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SERIJA:
MATEMATIKA
I FIZIKA
-
.N!! 412
SERlE:
-
MATH:£:MATIQUES
NI! 460
ET PHYSIQ
UE
(1973)
HOT ELECTRON TRANSPORT IN SEMICONDUCTOR
SPACE-CHARGE REGION*
441.
Dimitrije A. Tjapkin and Milan M. Jevtic**
ABSTRACT: The distribution function (I) of hot carriers in space-charge region
(SCR) is studied with help of steady-state
Boltzmann equation. Introducing
new variables the partial differential equation reduces to an ordinary one.
Assuming the appropriate
form for the colIision term and introducing the
current density, the comparative results are given for 1 for the case of the
uniform SCR (Figs 1 and 2).
1. Introduction
In space-charge region (SCR) there exists a built-in electric field of an
order of magnitude that would, as an external field, provoke in a homogeneous
semiconductor the well-known effects treated by the hot-electron theory. Consequently, the space charge and a strong non-uniform electric field constitute
conditions for a specific electron transport compared to the one in a homogeneous
semiconductor exposed to a strong external field. The problems of hot-electron
transport in homogeneous semiconductors have been widely analysed (e.g. see
books [1, 2, 3]). However, the results of these analyses cannot be directly
transferred to charge transport through SCRs. Despite a considerable theoretical
and practical importance, the later problems have been tackled to a noticeably
lesser extent [4, 5, 6, 7]. One of still unresolved problems is the form of EINSTEIN
relation since both drift and diffusion processes are present. This and other
problems related to carrier transport through SCRs have also become important
for the theory of domains [8].
In the papers published so far two approaches are suggested. The first
approach relies upon direct solving of BOLTZMANNequation [6, 9] to obtain the
distribution function (in [9] the time dependent equation is considered whereas
the present paper treats only the stationary case). The second approach makes
use of the notion of the equivalent electron temperature, Te [4, 5] in the
Maxwellian distribution function of energies Is. According to STRATTON[5] the
exact form of Is is unimportant for the majority of problems of practical
interest
-
an assumption
which is not immediately
obvious.
The correction
of the
* Presented June 30, 1973 by J. POP-JORDANOV.
The work reported herein was partially sponsored by Serbian Fund for Scientific
** (under Contract N2 2861; Institute of Phisics of the Belgrade University, Beograd).
Research
143
D. A. Tjarkin and M. M. Jevtic
144
distribution function by Te, which is only position dependent, reduces the variety
of energy-dependent forms of is. The equivalent temperature defined via a
kinetic energy of electrons « w » is not, in general, equal to Te determined by
generalized EINSTEINrelation (e. g. see [8]). The theory developed in [5] employs
EINSTEINrelation which contains Te determined through < W> thus leaving space
for further reevaluation of the ap)roach.
The difficulties predominantly of mathematical nature caused that the
approach of direct solution of BOLTZMANNequation is far less developed.
GORDEEV[6] gave a method of direct solution of DAVYDOV'Skinetic equation*
in high-field regions of p-n junctions. He assumes the solution in the form of
a combination of MAXWELL'Sand DRYVESTYN'Sforms of distribution functions,
but the definitions of the corres:'onding boundary conditio..s are not correct.
Substitution
of the symmetric
J
by the equilibrium
J
in the analysis
of electron
mobility in [10] is not quite correct. The mobility is also considered in [11]
and [12]. The conce;)t of Te is also ap:-,lied in the analysis of the space charge
limited currents [13] and GUNN effect [14]. A discussion of some analogies of
the space charge current equation and the energy balance equation is given
in [7].
The present paper deals with the problem of electro:! trans;Jort through
s;,ace charge regions of semiconductors, starting from BOLTZMANN(and POISSON)
equation. Special care is devoted to the collision term and to the introduction
of current in the calculations. The developed theory is applied to the case of a
region containing uniform s~ace charge.
2. Theory
Assume that in the SCR the energy spectrum is not affected by the
influence of electric field (E) and that the processes are such that the conditions
of uIicertailty of momentum ad position do not apply. Then it is possible to
use BOLTZMANNequation for J(k, r) in the form
(1)
v \1,1 +!LE \1d=
II
Of
()
0f c
valid under steady-state conditions in the absence of tem~erature gradients. In (1)
q-carrier charge (-e
for electrons,
+ e for holes),
Of
( )-
collision
operator.
Of c
In comparison with the homogeneous specimen where E is an externally applied
field, the field in the [resent treatment is the sum of the built-in and ap;.:J!ied
fields determined by POISSON'Sequation
(2)
Space charge density p de:-ends upon position (r) via dependences of concentrations n=n(r), Nd=Nd(r), ... . If the collision term (oJ/ot)c and p, were
known, the set of equations (I) and (2) defines the distribution function and
consequently the electron trans;Jort through an SCR.
* This equation is valid only in the case of acoustic scattering and can be derived
from BJLTZMANN equation [15, 18] if the collision term is expressed in the form given on
page 225 of [2].
Hot electron
By writing
(e. g. [2])
transport
in semiconductor
the collision term
and
space-charge
distribution
(3)
I(k, r)=Is(r,
W)+kxg(r,
(4)
(~~t =
(~t -kxg/T,
function
(
3
iJW
)
3ft
145
in the
form
of E.
Here
W),
one obtains for spheric energy surfaces in one-dimensional
qE
2W iJg iJls
2W ~
,
g+
+
=
It
region
~pace:
()
iJx
iJt c
(5)
qEItT iJIs
--+-~=-g,
m
iJW
where the positive direction
-
ItT iJIs
m iJx
of x coincides with the direction
Is - symmetric part of the distribution
anti symmetric part of f, l' (W)
function with respect to the wave vector k, kx g
m
scalar
effective
mass of carriers,
-
- momentum relaxation time in weak fields, and W - kinetic energy of carriers.
The collision term (iJls/iJl)c vanishes for elastic collisions or under conditions
of thermodynamic equilibrium. However, in the present case this term can by
no means be neglected.
By eliminating g from (5) there follows [15] the partial differential equation
for Is:
q2 E2,
2q, W iJE iJls
2 Wq2 E2 ~
(6)~J-J:'q2 £2~ iJ2is
+
+
+
3m
(
iJW2
-1-
m
3m
iJW
4WqE, _iJ21s
3m
iJWiJx
+
3m
qE'+2qEW
m
3m
(
)
~
)
iJx iJW
iJls+2W'
iJW iJx
3m
iJ21s=_
iJx2
iJls
()
iJt c
where the spacial dependence of the relaxation time is not considered,
i. e. iJ1'/iJx=O.
For the case of a field slowly varying in space the approximation whereby
all terms involving derivatives of Is in x in (6) are neglected [17], can hardly
give correct solutions, although the term containing iJE/iJx is completely retained.
Introducing new variables [16]
qcp
, Y = 1),
(7)
u= z+
kT
~
and normalizing Wand x as follows z = W/kT, 1)= x/w, from (6) one obtains a
new equation of the form
iJ2 Is
(8)
iJy2
where <p-
qEw
+
kT
(
1
electrostatic
iJT
iJu
+
3
potential:
include derivatives of Is in u, then
with u as a parameter.
10 Publikacije Elektrotehnickog
iJls
2(u-qcp/kT)
fakulteta
)~-.;;= -
3 mw2
2kT,(u-qcp/kT)
iJfs
(-;)t)c'
iJcp
-
= E. If the collision term does not
iJx
(8) becomes
an ordinary
differential
equation
D. A. Tjapkin and M. M. Jevtic
146
In the presence of different scattering mechanisms the collision term can
be written in the form
iJfs
iJfs
( )= L( ) ,
iJ t c
i
iJ t ci
where i denotes i-th scattering mechanism. In general the contributions of each
of them is different. As a criterion of the dominance of the electron-electron
scattering it is possible to use the inequality [8]
(9)
where
LD = (1::1::0
kT/8
7t ne2)1/2
-
DEBYE radius.
Typical
values
of concentrations
beyond which the electron-electron scattering prevails are in Ge 1014 cm-3 and
in Si 1013 cm-3.
There are no obvious reasons that the energy dependence of the collision
term in the SCR of a non-homogeneous semiconductor is different from this
dependence in a homogeneous semiconductor. Thus, in the first approximation
it is possible to use for (0Is/0 t)c the form obtained by using Is valid for a
homogeneous semiconductor in a strong electric field, for each scattering mechanism. In doing so there appears the field E which is also included in the
energy balance equation
(10)
JE=-
iJfs
j w ( ) dVk+~
iJt c 47t3
Vk
~(kxg)WdVk.
iJx .f 3 m
Vk
47t3
The current density J introduced in (10) is of basic impQrtance for the treatment
of electron transport in SCR since in an SCR there exists a built-in electric
field even under equilibrium conditions. This complicates the calculations but
alIows certain approximations, especially if the second term on the right hand
side of (10) is much smaller than the first. In the latter case (10) reduces to
(11)
For small current and weak fields (1 I) combined with (13) gives
E 2=-s E n2=~
(12)
.
,
enb fl.io
nb - carrier density at the edge of SCR (boundary between the
homogeneous and non-homogeneous regions), and [LiO- low-field mobility for
the i-th scattering mechanism. An application to a specific physical model defines
the spacial dependence of E.
If in the case of acoustic scattering (ols/ot)c is used from [2], the described
approximation gives
where
(13)
iJfs
=2v/(2mz)1/2.S(2s-z2+Z)J;,
l(kT)1/2
(Z+S)2
iJt Ca
()
s
wherevs-sound
velocity in the crystal, and s=E2/En2 with E/= 16v//(37t[La02).
The field variable s in (13) has to be replaced by the right hand side of eq. (12).
Hot electron
transport
in semiconductor
space-charge
region
147
3. Distribution function in the region of a uniform space charge
The theory of Section 2 will be applied to a particular physical model of
the SCR in a non-polar semiconductor. The electric field (E) and electrostatic
potential (r.p)are defined as follows:
qE
--=
kT
(14)
where
the
constant
-G 0 wy,
Go is determined
by
each
specific
of the SCR. For an abrupt asymmetrical p-n junction
field Em at y = 1, Go= eEm/kTw.
model,
and
w
having maximum
For acoustic scattering at moderate fields (s~ 1, u'}> Gow2y2),
~
and (13) the following differential equation is derived
d2 J.
--"--y
dy2
(15)
a
dl'
JS+(A
II
dy
0
y2+C
0
y)J;=O
S
-
width
electric
from (8)
,
where
(16)
fLaokTJw
B o = -~
16
C0 =
IlDe2v/'
Jaw(1-3~
fLaollbkTII'
The boundary conditions are chosen to obtain a well behaved solution at
the transition between non-homogeneous and homogeneous regions at y = O.
They can be expressed as follows:
dfs
is (u, 0) = Ae-u,
(17)
I
dy Iy~o =
0.
Here E = 0 at y = 0, and constant A is determined from the conditions assumed
to apply at the boundary of the homogeneous region
A =
lib 113
(2 7t mkT)-3/2,
2
After several transformations
the solution of (15) is obtained in the form
(18)
with
fs = V (u, y)fs (u, 0),
(19)
a
C
(1.=---4/1
II'
$ = $
12'
b= -
10'
(l!.-
a2
C=
~'~
2'
)
2 '
C02 + 20( C2
C3
~
'
--Ao,
u2
~=
Co.
C'
(
$ 2 = t $ l!.-+~,
2 2
t = - yC- y-c
(
2~
)'
3.
2'
t2
2' )
D. A. Tjapkin and M. M. Jevtic
148
.
"" (a, c, Z ) =
'I-'
, a
1 ,---
(20)
<1>10
=
(
.L.
2'
- 2 ~b
<1>
<1>
<1>~0=
iJ..
2'
c
z
'-.
I!
a(a+l)
+ -c(c+
Z2,
1) 2!
2
~o~
2 '
)
<1>
20
(iJ..2 + I '
2..
2'
=
a(a.d)
T
~
V/ C
102
(ac.2)
Z3
--
c(c+ I) (c+2)
<1>
(
_k.
2
3!
3.
+ }-2 , 2'
+ ... ,
102
)
2 '
~; 1;2)+ 4:2 <1> +%,
(~
<1>;0=-VC[<1> (i+-},
)
2 '
.};
1;2)]-
Here <1>
(a, c; z) is hypergeometric confluent function [19] of the independent
variable z, and parameters a and c. Distribution function (18) depends on y
and u, i. e. on "f)and z, and satisfies conditions (17) at y = O. It also satisfies
the transition from nonequilibrium to equilibrium conditions, for if J -+ 0, also
Co-+ 0, Ao -+ 0, a:-+ 0, and Is reduces to the Maxwellian form.
If Ao~ 0, then a:~ 0 and the following approximations can be used
(21 )
,
C~.ll
U
t~
-.ll
U
(y-
3 Co U2 ,
a2
)
-_C02 U3.
b~
a'
At moderate electric fields t2, to2<t;,I, and the corresponding functions
<1>
(a, c; z) can be approximated by two first terms when calculating Is.
The results obtained for an Si abrupt p-n junction are shown in Figs 1
and 2. Donor and acceptor impurity concentrations are assumed Nd= 1017cm-3
10
9.0
I
I
l..4.~.foo.
V(",OJ) ~83.5tS
1
0.9
5"; ::p-~~
J
08
07
/
/
/
/
/
8.0
4
j--
~.
'l0
.1.0
:
!
I
0-6
O.S
li'
0'1
0.3
/1120
V (<I,o.t)
1,
I
01
°
109
--\0
Fig. 1. Symmetric
part of the distribution
function
total energy (u.kT~ W + q 'P) for several positions y
in SCR at constant
~ 2.0
---
U.
(Is) and factor V versus
~
x/w (or field strengths)
space charge.
and No = 1012 cm-3, maximum electric field Em = 4 X 102 Vjcm and SCR width
are the curves
corresponding
to homogeneous
w:::: 10- 3 em. Also shown
semiconductors
(after [2]). Curve (5) of Fig. 1 obviously
is not an acceptable
approximation
since it does not reflect any current dependence.
If E2 in s is
Hot electron
transport
in semiconductor
space-charge
149
region
calculated using (I2) which contains current density, one obtains curves shown
in Fig. 2. Noticeable difference between these results and those obtained for a
homogeneous semiconductor confirm that the latter cannot be directly transferred
to the transport problems in SCRs. Fig. I also shows V (u, y) at y = 0.1 which
~=0.1.
1 - e-U.
2 - J = 10-'A/","
, - .J =10" Aim'
0.8
-
E'\; <'8)
- from
aHer rcf.IZ/'wi~:I>
-. -.-
'-
.=JE/en~}J
E..
4 - J;': IQ"2Alm<
0.7
06
0.5
0.4
0.3
0.2
0.1
O.OL
0.0,
0.1
0.5
10
Fig. 2. Symmetric part of the distribution
from eq. (18) and after ref. [21, current
5.
10,
20..u.
function versus total energy,
density being a parameter.
indicates that the "deformation" effect is considerable at low energies. At high
energies, despite the steep rise of V (u, y) final contribution of Is is of little
importance because of the very sharp falloff of exp (-u).
The result (I8) indicates that in general it is not possible to replace the
symmetric part of the distribution function by the equilibrium di~tribution
function in SCR in nonequilibrium conditions.
4. Conclusion
The problem of electron transport in nonhomogeneous semiconductors and
strong electric fields can be treated starting from BOLTZMANN(and POISSON'S)
equation. Aprlying approj)riate transformations it is possible to derive a new
equation for Is> eq. (8). By making suitable choice of the collision term (8)
reduces to an ordinary differential equation of second order. Applying the
energy balance equation (subject to low current densities and moderate electric
fields) current density has been introduced into calculations. This is of particular
importance for treating nonequilibrium processes in SCRs.
The developed theory is applied to the uniform charge SCR, defining the
boundary conditions which satisfy the transition from the nonhomogeneou;
to
homogeneous region. An ex)licit solution for Is is obtained, eq. (I8). The results
calculated from this Is (Figs 1 and 2) show that: (i) the symmetric part of
the distribution function differs increasingly from the equilibrium distribution
function as the built-in electric field or current density increase; (ii) the resulting
effect of high electric field is such that the low energy carriers undergo most
D. A. Tjapkin and M. M. Jevtic
150
of the induced changes; (iii) the Is dependence on energy and position becomes
much more complicated from that expressible via the equivalent temperature
approach.
Applications of the present theory to calculating EINSTEINrelation, mobility
current flow characteristics in SCR, etc. are being investigated. The results, if
significant, will be published in due course.
ACKNOWLEDGEMENTS.
Interesting discussions and useful suggestions of Prof.
Dr. J. POP-JORDANOV,Dr. Z. DJURIC, and Mr. M. SMILJANICare gratefully
acknowledged.
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G.
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