840
HUBERT M. JAMES AND ARTHURS. GINZBARQ
Vol. S i
BAND STRUCTURE I N DISORDERED ALLOYS AND IMPURITY SEMICONDUCTORS1” BYHUBERT M. JAMES AND ARTHURS. GINZBARG’~ Department of Physics, Purdue University, W. Lafayette, I d . Received March 2.9, 1868
The band structure of a one-dimensional model of a disordered alloy, or of a crystal with randomly placed impurities, is treated by a. method that takes full account of the random element of the problem, and does not involve the assumption that the deviation from periodicity is small. The method is based on the idea of counting the number of nodes in a solution of the wave equation with energy E, as a means of determining how many states of the system have energies lower than E. It requires solution of the wave equation only within a single cell of each type considered; thereafter, the calculation can be reduced to an algebraic form well adapted to machine calculation, or to a graphical form suited for the discussion of general properties of these systems. An illustrative calculation on a linear model shows how the sharply defihed “impurity band” of a crystal with regularly placed impurities is replaced by a broadened region of high energy-level density when the impurities are randomly arranged.
Introduction In discussing the conductivity of impurity semiconductors, it is commonly assumed that substitutional impurities introdwe localized acceptor or donator levels in the forbidden bands, while leaving unaffected the distribution of energy levels in the valence and conduction bands. In this approximation, the impurities affect the conductivity by changing the number of mobile carriers and by contributing to the scattering of these carriers, the latter effect being treated by perturbation methods. Similarly, many discussions of the properties of substitutional solid solutions, such as the alloys of the transition metals, are based on the idea that the distribution of energy levels is eqsentially fixed by the lattice structure, and that the composition of the alloy is of importance mainly in that it determines the number of electrons in the several energy bands.2 The change in conductivity due to scattering arising from the random arrangement of different types of atoms in a regular lattice has also been treated by N ~ r d h e i m again ,~ by perturbation methods. For some purposes, it is necessary to have more detailed information about the distribution of energy levels, or “band structure,” in alloys and impwe semiconductors. If these systems are treated as perfectly ordered, well known methods can be applied to the problem. Perfectly ordered alloys exhibit a Brillouin zone structure characteristic of their periodicity, and strictly forbidden energy bands-ranges of energy in which the density of energy levels is zero. Similarly, if one could modify a pure metal crystal by introducing a superlattice of substitutional impurities, the Brillouin zones of the original crystal would be subdivided in a way determined by the superstructure, and new forbidden bands might occur. The same effect can be produced by a superlattice structure due to periodic displacements of the original atoms. In the case of a linear array of identical atoms, replacement of every nth atom by an atom of another type would split each band of permitted energies
into n sub-bands, separated by more or less narrow forbidden bands. Linear models of this type have been discussed by Sawada,5 Saxon and HutnerlB Ginzbarg’ and Hoffman.* I n particular, Ginzbarg has shown that, in cases where a single impurity atom wouId give rise to a localized impurity state near a band edge, replacement of every nth atom by such an impurity would give rise to a sub-band relatively well separated from the others, and having its wave functions particularly large in the neighborhood of the impurity atoms. Such a band may be called an “impwity band.” Erginsoyg has also discussed a three-dimensional model of an impure crystal (tacitly assuming a regular arrangement of the impurities), with special attention to sub-bands associated with excited states of the impurities. The details of band structure derived for periodic systems will have little relevance to the cases of disordered alloys or semiconductors with randomly arranged impurities, where one must expect the energy sub-bands that would be produced by superlattice structure to be absent, or a t least strongly modified. Detailed discussions of such systems are almost completely lacking, and appear to be limited to the replacement of the disordered system by a supposedly equivalent ordered system. Mutolo has started from an electronic potential energy of the type assumed by Nordheim for a disordered alloy
(1) (a) Based, in part, on a thesis presented inpartial fulfillment of the requirements for the Ph.D. degree at Purdue University b y A. 8. Ginzbarg, June, 1949. (b) Shell Oil Co., Houston, Texas. (2) See, for instance, J. C. Slater, J . Applied Phys., 8 , 885 (1937). (3) L. W. Nordheim, Ann. Phyaik, [5]9,607,641 (1931). (4) (a) J. C. Slater. Phys. Rev., 84,179 (1951); (b) E. Katz, ibid.. 86, 495 (1952).
(6) D. S. Saxon and R. A. Hutner, Phillips Research Rep., 4, 81 (1949). (7) A. 8. Ginebarg, Ph.D. Thesis, Purdue University (1949). (8) T.A. Hoffman, Acta Physica. Acad. Sci. Hunua., 1, 175 (1951). (9) C. Erginsoy, Phye. Rev., 88, 893 (1952). (10) T. Muto, Sci. Papers Inst. Phya. Chem. Research ( T o k y o ) , SO, 99 (1936);ibid., 84,377 (1938).
1.
*
B(r)
1
+ +
+
Vn
= n
(T
- T”)
(1)
where Vn is the potential energy due to the ion actually occupying the nth lattice position; he has then converted it into the periodic potential energy of a “virtual crystal’’ by replacing each Vn by a
where Va) is the potential energy due to an ion of type a and p n a is the fraction of sites equivalent t o the nth that are occupied by a-ions. Muto takes ( 5 ) S. Sawada, Proc. Phys. Math. Sac. Japan, 24, 165 (1942).
I
Nov., 1953
BANDSTRUCTURE IN DISORDERED ALLOYSAND IMPURITY SEMICONDUCTORS
the virtual crystal as the unperturbed system, and treats the disorder as a perturbation. He notes that the average of the perturbation potential over the Bloch functions of the virtual system is zero, and concludes that the second-order perturbation will push a few levels away from the band, while leaving the band structure generally unchanged. Since Muto’s argument neglects the step of computing the correct zero-order functions and firstorder enwgy perturbation of a degenerate problem, and is nowhere quantitative, it does not seem to provide a satisfactory justification for use of the virtual crystal model. I n the case of impure semiconductors, especially, it is quite evident that the virtual crystal model is completely unsatisfactory. I n this paper we wish to treat the band structure of a one-dimensional model of a disordered alloy, or of a crystal with randomly placed impurities, by a method that takes full account of the random element of the problem and does not involve the assumption that the deviation from periodicity is small. The method reduces the problem to terms well suited for general graphical discussion and for machine calculations on special models. It is not evident how the method can be extended to three dimensions, but we hope that the understanding it gives of the one-dimensional problem will be helpful in practical cases.
84 1
and the method of treating it, to the case of more than two types of cell, and cells of unequal lengths. 3. Node Counting Since our method of treating this problem is based essentially on counting the zeros or “nodes” in a real solution of the wave equation
- -2ham-d2+ + V(Z)+(Z) = E$@) dx2
(8)
we shall need to take note of certain theorems concerning such nodes. It is a familiar fact that the pth stationary state wave function for any one-dimensional problem with discrete energy levels, such as the harmonic oscillator or the M-celled crystals considered above, has exactly (p - 1) nodes inside the (finite or infinite) domain of x. Let I)~(X) and 4b2(x) be any two real solutions of eq. 8 for the same energy E, not necessarily satisfying the boundary conditions of the problem. From the constancy of the Wronskian Wh,J.a)= +I(Z)tl.a’(Z) - $JXZ)+Z(S) (9) it follows that the zeros of $1 alternate with the zeros of $2, unless these solutions differ only by a constant multiplier. It is thus clear that the number of nodes of $1 in any given range of x can differ from the number of nodes of $2 by a t most 1. From this result it follows that the quantity 2. TheModel N ( E ) defined in eq. (6) is the limit, as M + a, We shall consider a particle moving in one di- of the average number of nodes per cell in any real mension with potential energy V ( x ) . We divide solution $(E; x ) of the wave equation for energy the range of x into cells of length a, such that E. As a practical matter, one can determine N ( E ) with arbitrary accuracy by counting the number of for the nth cell nodes of such a solution in a sufficiently large, but na 5 x < ( n + l ) a , n = 0, fl, f 2 , .. . (3) finite, number of cells. We assume that V ( x ) in the nth cell may have one of two alternative forms: it may be an A-cell, with 4. Reduction of the Node Counting Process to Algebraic Form v(Z) = v A ( Z - nu) (4) The process of counting the nodes in a solution of or a B cell with the wave equation will now be reduced to algebraic form. For this purpose, use will be made of the V ( S ) = VB(Z - na) (5) familiar process of constructing a complete solution We assume, moreover, that these two types of cells of the wave equation by joining together solutions occur in random order in the crystal, with relative’ valid in the component cells, with due regard for the probabilities p and 1 - p. continuity of # and its first derivative.” If the “crystal” is a finite one, with M cells enLet ~ A I ( Ex) , and y ~ z ( Ex,) be solutions of the closed between infinite potential jumps, the sta- wave equation for an A-type potential in the zeroth tionary-state energies of the system will be dis- cell crete. A.s M increases, their density-in-energy will increase proportionally. In the case of a periodic - 2-ham_d2?JAi _ dx2 VA(x)?./Ai(E,x) = E!/Ai(E,Z) (10) potential, the number of levels in a band will be M , which satisfy the following conditions a t the left and edge of the cell 1 number of energy levels N ( E ) iVl*Lim % / w i t h energy less than E (6) ?./AI(E,O) = 1 !/h(E,O) 0 (11) !/AZ(E,O) = 0 ?./.h(E,O)= 1 (12) will be the number of bands, plus a possible fractional part of a band, with energy less than E. We give names to the magnitudes and slopes of We shall take N ( E ) , and the corresponding band- these solutions a t the right edge of the cell, as follows density-in-energy
+
~
1
(7)
as characterizing also the band structure of the disordered crystal models. These quantities will depend only onthe forms of VA and VB, and the value of p . It is a trivial matter to generalize this model,
yAl(E,a) = d ( E ) y ~ i ( E , a )= &(E)
?./AdE,a) = &(E) yAz(E,a) & ( E )
1
(13)
Since the Wronskian of YAI and Y A ~is the same at z = 0 and x = a,one must have the following relation between the c’s (11) See for instance, H. M. James, Phys. Rev., 76, 1602 (1949), where the method is discussed in detail.
HUBERT M. JAMES AND ARTHUR S. GINZBARG
842
Cf1(E)&(E)- & ( E ) C(E) ~ E1
Vol. 57
given range of x can be reduced to counting the number of sign changes in the corresponding seWe similarly define solutions VB~(E, x) and.yBz!E, x) of the wave equation for a B-type potential in the quence of a’s, in the following way. It follows from our discussion of node counting Equations zeroth cell, and quantities &E). that there are only two possible values for the num(10) to (14) are then valid, with A replaced by B. ber of nodes of rC/(E,x)in any A-type cell. To deWe now consider a pure A-type crystal. We termine these numbers we consider an A cell, taken , - nu) and Y A ~ ( Ex ,- nu), two inhave, in ~ A ’ ( Ex. dependent solutions of the wave equation in the to be the zeroth cell, and restrict attention to a nth cell, and can express an arbitrary solution in value of E for which c,“, = ~ A ~ ( E ,isu )not zero. x ) p zeros inside the zeroth cell, that cell as a linear combination of these two special Let y ~ ~ ( E , have ones. If $(E, x) is an arbitrary solution of the counting the one a t x = 0. Possible solutions of the wave equation include yAZ(E,x) + EYA@,z), with crystal problem, one can then write ( p - 1) zeros in the zeroth cell, and y ~ z ( E , x )$(E,X) a n y A l ( E , X - na) bnYAz(E, X - nu) (15) EYA~(E,z) with p zeros in the zeroth cell, E being forna 5 2 < (n + l ) a , n = 0, fl,1 2 , ... sufficiently small. For very low E , p = 1; as E If one fixes two of the coqstants in this solution, say increases, p increases by 1 whenever &(E) = 0. uo and bo, one can determine the other constants, We therefore divide the domain of E into ranges and the complete form of the solution, by the condi- according to the value of p for an A cell: if the tion that $(E, x) be continuow and have a contin- pth zero of &(E), counted for E increasing, occurs uous first derivative at all cell boundaries, x = when E = E t , then the pth range of E lies between nu, n = 0, & 1,.... The conditions of continuity at Eiyl and E t . We can then state: If the energy x = (n 1)u can be expressed in matrix form as E lies in the pth range thus defined, $(E,x) can have only (p - 1) or p zeros in any A cell. The pth permitted energy band of a pure A crys(The dependence of the c’s on E is not made explicit tal lies inside the pth energy range thus defined, here, but should be remembered.) It follows that since for any energy in this band the average number of nodes per cell lies between ( p - 1) and p . If VA is symmetrical about the center of the cell, Et can be shown to coincide with the upper edge of a relation valid, since # 0, for negative as the pth permitted band or the lower edge of the well as positive n. If &(E) &(E) 2, there ( p 1)th permitted band; in other cases it may be exist solutions $(E, x) that change from cell to cell inside the forbidden band. by a multiplying factor eoi@); these are the Bloch To determine whether +(E,x) has (p - 1) or p solutions of the problem, and the energy E is in a zeros in a given A cell, say the nth, one has only to permitted band. If c t ( E ) &(E) > 2, on the note whether a n = +(E, nu) and a n + l = $(E, 1)u) have the same sign (even number of other hand, it can be shown that every solution (n increases exponentially either as x --t + a or as nodes) or opposite sign (odd number of nodes). x + - ; the energy is in the forbidden band. Let sn be +1 or - 1, according as a, and Un+1 have The band structure of the pure A crystal, including the same or opposite sign. Then the number of the form of p(E), is completely determined by the nodes of $(E,x) in the nth cell can be written as quantities (E). Next we consider a crystal containing both A and (21) B cells. Let $(E,$) be an arbitrary solution of the crystal problem. If the nth cell is an A cell, one In the same way, the domain of E can be divided can express $(E,x) inside this cell as a linear com- into ranges according to the value of p for a B cell, bination of VA~(E, x - nu) and g~z(E,.z- na); if separated by energies E,”for which c,”,(E) = 0. the nth cell is a B cell, one can express it as a linear We now consider rC/(E,x) for a miqed crystal, for combination of ~ B I ( Ex, - nu) and y~z(E,x - nu). an energy in the pAth range as defined for A cells, We write and in the pBth range as defined for B cells. Let &yI(n)),l ( E , x - nu) bnyI(n)lz ( E , x - nu) the sequence of numbers a, be constructed for a re+(E,X ) gion in the crystal containing N A A cells and NB (18) nu 5 2 < ( n l ) a , n = 0, *l, f2, . . . B cells, using eq. (19). If s n = 1 for N 6 A cells where I ( n ) is A or B according as the nth cell is an and for Ni3 B cells, the total number v of nodes in A cell or a B cell. Similarly, we can write the con- $(E,$) is easily seen to be tinuity conditions as v = NA(QAf NB(PB Ice\
P
(14)
+
+
IC$
+
0
(20)
(The generalization to n 0 and $(E,x) is concave toward the axis; it may increase or decrease in regions where E V ( x )is negative. Whatever the choice of K > 0, the value of q5 is fixed for certain values of 2: it is (m - 1/2)7r a t the mth zero r(.(E,x)for x > 0, and ma at the succeeding zero of \L’(E,x). The relation of +(E,x) to the problem of node counting is then obvious: if the phase of $(E,x) increases by Aq5 in a given range of x, the number of zeros of, rl/ in this range is A q 5 / ~ , to within 1. The advance in phase of $(E,x) can be computed in stepwise fashion as the sum of phase advances in successive cells. The initial phase 4i in the nth cell [z = na] and the final phase q5f [z = (n l)a] are given by
+
-
tan 4i = &/an tan 4f = ~b~+~/a,+l
(25)
The continuity condition, eq. (19), then yields the relation
0TI
I I 1
en
Fig. 1.-Schematic representation of &(4i)for a sequence of energies, E,, > &-I.
lower edge, inside, and at the upper edge of the first permitted band, respectively; E5 is in the second forbidden band; and Ee and ETare inside the second and third permitted bands, respectively. If the cell potential is symmetric and E p is an energy in a permitted band of a pure I crystal, &(Ep) and &(Ep) will have opposite sign. [See ref. 11, eqs. (2-17) and (2-36), and the condition for permitted bands.] It will then be possible to write tan 6 =
- KC~’I(E,)/C%E,) = C ! ~ ( E ~ ) / K C Z T ~(29) (E~)
by choosing If the cell potential is symmetrical about the center of the cell, one has c:1 ( E ) =
CZTZ
K(Ep)
(E)
[ref. 11, eq. (2-17)], and can write
= [-
d z ( E p ) / C ~ ’ l ( ~ ~ ) l ’ / ~(30)
Then eq. (27) becomes, for this energy only tan
where
+t
= tan(& 4- 6)
(31
844
HUBERT M. JAMES AND ARTHUR S. GINZBARG
VOl. 57
cording to eq. (30), the curve h(4i) becomes a straight line, and the phase advance in every cell is 6. It follows that N(E3) = 8 / n
(35)
which fixes the position of E3 inside the first permitted energy band. Next we consider an energy such as E6 in Fig. 1, for which &(&) oscillates back and forth across the line & = +i T. The step construction indicates that the phase of a solution advances continuously, by less than T in some cells, by more than n in others. Now & 4i is periodic in 4i with period T. It follows that if one lowers the curve &(q$) by n and repeats the step construction, one will obtain a t each step a phase advance less than the actual one by exactly n. This lowering of the curve makes the situation exactly like that shown in Fig. 2. It follows that the phase advance in M cells, in excess of Mn,is less than n, and that N(E6) = 1. If the energy is slightly raised, one has again the same situation; thus N ( E 6 e) = 0, and p(E6).= 0. We see, then, that Es, or any energy for which +E($i) oscillates across the line 4f= 4i T,is in the second forbidden energy region. In the same way, it can be shown that if &(+i) oscillates across the line & = & nn, the corresponding energy is in the (n l)at forbidden band. On the other hand, if &(+i) lies in the region n?r, the between & = q5i (n - 1)nand $f = +i corresponding energy is in the nth permitted band. N ( E ) can then be determined by eqs. (30), (32), (33) and (35). 6. Phase Advance in Mixed Crystals We now consider the phase advance in the mixed crystal model. Figure 4 shows an alternative construction to that previously illustrated, which employs a plot of & - & against &. Separate curves give the phase advance function for A cells and B cells; the energy considered in Fig. 4 clearly lies in
+
Fig. 2.-Construction for phase advance in a pure crystal; energy in lowest forbidden band.
One can write, more simply +f
=
4i
+8
(33 1
if 6 is the value of the arc tangent between (n - l ) n and n a when E , lies in the nth permitted band. For an energy in a forbidden band, no choice of K can lead to a linear relation between df and +i. The use of the phase advance curves &(&) will now be illustrated by consideration of a perfectly periodic potential, say a pure A crystal, for which there is the same functional relation between the initial and final phases in every cell. Figure 2 illustrates a graphical construction for 4i and & in a sequence of A cells, for a low energy E , such that the curve &(&) oscillates back and forth across the straight line & = +i, The construction follows in an obvious way from the fact that & for one cell is & for the next. It is clear that +f and $i approach a common limit where the curves &(&) and +f = 4i intersect. The total phase advance is less than T, no matter how long the sequence of cells; thus N ( E J = 0. If E is raised slightly, the curve &(&) will be raised slightly, but will still intersect & = +i; N ( E ) will vanish also for this higher energy. We thus see that p(EJ = 0: El lies in a forbidden energy band. The band in question is the lowest forbidden band, since [as follows from eq. (28)] the same situation exists for all lower values of E . As E rises, the curve ~$f(+i) rises monotonically, and the upper edge of this forbidden band, the lower edge of the first permitted band, is reached when &(&) ceases to intersect r#q = &,becoming tangent to it (Ezin Fig. 1). Figure 3 illustrates the same construction for an energy EI such that &(+i) intersects neither 4r = 4i nor +f = ~ $ i T. The phase of the solution advances steadily from cell to cell. If K is chosen ac-
-
+
+
+ +
+
+
$f - @ i
+
Of
,
1
O
Y 011
I
Qi
Qie
Fig. 3.-Construction for phase advance in a pure crystal; energy in lowest permitted band.
I
Fig. 4.-Construction
for phase advance in a mixed crystal.
the first permitted band for both pure A crystals and pure B crystals. Figure 4 shows the phase advance in an A cell followed by a B cell and two more A cells. It is evident that the phase will advance steadily in any sequence of A and B cells, but this does not imply that p(E) # 0 for the energy considered: to determine p(E) one must compare the phase advance in a long sequence of cells for adjacent values of E . It is quite possible, for instance, that in this case one would have p(E) = 0 for some regular sequence of A and B cells, but p(E) # 0 for a random sequence. Figure 5 illuitrates the more interesting case of an energy in the lowest forbidden band of a pure A
4
*
Nov., 1953
BANDSTRUCTURE IN DISORDERED ALLOYSAND IMPURITY SEMICONDUCTORS 845
If E = El, an isolated B cell cannot advance the phase beyond d,d, but two consecutive B cells, or a sufficiently closely spaced triple of B cells, can do this, as ia shown in Fig. 5(b). After the phase has increased beyond $Id, it will be carried nearly to cpc T by a sequence of A cells, for a net phase advance of T . It is thus clear that the phase of a solution of the random mixed crystal problem, for E = El, will advance continuously, a t a rate deter; mined by the probability of occurrence of pairs of adjacent B atoms and other short regions of relatively high B density; it is clear, moreover, that the rate of advance will depend on the relative values of &J - & for an A cell and of & - 4i for a B cell-that is, it will depend on E. We conclude, then, that neither N ( E J nor p ( E J is zero: El does not'lie in a strictly forbidden energy band of the disordered miTed crystal. The argupent also makes it clear that N(E1) will increase rapidly as n decreases. If E = E%,on the other hand, the phase advance of a solution will be reduced below the value of ?r per B cell by the occurrence of adjacent pairs or other closely spaced groups of B cells, essentially because maximum use is then not Qc $d made of the possibility of phase advance in A Fig. B.-Phase advance in mixed crystals, for energies near cells (see Fig. 5(a)). Again it is clear that the the energy Ei of an impurity level. actual phase advance depends on E , and increases as n decreases. We thus find N(E2) < l/n, crystal, but in the first permitted band of a pure B p(Ez) # O:Ez is not in a forbidden band for the crystal. The lower sketch corresponds to a value random mixed crystal, but fewer energy levels lie of E below the energy Ei of the localized impurity below it than would in a crystal with the same numstate produced by a single B cell in a matrix of A ber of impurities regularry spaced. cells, while the upper sketch corresponds to E2 > We shall show elsewhere how this graphical Ei. I n each case the value of K has been so chosen method of argument can make it clear that treatthat & - 4i = 6 is constant for B cells. I n both ments of isolated impurities or mixed crystals using cases the phase in a pure A crystal will converge on the Kronig-Penney model with &function wellse &, from the right or left, from any value in the or barriers can lead to quite atypical results. In period +d - IF < $i < 4 d . particular, we shall show that the hypothesis of We consider first an A crystal in which every Saxon and Hutner6-that energies forbidden in both nth A cell has been replaced by a B cell. If E = pure A and pure B crystals will be forbidden El, and n is large enough, the phase after any se; energies in any arrangement of A and B atoms in a quence of (n - 1) A cells will be so close to & substitutional solid solution-is not generally that the phase advance in a B cell will not carry cp valid, though it has been proved by Luttinger12 beyond The next sequence of (n - 1) A for this special Kronig-Penney model. Derivation cells will carry the phase back nearly to &,and so of Luttinger's result also becomes a trivial matter on; no matter how long the regular sequence of when this graphical method of argument is emcells, the phase will never advance beyond 4d. ployed. The phase advance per cell is zero, and will remain so even when E is slightly raised. We thus see that 7. Calculation of Phase Advance in a Long Sequence of Cells N ( E J = p(EJ = 0; E is in a forbidden energy Although it is easy to follow the phase advance band, with no energy levels below it. 1.fE = Ez, and n is large enough, the phase after any sequence in a few cells of a mixed crystal by the graphical of (n - 1) A cells will be so close to & that the ad- method of Section 6, this method becomes imvance in a single B cell will surely carry d, beyond practicably tedious when one needs to determine 4 d . The next sequence of (n 1)A cells Will carry the phase advance in a thousand or more cells, in order to get reasonably accurate values of N ( E ) . T ; the next B the phase forward, nearly to The calculation of phase advance can, however, be cell will then carry the phase beyond 4d IF, and so on. Thus the phase advance per B cell will be put into a form that is practicable, and indeed exactly T,and will remain so even when E is slightly much superior to the method of Section 4 when a raised. We thus see that N(E2) = l / n , p ( E 2 ) .= high speed computer is not available. This in0:Ez is in a forbidden energy band, but below it, volves the construction of a slide rule from which between E1 and E2, there is one energy level per B one can read the phase of a solution a t the end of a atom in the crystal. These intervening levels sequence of n similar cells, given the phase with make up an impurity band of the A crystal with which the solution enters this set of cells. regularly spaced B impurities. To prepare such a slide rule, it is necessary to exWe now consider an A-crystal in which one cell in press the phase as a smooth function of a continuous n, chosen a t random, has been replaced by a B cell. (12) J. M. Luttinger, Phillips Reaearch Rep., 6, 241 (1951).
+
- +
+
Vol. 57
HUBERTM. JAMES AND ARTHUR S. GINZBARG
846
parameter n (distance along the uniform scale of the slide rule) such that
n
-170 --160
=--I80
-
If
(We assume symmetrical cell potentials).
--I
-
-
-
+ Bmg
-
2 (41
f
4GXF-U
-
(40)
-I
Convenient special forms for tan 4(n) are theii found to be as follows: Permitted band c:~ =
tan+(%) =
-
c:2
COS
e
sin e tan
[a
+ ne]
f
-60
-100
-5
0
-
0
.4
-
-4
-30 0
-
(42)
The choice of CY is arbitrary. Scales of C$ as a functiod of distance n along the slide rule are shown in Fig. 6 , for an energy in a permitted band of a pure B crystal, and in a forbidden band of a pure A crystal. Placing the zero of the uniform n scale opposite the initial phase on the A or B scale, one finds opposite each value of n on the uniform scale the phase at the end of n cells of type A or B, as the case may be. The scale corresponding to the permitted energy (B) shows an indefinitely increasing phase, the rulings having a spacing periodic with period 180". For a forbidden energy (A) one ideally needs two infinite scales, - T to &, in which one to cover the range phase increases with n, and the other to cover the range t$,, to t$d, in which 4 decreases as n increases; Fig. 6 shows a portion of the former scale. Different scales must be constructed for each value of E t o be considered. If the mixed crystal is specified to consist of nl A cells followed by nz B cells, n3 A cells, and so on, and the initial phase of the solution is given, one can then determine the phase advance in the whole crystal in an obvious way. 8. An Illustrative Calculation To illustrate the method of calculation, we have computed the energy level distribution in an A crystal with randomly distributed B impurities, in a range of energies near the bottom of a permitted band in the pure A crystal. To simulate the situation in a semiconductor, the constants cij characterizing the A and B cells have been so chosen that the localized impurity state due t o an isolated B atom in the A crystal matrix would lie 0.01 e.v. below the permitted band of the pure A crystal. The atomic concentration of B atoms was taken to
-
-20
cL = cosh e
+ ne]
IO
-
-9
C:I
-70
-
(41)
Forbidden band tan +(n) = -5 sinh e tanh [a
.5
-120
(39)
where n21,z =
6
50
-130
Y ( n ) = Am:
80
-140
-then Y,, must satisfy the linear difference equation which is easily solved
-
-
-130
-
-120
-
-50
-
-0
0
-6
-3
-140
-80
-7
=I80 -I70 =_I88
-110 0
-1
-
-40
-- 3 0 -20 -
-IO -0
-I00 -0 Fig. &-Typical slide rules for rapid calculation of hase advance in a sequence of identical cells: +*(n), sc& for energy in forbidden band (one of two required); @(n), scale for energy in permitted band.
be l/Io, corresponding in this one-dimensional model to an atomic concentration of l/looo in a three- dimensional m ode1. The constants characterizing the individual cells were chosen as follows. It was assumed that '/E of the levels in a permitted band of a pure B crystal would lie below the band edge Eo of the pure A crystal, and that c z ( E o ) = - &Eo). Variation of the constants c: in the small energy range considered was neglected entirely. In the case of the A cells such neglect is not possible, since ( c t - I)/ (c; 1) and ch/ci', should be nearly linear functions of EO - E. To bring the impurity levels to the desired position, we chose
+
Nov., 1953
BANDSTRUCTURE IN DISORDERED ALLOYSAND IMPURITY SEMICONDUCTORS 847 calculation is its indication that, in an impure semiconductor, the distribution of energy levels near a band edge may deviate markedly from that in the pure semiconductor. Suph structure in the energy level distribution will be most important when the temperature is so low that IcT is small compared to the width of the impurity band, and when the Fermi level lies in this range of energy. This suggests the importance of studies, a t liquid helium temperatures, of the conductivity of semiconductors in which the number of carriers is controlled by careful balance of acceptor and donator impurities or imperfection^.^^^'^
APPENDIX: PROOF THAT
($$)+i
.
,
>0
-
To prove the stated result, it will suffice to show that +f This follows immediately from the definition of tan +, with K > 0 and the following more general Theorem: Let $(E,z) be a family of real solutions of any one-dimensional wave equation, all with the same value of the logarithmic derivative u(E,z) = $'(E,x)/\1.(E,z) (45) at the point x = x,. The value of the logarithmic derivatives a t any other fixed point 2 2 will then be a function of E only b tan (T)+~ > 0.
-6
'5 -4
uz
-3 -2
-I
.ol
I
I
I
0
= uz(E)
(46)
such that
Proof: Equation (46) states a well-known result. By the use of the wave equation, one finds easily d dx W$.(E1,x),rl.(Ea,z)l = 2m -(E2 - E J ! ~ ~ E I , ~ ) + ( E Z (48) ,Z) Ti2 The Wronskian in this equation vanishes for x = 21, since the two $Is have.there the same value of c . Integrating eq. (48), one obtains W[\1.(EI,d, \1.(Ez,a)l =
-
h2
(E2
- Ed
lF
dzrl..(E~,x)$(E2,x) (49)
Division by $( E I , ~$(Ez,zz) ) yields ~2(Ez)- ~ P ( E I=)
Division by eq. (47).
E2
- EI and
passage to the limit then yields
DISCUSSION H. J. VINK (N. V. Philips Gloeilampenfabrieken, Eindhoven, The Netherlands).-Are the results of your calculations independent of whether the total number of impurity levels is equal to or much greater than the total number of electrons available for these levels? H. M. JAitIES.-YeS. We have determined the density of energy levels by considering a one-electron problem with a non-periodic potential. Such a potent,ial would arise from the crystal lattice, the impurity ions, and any electrons which may be present in non-localized states. While we have assumed a special form for the effective potential, this does not seem to imply anything about the number of electrons in non-localized states which contribute to it. VmK.--In one of your figures i t was shown that the impurity band of donors and the conduction band were connected (13) C. 8. Hung and J. R. Gliessrnan, Phys. Reu., 79, 726 (1950). (14) C. S. Hung, ibid., 79, 727 (1950).
848
HUBERT M. JAMESAND ARTHUR S. GINZBARG
with each other. It' looked as if the two bands were overlapping. Does it follow that conduction in the impurity band always will be electron conduction, and never hole conduction, even for the case that the total number of electrons available for the impurity band is somewhat smaller than the total amount of impurity levels?
JAMES.-Iwould expect that one might have, effectively, hole conduction in a nearly filled impurity band, even though it is not sharply separated from the higher conduction band. However, the theory of conduction in such systems has not yet been worked out.
VOl. 57
ingly, one must expect that conduction by jumps of electrons from one impurity to another in a semiconductor-as distinguished from that due to electrons dissociated from impurities-will be quite negligible up to a critical concentration of impurities, and will thereafter rise very Tapidly. For high impurity concentration the impurity band will not be distinguishable from the conduction band, and the two conduction mechanisms will not be separable.
F. S E I T Z(University of Illinois).-I would think, on intuitive grouds, that you would have to have a critical density in a three-dimensional lattice before you would begin to find continuous threads of conductivity.
H. BROOKS (Harvard University).-It would seem that correlation effects should be much more important for a random distribution of impurities than in the ordinary band model, so that perhaps the legitimacy of applying one electron calculations to this type of problem is more uestionable than in the ordinary periodic potential case. h a v e you given any thought to this?
JAMEs.--This would certainly be the case if there were a sharp upper limit to the distance an electron could jump from one impurity atom to another. The problem resembles that of the gelation of rubber during cure. If you start with a liquid of long molecules and form bonds between them, you will a t first simply create more complicated molecules. However, when the concentration of bonds reaches the critical value of one per molecule, a gel is formed-essentially a molecule of infinite molecular weight, involving only a tiny fraction of the whole material, but forming a continuous structure from wall to wall of the container. Thereafter, the addition of more bonds consolidates more of the material into the gel, and ver rapidly increases the number of continuous paths througi the molecular network. Correspond-
JAMES.-Iwould expect correlation effects to be a t a minimum in periodic systems, to increase somewhat when potential fluctuations produce local increases in the density of untrapped electrons, and to be quite large in situations in which some of the electrons are to be regarded as trapped in localized states. A student of mine, Mr. G. W. Lehman, is currently working on a theory of semiconductors in which some of the electrons are treated as trapped by the attraction of positive ions, while others move in the conduction band under the perturbation of the impurity ions and the trapped electrons. One-electron orbitals are employed in this theory, but correlations are partially taken into account, in that localized and non-localized orbitals satisfy Hartree equations with different effective potentials.
