+ l)2 W]
x 16 -’2t+ 1’ 1;
c T a.mTace.
(2t + 1)W .
>
(3.14)
j(z,
W)=f!$i?.l
w. cox
312
for the current density per unit energy terms of j eqn (4.2) may be rewritten:
a*i
J(W) _.
s+ aw changing
interval.
the variables
In
(4.3)
’
to:
C$= Wj,
W = e’,
in turn reduces this to the adjoint
produces
(4.4)
diffusion
equation
^
s+gT=O T>O
Unfortunately the q-equation which results on separation of variables in this case does not appear to be readily reducible to a standard form and requires a systematic development before being of practical use. In the case of x = 0, b # 0 the equation (4.1) becomes, replacing W, X with W/p, X/y respectively
(4.5) w$$+
for C#J(Z,T). The BVP eqn (3.6) translates
again elliptic. This equation of the form:
#(O, T) = erh(er) JirnoemT+(z, T) = 0. We may again use the solution obtain
(4.6)
eqns (3.7) and (3.8) to
h(X
T > To
,= Jwxe’W-eT”)d7 the integral
with
W = e’
wj(z, w) = Jot e-Z2i4’n(waiw) (4.8)
J47r [ln( W. I WI’ so Joz =
,-z~:41n(Wo#w
WJ4n
(4.9)
[ln( W,/ W)13
This is the solution for acoustic phonon scattering which corresponds to Ridley’s result eqn (3.9) for non-polar optical scattering. Next consider the non-zero field but p = 0 case, again choosing E,, x, to make x, y both unity. There results:
3%
8%
+=-wax ax
axdw
+
;
1
constant. By substituting may be converted to the equation:
&!2+(2_ W)A!&*
=o,
dW
H(W)
_ W+’
(4.17)
and solutions well behaved for large W may be obtained if we restrict w < 2. These may be suitable for certain physical conditions.
PHONON
SCATTERING
$+
2f,=o.
2
(4.10) (cr*W*+aw,~-2,,W2~w+p2W:~
form with the variables:
<=x+w q=w
(4.16)
finite at W = 0. , F, denotes the confluent hypergeometric function. Therefore, we have solutions in analytical form and, subject to boundary conditions imposed on the rectangle a < X < b; c < W < d, we may have a reasonably tractable BVP. If this proves too complicated then we must resort to numerical methods. Also it is worth noting that, since the asymptotic behaviour of ,F, (a; b; X) for large X is -exXueb, we see that:
5. PIEZOELECTRIC
so
>
to canonical
(4.15)
In this case, eqns (2.18) and (2.22) in eqn (2.8) gives:
+ w
( Reduction
confluent
separation eqn (4.14) hypergeometric
II/ = ,F,(w; 2; W) = e’“H
with W, = e’O. Evaluating gives:
l?WZ
the
(4.14)
which has solution:
s
2
o
H = e-W$,
dW*
e-r2.‘4(r- n
32% --
WI,
w d*H ~+(W+2)~w+(2-+0 with
WI
w = woH(
solutions
where
(4.7)
If we use the boundary condition for high energy injection used by Ridley, eqn (3.4) with the qualification about uniqueness mentioned in Section 3, we have:
jk
has separable
$+mF=O = 0
r$(z, T) = J,z
(4.12)
w,~;+2fo4
w%+(z+
into:
-2ogW$+[(2&+
l)w+p1;;+,+0, (5.1)
Boundary value problems in hot electron transport where again G(,/?, y as in eqn (3.1) but with /3 = &
373
gives F(r) = e_OEand H satisfying:
and: ‘!2$+39~+(l--w~)H=0.
3M: ’ = ZE,r,r,,, The treatment is now familiar from Sections 3 and 4 and we only outline the results.
The substitution tion:
(5.11)
H = $1~ reduces this to the equa-
(5.12)
or=/?=0 The equation is
The solution of this which is finite at the origin, q = 0, is the zero order modified Bessel function Refs[l6,17].
w:gj+ wg+f+)
(5.2)
with x = X/y. In terms of the current density per unit energy interval. which in this case is
) = J0(2iJG
)
(5.13)
So the separable solution of eqn (5.9) finite at W = 0 can be written: rn “Xe~“‘w10(2&%) dw, (5.14)
(5.3)
i(X W)=-&(W%)
where we must take o > 0 to ensure that &(X, W) remains finite as X -+ co. Further, the asymptotic form of I,(z) is[18]:
eqn (5.2) can be rewritten: W2.?L!+W-!L_j=0
ax2
$ = &(2&G
aw
(5.4)
.
I,(z)
With the change of variables:
T=q,
j;
IzI+m,
this reduces to the adjoint diffusion equation: T>O
$$+FT=O
(5.6)
The solution corresponding to the Ridley high energy injection conditions eqn (3.4) is found using eqn (3.7) and (3.8) and the boundary conditions in the form:
’ .i(O,~)=~6($?JZT m
440, 7’) = -
lim @4(x, T- r0
T) = 0.
w,)
(5.7)
J,,xWfi
W) =
?r(Wi-
e-X*/2V4’- t+?
(5.8)
wy
With EC, x, chosen to make CC,y = 1 there results: 8%
aw2
2w2
<(2-c)n/2,t
-;),
= &-1
where 2Fo(a, b, x) is a generalized hypergeometric series which is finite at x = 0. It follows that for w > 0, e~“W1,(2~)-+0 as W + co. Thus the solution eqn (5.14) for f0 is finite as both X and W become infinitely large, and there is therefore a possibility of a solution to the final BVP of the type eqn (3.12) wherein A (co) would be determined from the integral equation: h(W) =
z A(w)eeUWZ,,(2m)dw.
(5.15)
s0
cr=O,/!I#O W$$+
azo,p=o
w2
-(2+t)rt/2
2Fo(;,;;
The possibility of solutions for other BVP’s should also not be excluded.
The result is: Ax,
+G
(5.5)
8th +w2a2fo
axaw
ax?
--2W~+3W~;+/,=O.
Using the canonical coordinates this becomes:
,a%
ah
{= X+
ah
q &.Y+3tl~+)l-@+fo=o.
Looking for separable solutions: fo(& ‘I) = E(r)H(rl),
(5.9) W, 9 = W
(5.10)
W’g+(W+
l)F;+fo=O,
(5.16)
on replacing Wand X by W/B and X/By respectively. This equation is elliptic. On separating the variables the W equation is, with separation constant w, w d2H d~‘+(W+l)g~+(l-mW’)H=O.
(5.17)
We have not been able to reduce this to any of the standard equations with well known solutions. 6. DISCUSSION
We have considered the sorts of BVP’s which arise on using the nearly isotropic approximation to the BTE for electron transport with or without a con-
314
w. cox
stant field, with specific collision integral approximations for the cases of non-polar optical, acoustic and piezoelectric scattering. In each case one obtains a second order partial differential equation for the symmetric part of the distribution function fO. If one uses only a first order (in phonon energy) approximation for the collision integral this equation is parabolic, whether or not a constant field is present, but otherwise it is elliptic. In the zero field case we can, following Ridley[6], obtain also an equation for the current density per unit energy interval; and in the first order collision integral approximation this may be converted to a diffusion equation (or its adjoint) and solved explicitly for simple boundary conditions. We have done this, extending Ridley’s results, for acoustic and piezoelectric scattering. The BVP arising when a field is present, or a second order collision integral approximation is used, is naturally more complicated, but in all but two cases (acoustic with field and first order collision approximation; and piezoelectric, zero field and second order collision approximation) one can obtain analytical separable solutions. We have outlined these solutions and the possible BVP’s which they can solve. We have not examined specific solutions in detail or considered their applications. These aspects are best treated in the context of specific applications and will be published elsewhere. What we have tried to do here is give an overview of the types of analytical BVP’s which are available for modelling nearly isotropic hot electron transport in the presence of a field and a diffusion gradient, under various conditions. Acknowledgement-The Bailey for many useful
author would like to thank E. J. discussions relating to this work.
REFERENCES 1. W. Paul (Ed.), Handbook of Semiconductors. Vol. 1. North Hohand, Amsterdam-(1982). 2. E. Gornik. G. Bauer and E. Vass (Eds). Proc. 4th Inf. Conf. Hot Electrons in Semiconductors: Physica. Vol. 1349. North Holland, Amsterdam (1985). 3. E. M. Conwell, High field transport in semiconductors. In: Solid State Physics, Suppl. 9. (Edited by F. Seitz and D. Turnbull). Academic Press. New York (1967). 4. 9. R. Nag. Theory of Electrical Transport in Semiconductors. &gamon Press, Oxford (1972). 5. P. J. Price. Solid-St. Electron. 21. 9 (1978). 6. 9. K. RidJey, Solid-St. Electron. 24, ‘147 (1981). J. Comput. Maths elect1 elec7. W. Cox, COMPEL-lnt. tron. Engng 4, 183 (1985). 8. J. M. Woodcock, J. J. Harris and J. M. Shannon, Proc. 4th Int. Conx Hot Electrons in Semiconductors: Physica, Vol. 1349, (Edited by E. Gomik, G. Bauer and E. Vass), p. I1 I. North Holland, Amsterdam (1985). 9. 9. R. Nag, Phys. Reo. 11B, 3031 (1975). 10. G. A. Baraff, Phys Rec. 133A, 26, (1964). II. J. G. Boulton. Ph.D. Thesis, University of Cambridge (1984). 12. A. Friedman, Partial Dtflerential Equations ofParabolic Type. Prentice-Hall, New Jersey (1964). 13. D. V. Widder, The Heat Equation. Academic Press, New York (1975). M. M. Smirnov and E. 9. Gliner, 14. N. S. Koshlyakov, Dtrerential Equations ofMathematical Physics. North Holland, Amsterdam (1964). 15. W. Cox and E. J. Bailey, in preparation. D@erentialgleichungen-Ltisungsmethoden 16. E. Kamke, and Losungen, Bande. I. Gewohnliche Differentialgleichungen. Chelsea, New York (1971). 17. 1. N. Sneddon, Special Functions of Mathematical Physics and Chemistry. Oliver and Boyd, Edinburgh (1961). 18. Y. J-. Luke, Integrals qf Bessel Functions. McGraw-Hilt, New York (1962).