DISPLACEMENT OF ELECTRON ENERGY LEVELSIN SEMICONDUCTORS
2031
Displacement of Electron Energy Levels in Semiconductors
as a Result of Interactions
by W. W.Harvey Ledgemont Laboratory, Kennecott Copper Corporation, Lexington, Massachusetts 091 73 (Received October 4, 1966)
Interactions among imperfections in semiconductors affect the occupancy of electron energy levels and lead to displacements of these levels relative to a k e d reference level. The conduction band edge and acceptor impurity levels are displaced downward and the valence band edge and donor impurity levels are displaced upward by attractive interactions. Consequently, the band gap and impurity ionization energies decrease with increasing concentration of imperfections; moreover, the rate of displacement of an ionized impurity level is considerably greater than for the neutral impurity level, so that their separation decreases rapidly. The general method for determining the contribution of interactions to the displacement of electron energy levels consists of writing a chemical equation for the occupation of the empty level by an electron or hole. This gives a relation between the electron electrochemical potential, or Fermi level, and the electrochemical potentials of the occupied and unoccupied levels. The relation is expanded in terms of concentrations and activity coefficients and compared with a corresponding expression obtained by application of Fermi-Dirac statistics to the occupancy of the given energy level. The dependence of the position of the level on the concentration-dependent activity coefficients of the chemical species constituting its occupied and unoccupied states is thereby derived.
Introduction While it is generally recognized that the intrinsic energy gap and impurity ionization energies in semiconductors are decreased by attractive interactions between mobile charge carriers and ionized impurities, the question of the directions and relative magnitudes of the shifts of band-edge energies and impurity levels does not seem to have been treated explicitly. For example, the width of the forbidden gap would be reduced by a downward displacement of the conduction band edge &c or an upward displacement of the valence band edge &V relative to a fixed reference level of energy. It could be of value to know not only whether &C and &v are shifted to higher or lower energies, but also the concomitant displacements of impurity levels. In a previous publication, the implicit concentration dependence of &c and &V was derived by comparison of thermodynamic and Fermi-Dirac statistical expressions for the electron and hole concentrations. The expressions obtained may be written
&c(T,2, 9) = fco(T) - e$
+ kT In Yn%/%'
(1)
Ev(T, 2, $1 = E v O ( T ) - e$ - kT In yP3tv/%O (2) where T denotes absolute temperature, J. the (macroscopic) electrostatic potential, and x represents concentration dependence. yn(T,x) and y P ( T ,x) are the chemical activity coefficients of electrons and holes and %(T, x) and %(T, s) are the effective densities of states in the conduction band and valence band, respectively. k o ( T ,J.)= t c o ( T ) - e$ and %"(T) are the limiting values of & and % at extremely low concentrations of carriers, impurities, and other imperfections and & v O ( T , $) = f v " ( T ) - e$ and %vo(T)are the corresponding limits of &V and XV. It is shown below that attractive interactions lead to a simultaneous downward displacement of EC and u p ward displacement of &v. Moreover, an extension of the treatment developed in the previous paper gives (1) W. W. Harvey, Phys. Chem. Solids, 23, 1545 (1962).
Volume 71, Number 7 June 1967
W. W. HARVEY
2032
precise expressions for the displacements of various impurity levels in terms of the concentration-dependent activity coefficients of the chemical species constituting the occupied and unoccupied levels. Briefly, the procedure consists of writing a chemical equation to represent the occupation of the empty level by an electron or hole. This gives an equilibrium relation between the electron electrochemical potential ( 55;( or Fermi level EF and the electrochemical potentials {j
pj
+ zje$ =
pjo
+ IcT In
cjyj
+
zje$
Interactions and Imperfections The electrons and lattice ions of a perfect crystal interact, but the effects of their interactions are fully accounted for by particulars of the band theory.2 The interactions of concern here are primarily those among imperfections, viz., electrons in normally empty bands, holes in normally filled bands, impurity atoms and ions, and lattice defects. These “chemical” interactions may, especially at high concentrations, modify the interactions between imperfections and the lattice and contribute, for example, to changes in the effective masses of the carrier^.^ The primary interactions, together with the modifications they induce, determine the magnitudes of the chemical activity coefficients of the imperfections; their effect on the effective densities of states is superposed upon that of an inherently nonquadratic form of the energy-wave vector relation &(k). I n addition to the chemical interactions, electrons and holes are subject to repulsive interactions, which may be called “quantum statistical.” These are due to quantum attributes, especially indistinguishability and the Pauli exclusion principle, not common to Boltzmann and Fermi-Dirac statistics. The quantum statistical interactions determine the values of the quantities f n and!, in the expressions’ (4)
and pf, =
3tV
exp[(Ev
- EF)/kTl
(5)
Three effects contributing to displacements of electron energy levels can thus be formally identified (Table I). The Journal of Phyeical Chemktry
“Chemical” (including electrostatic) interactions
Pertain to imperfections generally, determine y’s, contribute to 6 % ~ and s % ~ , affect occupancy and position of electron energy levels generally
Quantum statistical interactions
Pertain to electrons and holes, determine f,, and fP, affect occupancy but not position of &c and &v
Nonquadratic form of
Pertains to electrons and holes, contributes to 6% and a%v, affects occupancy and position of &C and EV
(3)
and hence concentrations cj and activity coefficients rjof the occupied and unoccupied levels. From a comparison of the equilibrium thermodynamic relation with a corresponding expression obtained by application of Fermi-Dirac statistics to the occupancy of the impurity level, the desired functional variation of its energy can be deduced.
nfn = % exp[(EF - Ec)/kTl
Table I : Effects Contributing to Displacements of Electron Energy Levels
Chemical Activity Coefficients Considering individual imperfections as particles of solute, each Ti approaches unity as the total concentration of imperfections becomes very small. At finite concentrations the change in free energy AFT,vfor a chemical reaction involving imperfections deviates from the value it would have in the absence of chemical interactions by an amount 6AF = kTZvi In Ti
(6)
where the v’s are the coefficients, negative for reactants, in the chemical equation. The deviation of AF from its ideal value may also be written 6AF =
‘/22viwi
(7)
where wi is the energy of the resultant interaction between an imperfection of the ith kind and the aasemblage of imperfections. The factor 1/2 prevents counting each interaction twice. For imperfections with a net charge zie, the interaction energies may be written wi
= 2ielj.i’
(8)
where $if is the potential of the resultant field of all of the imperfections but one, which acts on the one. For free carriers and ionized shallow impurities in germanium4 and silicon16#if can be approximated by the Debye-Hiickel screened coulombic potential, -ZieK/ c(1 K U ) . ~In the following development, $i’ is as-
+
( 2 ) L. Pinoherle. Proc. Intern. School Phys. “Enrico F m i “ (Varenna, Italy), 22, 3 (1963). (3) V. L. Bonoh-Bruevich and A. G. Miromv, Somet Phy8. Solid State, 3, 2194 (1962). (4) W. W. Harvey, Phys. Rev., 123, 1666 (1961). ( 5 ) w. w. Harvey, Proc. Intern. conf. Semicond., Paris, 196.4, 839 (1964).
2033
DISPLACEMENT OF ELECTRON ENERGY LEVELSIN SEMICONDUCTORS
sumed to be opposite in sign (net attraction) and approximately proportional to the charge zie on the imperfection. The symbol $' without subscript denotes the magnitude of an average interaction potential for singly charged imperfections.
t
Po
Displacement of the Band Edges I n some earlier literature, the energy of exciting an electron from the level EV to the level & (ie., the Gv) was conwidth of the forbidden gap AGG = &C sidered to differ from its value AEG' in the absence of interactions by an amount et,bn', defined as above. Equations 1, 2, and 6-8 reveal that an approximation is involved. Thus, if the concentration dependence of 3 t ~and %V can be neglected, consistent with the eff ective-density-of-states formulation' leading to eq 4 and 5, the band-edge energies are displaced according to
-
GC =
&O
EV = &v'
+ '/z(-&+') = &' - '/&' - ' / z ( + e g p ' ) = Ev'
+ l/ze$'
(9)
Displacement of Impurity Levels For the ionization of a donor impurity =
D+
+ e-;
l~= {Dt
+P
(11)
the electron electrochemical potential may be formulated as
- lD+ = PDo - PD+' + kT In (CD/CD+)(YD/YD+) - e$ (12) with kT In rD/^/Dt = '/Z(% - e h t ' ) . The activity
(=
R-
------Interaction Energy
('4
(a)
Figure 1. (a) Displacement of band-edge energies; (b) variation of electron energy levels with interaction energy.
cumstance, the concentration of occupied donor levels will be given by the usual expression
(10)
The variation in the width of the gap is determined only in part by the interaction of the electron with its "atmosphere" of imperfections, the remainder being determined by the interaction of the hole. In general, the displacement of an electron energy level involves both the unoccupied and occupied conditions. The important point to make now is that attractive interactions result in a simultaneous downward displacement of & and upward displscement of &v, relative to a fixed reference level (Figure la).
D
fv
+
('/ZgD>
exp[(ED - &)/kTIf (13)
where ND = CD C D + is the donor concentration. The preexponential factor l/z makes allowance for the occupation of an empty donor level by an electron with either sign of spins and gD is the statistical degeneracy of the ground-state donor leveLQ Equation 13 can be rearranged to EF = ED
- IcT In 2gD + IcT In CD/CD+
(14)
where (ND/cD)- 1 has been replaced by CD+/CD. Note that use of Boltzmann statistics not only introduces no simplification, but actually leads to an incorrect result, giving CD/ND rather than CD/CD+ in the logarithm. By equating { and EF, which are identicallo if measured from the same reference level, the following functional dependence of the donor energy level is obtained
&D(T,x, $1
+ IcT In
=
b o ( T ) - e$
=
ED"(T, J.) 4- '/Z(WD- eJa+') =
{D
coefficient of the neutral donor atom is diflFicult to evaluate, but if the interactions are predominantly electrostatic, YD is probably close to unity up to reasonable concentrations of neutral donors CD. Hence, kT In T D / r D t will be approximately equal to +'/*e$'. Randomly distributed impurities will introduce impurity levels covering a range of energies. If the range is sufficiently narrow, as might be expected in the absence of association and clustering, the occupancy will approximate that of a single, ND-fold degenerate level situated a t the mean level of energy ED. I n this cir-
+
CD = N D / ( ~
ED'
YD/YD+
+ l/ze$'
(15) (154
Equation 15 follows from the consideration that the standard chemical potentials PD' and PD+' are functions of temperature only (eliminating pressure as a possible variable). The donor level is thus displaced P. Debye and E. Huckel, Physik. Z.,24,185 (1923). (7) R. A. Smith, "Semiconductors," Cambridge University Press,
(6)
Cambridge, 1959,p 77. (8) See ref 7, p 88. (9) T. H.Geballe, "Semiconductors," N. B. Hannay, Ed., Reinhold Publishing Corp., New York, N.Y.,1959,pp 338,340. (IO) N.B. Hannay, ref 9,p 26.
Volume 71, Number 7 June 1887
W. W. HARVEY
2034
Table II: Displacement of Energy Levels Relative to a Fixed Reference Level Displacement Energy level
Interactions
ED .t
-
EA EV
-kT In (
+ e+;
{A
=
{A-
-{p
=
/.LA-'
- PAo kT In
+ tp
(16)
+ (CA-/CA)(YA-/YA)
- e$
(17)
The concentration of singly ionized acceptor levels is given by an expression of the same form as eq 15 except that the spin factor becomes 2 rather than '/2 CA-
=
N A / ( -k ~ ( 2 / 9 ~ exp[(EA ) - & F ) / ~ T ] }(18)
or EF = E A - LT In (gA/2)
+ kT In
CA-/CA
(19)
Thus, the energy of an acceptor level varies as
EA(T,5, $4 = ~ A O ( T ) =
EAO(T,
+
- e# kT In YA-/YA $1 - '/a(e$A-' + WA) c EA' - l/ze#'
(20) (2Oa)
corresponding to a downward shift of the acceptor level. Displacements of impurity levels and band-edge energies with increasing energy of interaction are shown schematically in Figure l b and catalogued in Table 11. All of these levels vary as -e$ and may also exhibit an inherent variation with temperature. It is interesting that donor and acceptor levels move apart, their separation being governed by GD
- EA = LT In (4gD/gA)
+
kT In
(CD+/CD)(CA-/CA)
(21)
Extension of this treatment to multiple ionization of impurities will merely be indicated. Successive stages of ionization should be formulated separately as, for example, selenium in germanium, Se = Se+
+
The Journal of Physical Chembtry
'/Z(
-'/&'
- e!bD+')
'/Z(e$'D+'
- 2e$DZt)
-%$'A*-'
+ +
e$'A-')
WA)
- '/ze+p'
-kT In Xv/%v")
so that =
'/%(WD
yp
upward by attractive interactions involving the positively charged donor ion. A similar treatment applies to acceptor impurities, for which A = A-
Approx
- '/%e$'*'
(+kT In Xc/Xc") kT In Y D / y D + kT In YD+/YD" kT In y A a - / y A kT In Y A - / Y A
D, D + D+, DZ+ A-, A2A, Ae+
ED
Physical
kT In yn
e-
EC
EA
Thermodynamic
+'/ze$' +a/ze$'' 3/ze$'
-
-'/ne+' +'/ze$'
+
e- and Se+ = Se2+ e-, consistent with a separation of the levels for the neutral and the singly ionized impurity. However, the displacement of the level of the singly ionized impurity as a result of interactions will be greater than for the level of the neutral impurity, owing to the greater charge of the unoccupied level. Where the interaction potentials are of the Debye-Huckel screened variety, the factor is 3 (Table 11). Consequently, the separation of neutral and ionized impurity levels should decrease rapidly with concentration (Figure lb).
Discussion The treatment given above may not apply quantitatively to certain problems of technological importance, notably semiconductors at temperatures giving high intrinsic concentrations of carriers (i.e., low-gap materials at ordinary temperatures and higher gap materials at elevated temperatures") and heavily doped semiconductors, such as are employed in lasers and tunnel diodes. The basic assumptions become untenable for very high-impurity contents; for example, the treatment ignores the spreading of the impurity ground-state energies and the tailing of the band edges, the existence of excited states, limitations in the applicability of Fermi-Dirac statistics, and nonparabolicity of the energy bands. A random spacial distribution would result in a spread of impurity states giving, in effect, an impurity band. The proper density-of-states function would then have to be formulated in order to calculate occupancies. The neglect of excited impurity states lying between the ground state and the conduction and valence band edges could be serious; the band edges themselves undoubtedly become smeared at high doping levels. Fermi-Dirac statistics should still apply, provided that the electron energy levels in the distribution undergo identical displacements as a result of the interactions. (11) F. J. Morin and J. P. Maita, Phys. Rev., 94, 1525 (1954).
DISPLACEMENT OF ELECTRON ENERGY LEVELSIN SEMICONDUCTORS
On the other hand, for distinctly nonparabolic energy bands, the quantum statistical and the chemical contributions to the activity coefficients are not separable as is implied in eq 4 and 5. In view of the indicated deficiencies of the treatment, only qualitative significance can be attached to the extrapolation of calculated energy level displacements to points of crossover, as in Figure lb. (In the case of electrically active impurities, crossover corresponds to complete ionization; in the case of free carriers, to transition from semiconducting to metallic behavior.) However, solid-state physicists are making great strides in formulating and estimating the effects mentioned, and the treatment provides a framework for transposing their results into a thermodynamic formulation. The principal conclusion drawn from this work should be broadly valid: attractive interactions displace the relevant electron energy levels in the directions indicated, leading to an increase in the degree of ionization of impurities and, eventually, to overlapping of impurity levels and band edges. The approach to metallic behavior is a consequence of a net attractive interaction experienced by the free carriers. While statistical degeneracy may set in at high concentrations of imperfections, it is not a causative factor. On the contrary, the exclusion principle operates in opposition to attractive interactions among the carriers and other imperfections.
Appendix Proof That the Displacements of &C and EV Are O p posed. It can be shown that of three possible sets of displacements which diminish the intrinsic gap by
2035
approximately e$’, only that can occur whereby Ec is decreased by ye$’ and Ev is increased by (1 - y)e$’, Thus, if the edge of the conduction band with y = is shifted to the higher energy &vo CY and the edge of the valence band is shifted simultaneously to the higher energy Evo (CY e$’), the sum &C &V will be increased by 2 a e$’. If the shifts are to the lower energies k 0 - (8 e$’) and &v0 - 8, the sum will be decreased by 28 e$’. However, comparison of the expression
+
+
+ + + + +
~ E = F Ec
+ Ev + k T In ( n / P > C f n / f p ) k T In %/%V
(22)
obtained by combining eq 4 and 5, with its thermodynamic equivalent
2t =
P’nO
+ P,O - 2e$ + k T + In (n/p>Vn/fP>+ k T In Y J Y P
(23)
+
reveals that &C &V varies as k T In yn/yR. Equation 23 is obtained by applying the equilibrium condition r,, = -tR to eq 3 for electrons and holes. For predominantly electrostatic interactions, the activity coefficients of singly charged species would be comparable in magnitude, so that the ratio y n / y Rwill be close to unity. (Actually, as shown in ref 12, y R includes the activity of the semiconductor; this latter will be given closely enough by the mole fraction for most cases of interest.) Accordingly, the permissible change of &C EV is not 2 a e$’ or -28 - e$’, but 1/2(-e$n’) I/Ze$,’ = 0.
+
+
(12) W. W. Harvey, Phys. Chem. solids, 24,701 (1963).
Volume 71, Number 7 June 1.967
