R. K. JAIN:
Fermi Level, Carrier Concentration, and Einstein Relation
221
phys. stat. sol. (a) 42, 221 (1977)
Subject classification: 13.1 and 14.3; 22.1.1; 22.1.2
Laboratorium Fysica en Elektronica van de Halfgeleiders, Departement Elektrotechniek,
Katholieke Univeraiteit Leuven, Heverleel)
Calculation of the Fermi Level, Minority Carrier Concentration,
Effective Intrinsic Concentration, and Einstein Relation
in n- and p-Type Germanium and Silicon
BY
R. K. JAIN~)
For the basic understanding of the physics of any semiconductor device an up-to-dateknowledge
of the various basic parameters is highly essential. Theoretical calculations are made of the Fermi
level, minority carrier concentration, effective intrinsic concentration, and Einstein relation in
n- and p-type germanium and silicon a t 300 K. The reported work is based on the recently developed new transport theory of heavily doped silicon by van Overstraeten et al.. It is found that
the results are significantly different from the classical results.
Fur ein grundlegendes Verstandnis der Physik eines Halbleiterbauelements ist eine vollstandige
Kenntnis der verschiedenen grundlegenden Parameter sehr wesentlich. Es werden theoretische
Berechnungen durchgefiihrt fur das Ferminiveau, die Minoritiitsladungstriigerkonzentration,
effektive Eigenleitungskonzentration und Einsteinrelation in n- und p-leitendem Germanium und
Silizium bei 300 K. Die Arbeit basiert auf einer kurzlich von van Overstraeten et al. entwickelten
Transporttheorie fur stark dotiertes Silizium. Es wird gefunden, daB die Ergebnisse sich bedeutend
Ton den klassischen Ergebnissen unterscheiden.
1. Introduction
A lightly doped crystal is theoretically characterized by a perfect periodicity of the
lattice. I n this case i t can be proved that the edges of the conduction and valence
bands are well defined and that the density of quantum states obeys a square root
dependence on energy. Since the concentration of the impurity atoms is strongly
diluted, the wave functions associated with the electrons of the impurity atoms do
not overlap. Consequently the energy levels of the impurity atoms are discrete.
Under such circumstances the semiconductor crystal is called nondegenerate.
At higher impurity concentrations the impurity atoms are beginning t o interact
with each other so that the wave functions of their associated electrons are going t o
overlap. This causes a splitting of the impurity energy levels, which a t higher concentrations results in the formation of a n impurity band [l]. As the impurity concentration increases, the electron concentration also increases and the interaction between
electrons and atomic cores may not be neglected any more. The result of this interaction is a modification of the lattice periodicity and consequently a change in the
number of allowed states. Instead of a well defined conduction band edge, we now
have a band tail. The equation of this modified conduction band has been calculated
in [2, 31. Therefore, a t very high impurity concentrations the semiconductor crystal
is called degenerate, and for optimum impurity concentratlions it is called semidegenerate.
-
Kardinaal Mercierlaan 94, 3030 Heverlee, Belgium.
Present address: Laboratorio de Electrhnica, Instituto Venezolano de Investigaciones Cientificas, Apartado 1827, Caracas, Venezuela.
l)
2)
222
R. K. JAIN
The purpose of the present work is to report tho theoretical calculations of the Fermi
level, minority carrier concentration, effective intrinsic concentration, and Einstein
relation in n- and p-type germanium and silicon at 300 K. For the basic. understanding
of the physics of any semiconductor device an up-to-date knowledge of the various
basic parameters considered here is highly essential. The reported work is based on
the recently developed new transport theory of heavily doped semiconductor by
van Overstraeten et al. [4].Heavy doping effects are mainly characterized by considering the dependence of the density of quantum states or impurity states on the
iiiipurity concentration. Classical calculations never consider such a dependence.
It has been found that our results based on position-dependent hand structure phenomena are significantly different from the well-known classical results available until
now.
2. Sumniary of the Basic Approach
The important basic parameters, namely the Fermi level, minority carrier concentration, effective intrinsic concentration, and Einstein relation, have been evaluated
using the basic approach. This approach is nothing but to consider heavy doping
effects in a quantitative manner. van Overstraeten et al. 141 have recently described
this approach in detail by considering the mathematical equations describing the
formation of band edge tailing and impurity band after Morgan [l] and Kane [a].
We have also considered the same method to evaluate the parameters considered
in the present work in n- and p-type germanium and silicon a t 300 K. Therefore, we
have omitted the detailed mathematical description of the basic approach and considered the final results and their physical significance. We have considered phosphorus
and boron as donor and acceptor impurities, respectively, but the present results can
also be considered for arsenic, since its energy level is close to the phosphorus level.
All calculations have been done using IBM 3701158 computer.
3. Calculation of the Fermi Level
The Fermi level is a unique important parameter which often occurs a t several
stages in semiconductor physics. It can be calculated accurately if the energy band
structure of a heavily doped semiconductor is well understood. We have calculated
the Fermi level by solving numerically the neutrality condition in a doped semiconductor material as
n - p = N D-NA,
(1)
where n and p are electron and hole concentration, respectively, and can be defined
in terms of concentration-dependent density of states :
where ,o,(E) and ,o,,(B)are respectively the total electron and hole density of states
and f ( E ) is the Fermi-Dirac distribution function. N , and N , are the donor and
acceptor concentrations, respectively. I n Fig. l a we have plotted the Fermi level
dependence on impurity concentration in germanium, while Fig. l b shows these
results in silicon. From Fig. 1 we observe that even at high concentrations the Permi
level is in the band gap (unlike classical results) and is approaching the nondegenerate
impurity level.
Fermi Level, Minority Carrier Concentration, and Einstein Relation
(atoms/cm3)
-
mIC” lor8
223
-
u8-N~lotoms/cm3)
loTy
1 0 ~loz1
~
a
%-Nfllotom/cm3/
-
b
Fig. 1. Theoretical plots of the Fermi level EF as a function of the net donor and net acceptor
concentration considering position-dependent band structure at 300 K a) in germanium, b) in
silicon
4. Calculation of the Minority Carrier Concentration
The minority carrier concentration can be calculated easily and accurately if we
have a knowledge of the Fermi level position and then use either equation ( 2 ) or (3).
As discussed earlier, first the E , (Fernii level) has been evaluated for a given set of
impurity concentrations and then the carrier concentrations have been calculated.
Fig. 2a shows the plots of the hole (minority carrier) concentration versus net donor
or electron (majority carrier) concentration in germanium and silicon, while Fig. 2b
shows the plots of the electron (minority carrier) concentration versus net acceptor
or hole (majority carrier) concentration. Tor comparison we have also plotted the
classical results (p(or n) = nf/n ( o r p ) ) in Fig. 2, where ni is the intrinsic carrier
concentration and has been considered after Morin and Maita [ 5 ] . From Fig. 2 we
observe that our results based on the heavy doping approach are completely different
I lo?
I
electronconcenkotianfcni9
-
Fig. 2. a) Theoretical plots of the minority carrier (hole) concentration as a function of the net
donor or electron concentration taking into account heavy doping effects at 300 K in germanium
(-)
and silicon (- - - -). (a)N A = 1017, (b) 1Ol8, (c) 5 X 1Ol8, (d) lo1$atoms/cm3; (e) classical
results. b) Theoretical plots of the minority carrier (electron) concentration as a function of the net
acceptor or hole concentration taking into account heavy doping effects a t 300 K in germanium
() and silicon (- - - -). (a) Nn = lo1”,(b) lola, (0) 5 x lola, (d) loxsatoms/crn3; (e) classical
results
224
R.K. JAIN
from classical results, although a t low donor and acceptor concentrations they are
the same. At high impurity concentration we found a big increase in the minority
carrier concentration, while classically i t decreases. Classical results also do not find
any effect of the compensating impurity on the minority carrier concentration. It
must be remarked here that there is not much difference in the results considered in
Fig. 2 a and b.
5. Calculation of tho Effoctivo Intrinsic Concentration
The intrinsic concentration is also one of the important basic parameters. Classically
i t is only a function of temperature, but we have found its impurity conc-entration
dependence and therefore we call i t effective intrinsic concentration ni,. We have
calculated this quantity in a usual way: nie = (np)'l2.Fig. 3 a shows the plots of the
njeversus the net donor concentration in germanium and silicon, while Fig. 3b shows
these results, but as a function of the net acceptor concentration. Again we find that
there is not much difference in the results plotted in Fig. 3 a and b. For comparison,
in Fig. 3 we have also shown the classical Morin and Maita's [ 5 ] results for the in1 rinsic
concentration. We observe that the heavy doping effects predict a strong impurity
concentration dependence of tho intrinsic concentration a t higher doping levels.
For lower donor and acceptor concentrations our results are in quite good agreeriicnt
with the classicla1 results [5].
6. Calculation of the Einstein Relation
Einstein's [(i]
work on diffusion about make than seventy years ago led to a
fundamental relation between the diffusivity and mobility of charged carriers, which
is called Einstein relation. It has been found that Einstein's equation is of great
importance in semiconductor physics for the device analysis and desgin and can be
defined in a usua,l way as
D
kT
_ --P
t
1027
q
t
A
7OZ71
.I*
/7
/
II
/i
net phu.iphurus Concentration lafuinsJcm')
-
-
net boron concenfrofioii lofoms/cm3/
--_-
3. a,\ ThPnret,inn.l =.
rilnts
intrinnir
- __ of
- - the
.
.
.- effective
___ - - __ .
._
_
_.__
.- - cnnoent.m.t,inn
- ___ - - __ __ -_____ w. -. :'~a~s--R.- fnnntinn nf
-- t,h6+
"-- _net.
__"
phosphorus concentration at 300 K in germanium () and silicon (- - - -). (a) N A = lo1',
(b) 1018, (c) 5 x 10'8, (d) 101s atoms/cm3; (e) Morin and Maita 151. b) Theoretical plots of the
effective intrinsic concentration nie as a function of the net boron concentration at 300 K in germanium (-)
and silicon (----).
(a) N D = IOl7, (b) 10l8, (c) 5 X lo1",(d) lo1$ atoms/cms;
(e) Morin and Maita [5]
Fermi Level, Minority Carrier Concentration, and Einstein Relation
225
where D and p are the diffusion coefficient and mobility of the charged carrier,
respectively, b is the Boltzmann constant, T the absolute temperature, and q the
electron charge. Equation (4)is valid only for nondegenerate semiconductors. In the
last two decades enough modified relations for the Einstein relation have been
emerged for the case of degenerate semiconductors “7 to 161. Anyhow, the analysis
presented in these previous references is based on the assumption of the parabolic
density of states function, i-e., no heavy doping effects have been considered. I n their
recent work on the transport theory of heavily doped silicon van Overstraeten et al.
[4] have shown qualitatively that the ratio of the electron diffusivity t o mobility in
a degenerate semiconductor material can be defined as
+I Qe(E)f ( E )dE
De
- - _-_ 1
pe
--a3
[
(5)
.
( d / d ~ , ) --m
l m Q e ( E/) ( E )d ~ ]
Similarly a relation for the ratio of the hole diffusivity t o mobility (Dh/ph)can also
be written. The generalized Einstein relation given by (5) can also be written simply
in this way as
kT
De
-_-P.2
4
j.
-w
1
+w
1
1
+ exp ( ( E - E,)/~T)dE
( 6)
exp ( ( E - E l ? ) / k T )
dE
Q ~ ( E[I+
) exp ( ( E - E,)/ET)12
--m
Fig. 4 a shows the plots of the electron diffusivity t o mobility ratio as a function
of net donor concentration in germanium and silicon, evaluating equation (6) numerically. Similarly Fig. 4b shows the plots of the hole diffusivity t o mobility ratio as
a function of net acceptor concentration in germanium and silicon, From Fig. 4 we
observe that the hole diffusivity t o mobility ratio is more than the electron diffusivity
t o mobilitv ratio. We also find that the carrier diffusivitv t o mobilitv ratio is more
for the case of germanium as compared to silicon. All tLe results p<otted in Fig. 4
1
Fig. 4. a) Theoretical plots of the ratio of the electron diffusivity to mobility as a function of the
net donor concentration in (1) germanium and (2) silicon at 300 K based on heavy doping approach.
N A = 101’ atoms/cms. b) Theoreticalplots of the ratio of the hole diffusivity to mobility as a. function of the net acceptor concentration in (1) germanium and (2) silicon at 300 K based on heavy
doping approach. N D = 10’’ atoms/cm3
15 physica (a) 42/1
226
R. K. JAIN:
Fermi Level, Carrier Concentration, and Einstein Relation
approach the classical value of k T / q a t low impurity concentrations. The asymptotic
lines drawn on the results shown in Fig. 4 give a n approximate idea of the order of
the impurity concentration after which the classical Einstein relation starts failing.
It must also be remarked t h a t the impurity concentration dependence of the Einstein
relation predicted by Lindholm and Ayers [ 111, assuming parabolic density of states,
is more pronounced than our results plotted in Fig. 4, which is due t o the fact that
we have considered heavy doping effects.
7. Conclusions
I n the present work various important basic parameters of the scrniconductor
physics have been evaluated theoretically considering position-dependent band
structure in the two important elemental semiconductor materials. It is found that
our results are quite different from the available classical results. It is hoped that
the present concentration dependence of the different parameters niay lead to a better
explanation of the experimental results; this has been reported recently [ 17 to 191.
Future efforts using the approach described here are in progress and will be considered
in subsequent papers.
Acknowledgement
The author would like to thank Prof. Dr. R. van Overstraeten for inany stimulating
discussions and his encouragement.
Rcforences
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The Electronic Theory of Heavily Doped Semiconductors, Elsevici
Publ. Co., New York 1966.
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H. DE MAN,and R. MERTENS,
IEEE Trans. Electron Devices 20
290 (1973).
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and R. W. AYERS,
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and A. M. MAZZONE,
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and D. P. PARUI,
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and R. VAN OVERSTRAETEN,
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[19] R. K. JAIN
and R. VAN, OVERSTRAETEN,
IEEE Trans. Electron Devices 21, 155 (1974).
(Received April 7 , 1977)