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Exciton energy levels in germanium and silicon
J. Phys.
Chem. Solids
EXCITON
Pergamon
Press 1960. Vol. 13. pp. l-9
ENERGY
LEVELS
Printed in Great Britain.
IN
GERMANIUM
AND
SILICON T. P. MCLEAN Royal Radar Establishment,
Malvern, England
and R. LOUDON Clarendon Laboratory, (Received
Oxford, England
17 July
1959)
Abstract-The energies of the lowest-lying levels of the direct exciton in germanium and the indirect excitons in germanium and silicon have been calculated in the effective-mass approximation by a variational procedure. These energies are compared with the experimental values obtained from optical absorption experiments.
1. INTRODUCTION
EFFECTS attributable to the formation of excitons have recently been observed by a group at R.R.E.(l) in the optical absorption spectra of both silicon and germanium and by a group at Lincoln Laboratory(s) in the spectrum of germanium. These observations allow values of the binding energy of the lowest-lying exciton states to be determined. The values found are in rough agreement with estimates made by DRESSELHAUS(~)on the basis of an effective-mass approximation for the exciton. The description of exciton energy levels by means of an effective-mass approximation was first proposed by WANNIER(~) and developed by him for the case of simple valence and conduction bands. DRESSELHAUS(3) has reformulated this treatment and extended it to cover the cases in which the band extrema are degenerate and occur at arbitrary points in the Brillouin zone. It follows from this work that the degeneracy of the valenceband edge in germanium and silicon leads to a description of the exciton system in these substances by a set of coupled differential equations which are very similar to the equations describing acceptor impurities in germanium and silicon. (5) These equations have recently been solved by a variational method.(s* 7) We present in this paper A
the details and results of a similar calculation of the exciton energy levels. The direct exciton states have a degeneracy similar to that found in the acceptor states.(s) This degeneracy is partially lifted for the indirect excitons as the electrons bound in these states are associated with several equivalent conduction-band minima with anisotropic mass. A detailed group theoretical discussion of these excitons is given here by a method which is general and could be used in more complicated cases. In germanium two types of exciton have been observed, one type being produced by direct optical transitions and the other by indirect optical transitions involving the creation or annihilation of a phonon. We shall refer to these as direct and indirect excitons respectively. In silicon only indirect excitons have been observed. We shall limit our considerations to these three cases. 2. THE
EXCITON
HAMILTONIAN
The exciton consists of a hole in the valence band close to k = 0 bound to an electron close to a conduction-band minimum at wave-vector k = kc. In both germanium and silicon the valence band is threefold degenerate at k = 0 when spin is ignored, and when spin is taken into account the sixfold degenerate band is split by spin-orbit interaction into a fourfold and a twofold band with
2
T.
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MCLEAN
the fourfold band uppermost. The conduction bands have only spin degeneracy, and since we shall not take into account the spin-orbit interaction between the electron and hole, we may ignore the conduction-band spin and take the bands to be non-degenerate. The wave-function of an exciton at the excitonband minimum may be written:
where t,hck,(re) and y5zO(rh) are the Bloch functions for the conduction- and valence-band edges, re is the position of the electron, rh that of the hole and :
r = r,-rh.
(2.2)
The i summation runs over the degenerate valenceband states at k = 0, and the subscript II denotes the internal state of the exciton. The functions l&Jr) describe the relative motion of the electron and hole. We are going to classify the exciton states group theoretically, so we require to know the group of transformations under which the exciton Hamiltonian is invariant. The Hamiltonian consists of a sum of three terms representing the electron-hole interaction energy and the effective-mass kinetic energies of a hole close to k = 0 in the valence band and an electron close to k = kc in the conduction band. The presence of the kinetic-energy terms in the exciton Hamiltonian requires that the transformations applied to re and rh must be contained in the symmetry groups associated with the points k = ke and k = 0 respectively. Further, the interaction energy depends on the electronhole separation 1t-1,and this term is maintained invariant only if the transformations simultaneously applied to the electron and hole co-ordinates re and rh are identical. We therefore have the result that the group of the Hamiltonian includes those operations in which a transformation contained in the intersection of the groups of the k-vectors of the valence-band maximum and the conductionband minimum is simultaneously applied to the electron and hole co-ordinates. For the cases we shall consider, the group of kc, G say, is either equal to that of k = 0 or is a subgroup of it, so that the required intersection is just the group G itself. Now consider the transformation properties of
and
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the exciton wave-function (2.1). Under the operations of the group of the Hamiltonian, it is clear that the t&(rh) and aJcek,(re) are basis functions for representations of the group G, the former functions generating a representation corresponding to the point k = 0 and the latter a representation associated with the point k = kc. It remains to consider the effect of these operations on the functions F&(r). These functions satisfy a set of coupled differential equations derived by &ESSELHAUS(~), which are defined as follows for the case of germanium and silicon. The valence bands are described for small values of k by the well-known 6x 6 matrix given, e.g., by KOHN (Ref. 5, equation (8.4)) and which we denote by H”(k), while the conduction-band energy is represented by Hc(k-kc). We take the interaction between the electron and hole to be of the Coulomb type modified by the static dielectric constant K, i.e. -9/u. Then the Hamiltonian for the relative motion of the electron and hole is:
HF = [
Hc(--iv)-; &j-Hpiv) 1
(2.3) where V is the gradient operator in r-space, and the F&(r) satisfy the set of equations :
7 HpFnjkc(r) = &d%&(r).
(2.4)
It is clear from the definition (2.2) that under the operations of the group of the exciton Hamiltonian r experiences only the rotational parts of the operators of G, since the same translation applied to re and rh cancels when the difference is taken to obtain the transformed value of r. The Hamiltonian (2.3) for the relative motion of the electron and hole is therefore invariant under the operations of the point-group, G, say, isomorphic with the space-group G, these operations being applied now to the co-ordinate r. Hence, under the operations of the group of the total exciton Hamiltonian, the wave-functions F&(r) for the relative motion are basis functions for representations of the point group GO. However, space-group irreducible representations corresponding to the point k = 0 can be formed from those of the isomorphic point-group by associating a factor unity with the translational part of the space-group operation whose representation is required.@) The
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functions F&(r) therefore form a basis for a k = 0 representation of the space-group G, where it is important to remember that the transformations of this group are applied not to the co-ordinate r but to the co-ordinates rc and rh. We have shown that under the operations of the group of the exciton Hamiltonian the three components of the exciton wave-function (2.1) form bases for representations of the group G of the wave-vector of the conduction-band minimum, the first two components generating representations corresponding to the point k = 0, and the third component giving a representation associated with the point k = ke. The transformation properties of the total exciton wave-function Yn,&re, rh) are determined by calculating the direct product of the three representations. We now consider the form of the Hamiltonian for the relative motion of the electron and hole for each of the three cases in which we are interested. (a) The direct exciton in germanium For the case of germanium the splitting between the fourfold and the twofold degenerate valence bands is about 0.3 eV,fsf which is about 300 times the observed exciton binding energy. We may therefore neglect the contribution to the exciton wave-function from the twofold degenerate band. Hence we take just the upper left-hand 4 x 4 part of the matrix W(k), which describes the full valence-band structure in germanium for small values of k, and denote this 4 x 4 matrix by Hl{k). The conduction band is spherical for small values of k and its energy is: h2b2
where m* = 0*037m is the effective mass, and is known from magneto-absorption measurements,(ls) m being the free electron mass. We are measuring energies from the valence-band edge, so that E0 is the energy gap at k = 0, i.e. the direct energy gap. Substitution of these expressions for H*(k) and He(k) into (2.3) gives the Hamiltonian for the relative motion of the direct exciton in germanium. The binding energy of the exciton in state n is &l-J%*. The Bloch functions for the four degenerate valence bands at k = 0, i.e. #z,(r) i = 1, 2, 3, 4, form a basis for the irreducible representation
AND
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3
I’* of the full cubic double group,(lr) while the conduction-band Bloch function t,&(r) transforms as I’; neglecting spin.(la) We use the notation for irreducible representations given by BOUCKAERT et c&(13) and ELLIOTT@*), and we shall also make use of the tables in these two articles. We note that the product of Bloch functions in (2.1) transforms like rixr; = I’;, which is a four-dimensional irreducible representation. To obtain the representation by which the total exciton wave-function transforms, I?; must be multiplied by the appropriate representation for F,,(r), whose nature we consider in Section 3. (b) The indirect exciton in germanium The conduction band in germanium has minima at the zone edges in the (111) and similar directions. The indirect exciton consists of an electron in the conduction band at one of these minima and a hole in the valence band at k = 0. The constant-energy surfaces close to the (111) direction minimum are ellipsoids of revolution about the (111) direction, and it is convenient to take a Cartesian co-ordinate system (1, 2, 3) in which the 3-axis points along this direction, For k close to ke, the conductionband energy is then :
where ml = 160m and mt = 0.0813m are known from cyclotron-resonance measurements and Ak= k-kc; EC is the minimum energy gap between the and conduction bands-the indirect valence energy gap. For Hv(k) we can again neglect the contribution to the exciton wave-function from the twofold degenerate valence bands and take just the upper left-hand 4 x 4 block of the complete valence-band matrix. The co-ordinate axes (x, y, z) for Hi(k) are normally taken in the crystal directions (loo), (010) and (001). For the indirect exciton Hamiltonian it is more convenient to apply to q(k) a unitary transformation which alters the reference axes to (1,2,3). We denote this transformed matrix by Z?;(k). Substitution of the above valence- and conduction-band Hamiltonians into (2.3) gives the exciton Hamiltonian. In state TJthe exciton binding energy is E~-Enko where the Enac are the eigenvalues of this Hamiltonian. The energy levels of
T.
4
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the excitor-is with wave-vectors corresponding to the other conduction-band minima are degenerate with the &,. The Bloch functions at ke form bases for the irreducible representations of the symmetry group L, which is a subgroup of the full cubic group I’. The L representations corresponding to k = 0 and k = kc are different, and we must therefore distinguish between them by quoting the k-vector. The compatibility relation between I’: and L representations is :
L:(O) is two-dimensional
and L:(O) and L:(O) are both one-dimensional, being degenerate by time reversal. The conduction-band Bloch function at kc transforms as the scalar representation Ll(kc), so that the product of Bloch functions in (2.1) transforms either as L,+(kc)+L$(kc) or as Li(.kc). Apart from accidental degeneracy, the exciton states in this case cannot have a degeneracy greater than twofold. (c) The indirect exciton in silicon In silicon the experimentally determined binding energy of the indirect exciton is about 0.01 eV, which is about one-third of the estimated spinorbit splitting of the valence bands at k = 0. The twofold degenerate valence bands may therefore make a significant contribution to the exciton state. On account of this one should use the complete 6x 6 valence-band matrix P(k) in forming the exciton Hamiltonian. However, this leads to rather and we have therefore lengthy computations, carried out calculations in two extreme limits, firstly proceeding as in the case of germanium and considering only contributions from the top four valence-band states (infinite spin-orbit splitting), and secondly taking the spin-orbit splitting to be zero, which corresponds to neglecting the valenceband spin. The conduction-band minima in silicon lie in the (100) and similar directions inside the first zone. Near the minimum in the (001) direction situated at k = kc, the surfaces of constant energy are ellipsoids of revolution about the (001) direction, the energy being given by:
Hc(k-kc)
= &+g(Ak;+Ak;)+2k: t
2
(2.7)
and
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where ml = 0.9% and mt = 0.19m are known from cyclotron-resonance measurements, and the co-ordinate axes (x, y, Z) are defined in Section 2(b). Assuming infinite spin-orbit splitting, we take for the valence-band matrix in (2.3) the matrix Hi(k) defined above. The Bloch functions at kc now form bases for irreducible representations of the A subgroup of the full cubic group, the conduction-band Bloch function transforming as AI( The A representations corresponding to k = 0 and k = kc are not identical, so we display the k-vectors of the representations in parenthesis. The compatibility relation between I’: and A group representations is :
~;+&(o)+&(o). If we assume zero spin-orbit splitting, we must take for the valence-band matrix in (2.3) the 3 x 3 matrix which describes the valence-band edge in the absence of spin-orbit coupling(r5) and which we shall denote by Hi(k). In this case the three degenerate valence-band Bloch functions at k = 0 form a basis for the full cubic group representation r&. When the symmetry group is restricted to A, this representation decomposes as follows :
Since the conduction-band Bloch function transforms as the scalar representation, the symmetry character of the product of Bloch functions in (2.1) is the same as that of the valence-band Bloch functions alone, except that the k-vector of the representation is changed from zero to kc. The exciton states again cannot have degeneracy greater than twofold. 3. THE VARIATIONAL
PROCEDURE FUNCTIONS
AND
TRIAL
In none of the three cases is it possible to solve equations (2.4) exactly for the F&(r), and we therefore adopt a variational procedure taking trial functions for the Ft. Much labour can be avoided by a suitable choice of trial functions, and we construct our trial functions by using the method of SCHECHTER(~). We write the FL&r) in a column matrix which we note by Fnkc(r), and we take as trial function for this matrix a linear sum of column matrices, the coefficients in the sum being
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variable parameters. This is the Ritz-Rayleigh variational procedure. We are most interested in the ground state of the exciton for each symmetry type, and we expect that the functions Fnk,(r) will be dominated by a term of s-like symmetry for such states. However, the form of the exciton Hamiltonian suggests that there should be some admixture of higher angular-momentum states. For example, without these higher states one effectively neglects the fluted nature of the valence bands. Now the exciton Hamiltonians are even under inversion, so that they do not mix states of different parity. Hence the functions included in the ground-state trial function must all be even under inversion. After the s-term we expect that terms of d-like symmetry will be the most important, and in fact we shall not include in the sum matrices of higher angular momentum than d. Taking only the s-term in the trial function, the Fnkc(r) will transform like scalars, and hence the total exciton wave-function as given by (2.1) will transform in the same way as the product of valenceand conduction-band Bloch functions. For example, in the case of the direct exciton in germanium Y&re, rh), where n refers to the ground state, must transform like IWhen matrices of d-like symmetry are included’fn the trial function, these matrices must be such that Y&r,, r-h) still transforms as Is. This fact enables us to restrict the number of d-terms which need be included in the trial function. The number of such d-terms may be determined as follows, still taking the direct exciton in germanium as an example. Under the operations of the cubic point-group, the five
1
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SILICON
angular-momentum d-functions form a basis for the point-group irreducible representations corresponding to rre and I&. Hence, using terms of dsymmetry in the trial function, the FAo(r) form a basis for the irreducible representations I?12 and r' of the full cubic space-group. We emphasize 25. again that the transformations of this group are to be applied simultaneously to re and rh, not to r. It follows that the products of three functions appearing in the exciton wave-function (2.1) form bases for the following representations :
G+r&)xrg
= 2(r,+r;)+3ri.
The representation Is occurs three times in the reduction of the direct product. This shows that there are three different matrices of d-symmetry which can be included in the trial function for Fno(r) such that the total exciton wave-function has l?, symmetry. The actual determination of the d-type matrices can be carried out by standard group theoretical methods involving the use of projection operators, or by a simpler method described by SCHECHTER(~). For the radial parts of the trial functions we take exp(--v/rr) for the s-terms and exp(-r/ra) for the d-terms. We consider the algebraic form of the trial functions separately for the three cases in which we are interested. (a) The direct exciton in germanium The number of d-terms in the trial function has been determined above, and the complete function is found to be :
_ _ 1 0 Cl
0
0 _ _
0
iz(x+iy) +c3
z2--_:(x2+y2) 0
exp(--/yl)+cz
ixy 0
ti(3/2)(x2-Y2)
I l-
5
O
-I
I 1 0
iz(x+
iy)
- 2ixy
id( 3)z(x - iy)
exp(-y/y&
6
T.
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MCLEAN
and
Here the four c’s are variational parameters and also rr and rs. The last matrix in the sum is not coupled to the s-matrix by the exciton Hamiltonian, but only to the other d-matrices, so that its effect on the exciton binding energy is small. There are, of course, three other functions similar to the above, but since they are degenerate with it they need not be considered. We note that an exciton state of this type can be produced optically, for the matrix element governing such transitions contains the ground-state wave-function of symmetry Fr and the gradient operator transforming as I’rb, so that the final state wave-function must transform as Fr X Frs = r15 for the matrix element to be non-zero. The electron spin function must now be included and transforms like 01,s of the rotation group, so that the total exciton wave-function has symmetry F, x D1,2 = I?;,+r,5+r20. This contains F1a, SO that optical transitions into this exciton state are allowed. (b) The indirect exciton in germanium We have shown above that the products of valence- and conduction-band Bloch functions in this case form a set of basis functions for Ll(k,)+ Ll(kc)+Ll(kc). The angular-momentum d-functions can be used to form a basis for the irreducible representations &(0)+2&(O). The products of these two sets of functions therefore form bases for:
[-b(o)+2h(o)]
X [L~(k,)+L:(k,)+L~(k,)]
= 3[L~(k,)+L:(k,)l+7L~(k,). There are therefore three matrices of d-symmetry which can be included in the trial function for the Li(k,)+Li(k,) exciton and seven such matrices for the case of the Lz(kc) exciton. We shall not exhibit here the actual form of the trial functions, but will confine ourselves to giving the result of the variational calculation below. It is easily verified that both these exciton states can be excited by indirect transitions involving either of the two acoustical phonons at the L symmetry point, but not the optical phonons. (c) The indirect exciton in silicon (i) Infinite spin-orbit splitting. There are two twofold degenerate exciton levels having symmetries As(k,) and A,(ke). The angular momentum
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d-functions can be used to form a basis for Ar(O)+Az(O)+AL(O)+As(O). We have:
[~~(~)+~z(O)+~~(~)+~~(~>l x Ps(kc)+&(ke)l = %(kc)+5&(kc) SO that for each exciton there having the correct symmetry clusion in the trial function. types of exciton level can be transitions involving any type symmetry direction.
are five d-matrices properties for inEither of the two excited by indirect of phonon in the A
(ii) Zero spin-orbit splitting. The two exciton levels are a singlet Ak(kC) and a doublet As(kc). Now: [Al(O)+As(O)+A;(O)+Aa(O)]
x [A;(k,)+As(ke)]
= 2Al(k,)+2A;(k,)+Az(ke)+2A;(ke)+4As(ke) so there are two d-type matrices for the Ag(k,) exciton trial function and four such matrices for the A5(ke) exciton. Either exciton level can be excited by indirect transitions involving any type of phonon in the A symmetry direction, with the one exception that the Ab(kJ level cannot be excited by a transition involving the longitudinal acoustic phonon. 4. RESULTS Minimizing the energies, corresponding to the various trial functions, with respect to the parameters cd, i = 1,2 . . . n, ~1 and rs involves finding the minimum latent root of a certain nxn determinant (whose elements are functions of rr and rs) which can be produced by variation of ~1 and 7s. This minimization procedure was carried out numerically in each case on the R.R.E. electronic computer. The results are shown in Table 1 for germanium and Table 2 for silicon. For the direct exciton in germanium the calculated binding energy of 0.0014 eV is only about 10 per cent larger than the value obtained by taking only the s-part of the trial function, so that one would expect the inclusion of higher angular-momentum states than d-states to produce a negligible increase in the binding energy. This value agrees reasonably with the R.R.E. experimental value of O*OOll& 0.0001 eV, but is in disagreement with the Lincoln Laboratory value of 0*0025&0.0004 eV. However, both of these values may be in error; the R.R.E.
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Table 1. The binding energies of excitons in germanium, using both -s-type and- (s+d)-type trial functions and band parameters quoted by KoHN(@, and the corresponding values of the variational parameters r1 and r2.
= Binding energy (eV)
__ Direct exciton-s-type function Direct exciton-sand dtype functions
(2) .-
0.00127
350
-
0.00138
320
220
0.00244
180
-
0.00347
120
87
0.00288
150
85
._
Indirect exciton-s-type function Indirect exciton-sand d-type functions w:+q Indirect exciton--sand d-type functions
L$
=I
AND
7
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value was obtained from an analysis of the absorp-
tion spectrum appropriate to a material having simple band edges, the error quoted here merely being due to a scatter in the values obtained at different temperatures; the Lincoln Laboratory value was obtained more directly, but from a specimen in a state of strain which has been shewn to influence considerably the value of the energy gap.(le) This strain resulted from the difference in thermal contraction of the thin slab of germanium and the glass mounting to which it was cemented. On cooling to liquid-helium temperatures, a uniform isotropic strain in the plane of the germanium slab is thus set up. In the presence of a strain of this type, the valence-band degeneracy at k = 0 is lifted, giving two valence-band edges and correspondingly two groups of exciton states. At sufficiently small values of k, the surfaces of constant energy for the two valence bands are ellipsoids of revolution about the direction of the slab normal, being prolate ellipsoids for one band and oblate ellipsoids for the other. If we assume that
Table 2. The binding energies of the indirect excitons in silicon, using both s and (s+d)-type trial functions and band parameters quoted by KOHN@), and the corresponding values of the variational parameters rl and r2 for both zero and infinitely large spin-orbit coupling and for both positive and negative values of the eflective mass parameter (L-M).(15)
_
s-type function
(L-W><0
s- and d-type functions
(L--M)<0
Binding energy (eV)
2,
0.0120
50
-
A;
0.0132
44
33
A5
0.0137
42
30
0.0145
39
28
0.0133
44
30
j
(%
s- and d-type functions
s-type function
(L-M)><0
s- and d-type functions
(L-M)<0
s- and d-type functions
(L-W>0
0.0120
50
i -
A6
0.0130
44
31
A7
1 0.0124
47
33
A6
0.0124
47
33
A7
0.0130
44
31
8
T.
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the splitting between the bands at k = 0 is large compared with the exciton binding energy, then it is this region of ellipsoidal energy-dependence which is important in determining the hole effective mass for use in the exciton wave-equation. Taking the conduction-band effective mass at k = 0 to be unaltered by the strain, the exciton functions F&r) then satisfy an equation similar to that for a donor state in germanium or silicon, and the ground-state binding energies can be calculated by the same method as has been used for the donor problem.(l7) It is found that the excitons due to both bands have approximately the same binding energy of about 0.0013 eV. Since the measured value of the valence-band splitting in the Lincoln Laboratory experiments was 0.005 eV,(ls) the exciton binding energy is as large as one-quarter of the splitting, so that the assumption on which the above estimate is based is not very well satisfied. However, the calculation indicates that the difference between the binding energies observed by the two experimental groups is not likely to be entirely accounted for by the strain in the Lincoln Laboratory specimen. As for the accuracy of the calculated binding energy, we may note that a similar theory applied to the ground states of acceptors in germanium(T) gives a binding energy which is about 20 per cent smaller than that measured experimentally, and the discrepancy is thought to be largely due to breakdown of effective-mass theory in the immediate vicinity of the impurity atom. However, the mean radius of the direct exciton wave-function in germanium is almost ten times that of the groundstate acceptor wave-function, so that effects due to the breakdown of effective-mass theory when the electron and hole are close together would be expected to be correspondingly smaller. Further, for the exciton there is no distortion of the lattice analogous to that introduced by the inclusion of the impurity atom which renders the effectivemass theory less accurate for the case of an acceptor. A better agreement between the calculated and experimental results for the direct exciton in germanium might therefore be expected. For the indirect exciton in germanium, the calculated ground-state binding energy is 0.0035 eV, and an excited state lies above this by an energy 0.0006 eV. The two binding energies represent substantial increases over the values obtained by
and
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using only an s-type trial function. Experimentally, the R.R.E. value for the ground-state binding energy is 0~0027f0~0004 eV, with an excited state at 0~0010~0~0001 eV above this, while for the same quantities Lincoln Laboratory measures 0.0033 f 0.0004 and 0.0010 eV, respectively. The reported ground-state binding energies are therefore consistent with a value of about 0.0030 eV, and the splittings between the two levels agree at 0.0010 eV. There is fair agreement between experimental and theoretical values for the ground-state binding energy, but for the splitting between the two levels the calculated value is significantly lower than that measured experimentally. We note that the energies calculated with the present trial functions are larger than the values of 0.0025 and 0.0033 eV calculated by the Lincoln Laboratory,@, 10) using the trial function exp{ - [u?za+ba(~a+ys)]}. For the various cases considered for the indirect exciton in silicon, the inclusion of the d-states in the trial function produces at most an increase of about 20 per cent over the binding energy found by using only an s-type trial function. The calculated ground-state binding energy is about 0.0120.014 eV, and the excited state lies above this by about 0~0005-0~0012 eV. These results are to be compared with the values measured at R.R.E. of about 0.01 eV for the binding energy, and 0.0055 eV for the splitting. The agreement between the calculated and observed ground-state binding energies is fair considering the uncertainty in the latter, but the agreement between the splittings is so bad as to suggest that the excited state observed experimentally is not that whose energy we have calculated. The mean radii of the indirect exciton wavefunctions are smaller, particularly in silicon, than that of the direct exciton in germanium, and, on the basis of the above remarks concerning the breakdown of effective-mass theory when the electron and hole are close together, we should expect the calculated values for these indirect exciton binding energies to be less accurate than the calculated binding energy of the direct exciton in germanium. However, the mean indirect exciton radii are larger than the ground acceptor state radii in the same semiconductor by a factor of about three, so that the accuracy of the exciton calculations would be expected to be correspondingly better.
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2. ZWERDLINGS., ROTH L. M. and LAX B., Phys. Rev. 109, 2207 (1958); J. Phys. Chem. Solids 8, 397 (1959). BUTTON K. J., ROTH L. M., KLEINER W. H., ZWERDLING S. and LAX B., Phys. Rev. Letters 2, 161 (1959). 3. DRE~SELHAUS G., J. Phys. Chem. Solids 1, 14 (1956). 4. WANNIER G. H., Phys. Rev. 52, 191 (1937). 5. KOHN W., Solid State Physics 5, 257 (1957). 6. KOHN W. and SCHECHTERD., Phys. Rev. 99, 1903 (1955). 7. SCHECHTERD., Thesis, Carnegie Institute of TechAcknowledgements-The authors wish to thank the nology (1958). Lincoln Laboratory for pre-publication copies of their 8. KOSTER G. F., Solid State Physics, 5, 173 (1957). results. Their thanks are also due to Dr. D. P. JENKINS 9. KAHN A. H., Phys. Rev. 97, 1647 (1955). for help in programming some parts of the calculation 10. ROTH L. M., LAX B. and ZWERDLINGS., Phys. Rev for the computer. to be published. This work was principally carried out in the summer 11. ELLIOTT R. J., Phys. Rev. 96, 266 (1954). of 1958, when one of us (R.L.) was a Vacation Student 12. HERMAN F., Phys. Rev. 95. 847 (1954). at the Royal Radar Establishment. He would like to take 13. BOUCKAERTL. P., SMOLU&OW&U g. and WIGNER this opportunity to thank Dr. G. G. MACFARLANEand E.. Phvs. Rev. 50. 58 (1936). his group for a pleasant and stimulating time. 14. ELLIOTT-R. J., Phyi. Rev. 96: 280 (1954). This paper is published by permission of the ConG., KIP A. F. and KITTEL C., Phys. 15. DRESSELHAUS troller, Her Britannic Majesty’s Stationery Office. Rev. 98, 368 (1955). 16. MACFARLANEG. G., MCLEAN T. P., QUARRINGTON REFERENCES J. E. and ROBERTSV., Phys. Rev. Letters 2, 252 (1959). 1. MACFARLANEG. G., MCLEAN I?. P., QUAFUUNGTON 17. KOHN W. and LUTTINGERJ. M., Phys. Rev. 98, 915 J, E. and ROBERTSV., Phys. Rev. 108, 1377 (1957); Proc. Phys. Sot. Lond. 71, 863 (1958); Phys. Rev. (1955). 18. KLEIN& W. H. and ROTH L. M., Phys. Rev. Letters 111, 1245 (1958); J. Phys. Chem. Solids 8, 388 2, 334 (1959). (1959).
For each of the three types of exciton there is fair agreement between the measured and calculated values of the ground-state binding energies. A more detailed comparison of theory with experiment must await more accurate experiments on silicon and a reconciliation of the two experimental groups whose differing observations of the direct exciton in germanium we have discussed.
Chem. Solids
EXCITON
Pergamon
Press 1960. Vol. 13. pp. l-9
ENERGY
LEVELS
Printed in Great Britain.
IN
GERMANIUM
AND
SILICON T. P. MCLEAN Royal Radar Establishment,
Malvern, England
and R. LOUDON Clarendon Laboratory, (Received
Oxford, England
17 July
1959)
Abstract-The energies of the lowest-lying levels of the direct exciton in germanium and the indirect excitons in germanium and silicon have been calculated in the effective-mass approximation by a variational procedure. These energies are compared with the experimental values obtained from optical absorption experiments.
1. INTRODUCTION
EFFECTS attributable to the formation of excitons have recently been observed by a group at R.R.E.(l) in the optical absorption spectra of both silicon and germanium and by a group at Lincoln Laboratory(s) in the spectrum of germanium. These observations allow values of the binding energy of the lowest-lying exciton states to be determined. The values found are in rough agreement with estimates made by DRESSELHAUS(~)on the basis of an effective-mass approximation for the exciton. The description of exciton energy levels by means of an effective-mass approximation was first proposed by WANNIER(~) and developed by him for the case of simple valence and conduction bands. DRESSELHAUS(3) has reformulated this treatment and extended it to cover the cases in which the band extrema are degenerate and occur at arbitrary points in the Brillouin zone. It follows from this work that the degeneracy of the valenceband edge in germanium and silicon leads to a description of the exciton system in these substances by a set of coupled differential equations which are very similar to the equations describing acceptor impurities in germanium and silicon. (5) These equations have recently been solved by a variational method.(s* 7) We present in this paper A
the details and results of a similar calculation of the exciton energy levels. The direct exciton states have a degeneracy similar to that found in the acceptor states.(s) This degeneracy is partially lifted for the indirect excitons as the electrons bound in these states are associated with several equivalent conduction-band minima with anisotropic mass. A detailed group theoretical discussion of these excitons is given here by a method which is general and could be used in more complicated cases. In germanium two types of exciton have been observed, one type being produced by direct optical transitions and the other by indirect optical transitions involving the creation or annihilation of a phonon. We shall refer to these as direct and indirect excitons respectively. In silicon only indirect excitons have been observed. We shall limit our considerations to these three cases. 2. THE
EXCITON
HAMILTONIAN
The exciton consists of a hole in the valence band close to k = 0 bound to an electron close to a conduction-band minimum at wave-vector k = kc. In both germanium and silicon the valence band is threefold degenerate at k = 0 when spin is ignored, and when spin is taken into account the sixfold degenerate band is split by spin-orbit interaction into a fourfold and a twofold band with
2
T.
P.
MCLEAN
the fourfold band uppermost. The conduction bands have only spin degeneracy, and since we shall not take into account the spin-orbit interaction between the electron and hole, we may ignore the conduction-band spin and take the bands to be non-degenerate. The wave-function of an exciton at the excitonband minimum may be written:
where t,hck,(re) and y5zO(rh) are the Bloch functions for the conduction- and valence-band edges, re is the position of the electron, rh that of the hole and :
r = r,-rh.
(2.2)
The i summation runs over the degenerate valenceband states at k = 0, and the subscript II denotes the internal state of the exciton. The functions l&Jr) describe the relative motion of the electron and hole. We are going to classify the exciton states group theoretically, so we require to know the group of transformations under which the exciton Hamiltonian is invariant. The Hamiltonian consists of a sum of three terms representing the electron-hole interaction energy and the effective-mass kinetic energies of a hole close to k = 0 in the valence band and an electron close to k = kc in the conduction band. The presence of the kinetic-energy terms in the exciton Hamiltonian requires that the transformations applied to re and rh must be contained in the symmetry groups associated with the points k = ke and k = 0 respectively. Further, the interaction energy depends on the electronhole separation 1t-1,and this term is maintained invariant only if the transformations simultaneously applied to the electron and hole co-ordinates re and rh are identical. We therefore have the result that the group of the Hamiltonian includes those operations in which a transformation contained in the intersection of the groups of the k-vectors of the valence-band maximum and the conductionband minimum is simultaneously applied to the electron and hole co-ordinates. For the cases we shall consider, the group of kc, G say, is either equal to that of k = 0 or is a subgroup of it, so that the required intersection is just the group G itself. Now consider the transformation properties of
and
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the exciton wave-function (2.1). Under the operations of the group of the Hamiltonian, it is clear that the t&(rh) and aJcek,(re) are basis functions for representations of the group G, the former functions generating a representation corresponding to the point k = 0 and the latter a representation associated with the point k = kc. It remains to consider the effect of these operations on the functions F&(r). These functions satisfy a set of coupled differential equations derived by &ESSELHAUS(~), which are defined as follows for the case of germanium and silicon. The valence bands are described for small values of k by the well-known 6x 6 matrix given, e.g., by KOHN (Ref. 5, equation (8.4)) and which we denote by H”(k), while the conduction-band energy is represented by Hc(k-kc). We take the interaction between the electron and hole to be of the Coulomb type modified by the static dielectric constant K, i.e. -9/u. Then the Hamiltonian for the relative motion of the electron and hole is:
HF = [
Hc(--iv)-; &j-Hpiv) 1
(2.3) where V is the gradient operator in r-space, and the F&(r) satisfy the set of equations :
7 HpFnjkc(r) = &d%&(r).
(2.4)
It is clear from the definition (2.2) that under the operations of the group of the exciton Hamiltonian r experiences only the rotational parts of the operators of G, since the same translation applied to re and rh cancels when the difference is taken to obtain the transformed value of r. The Hamiltonian (2.3) for the relative motion of the electron and hole is therefore invariant under the operations of the point-group, G, say, isomorphic with the space-group G, these operations being applied now to the co-ordinate r. Hence, under the operations of the group of the total exciton Hamiltonian, the wave-functions F&(r) for the relative motion are basis functions for representations of the point group GO. However, space-group irreducible representations corresponding to the point k = 0 can be formed from those of the isomorphic point-group by associating a factor unity with the translational part of the space-group operation whose representation is required.@) The
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functions F&(r) therefore form a basis for a k = 0 representation of the space-group G, where it is important to remember that the transformations of this group are applied not to the co-ordinate r but to the co-ordinates rc and rh. We have shown that under the operations of the group of the exciton Hamiltonian the three components of the exciton wave-function (2.1) form bases for representations of the group G of the wave-vector of the conduction-band minimum, the first two components generating representations corresponding to the point k = 0, and the third component giving a representation associated with the point k = ke. The transformation properties of the total exciton wave-function Yn,&re, rh) are determined by calculating the direct product of the three representations. We now consider the form of the Hamiltonian for the relative motion of the electron and hole for each of the three cases in which we are interested. (a) The direct exciton in germanium For the case of germanium the splitting between the fourfold and the twofold degenerate valence bands is about 0.3 eV,fsf which is about 300 times the observed exciton binding energy. We may therefore neglect the contribution to the exciton wave-function from the twofold degenerate band. Hence we take just the upper left-hand 4 x 4 part of the matrix W(k), which describes the full valence-band structure in germanium for small values of k, and denote this 4 x 4 matrix by Hl{k). The conduction band is spherical for small values of k and its energy is: h2b2
where m* = 0*037m is the effective mass, and is known from magneto-absorption measurements,(ls) m being the free electron mass. We are measuring energies from the valence-band edge, so that E0 is the energy gap at k = 0, i.e. the direct energy gap. Substitution of these expressions for H*(k) and He(k) into (2.3) gives the Hamiltonian for the relative motion of the direct exciton in germanium. The binding energy of the exciton in state n is &l-J%*. The Bloch functions for the four degenerate valence bands at k = 0, i.e. #z,(r) i = 1, 2, 3, 4, form a basis for the irreducible representation
AND
SILICON
3
I’* of the full cubic double group,(lr) while the conduction-band Bloch function t,&(r) transforms as I’; neglecting spin.(la) We use the notation for irreducible representations given by BOUCKAERT et c&(13) and ELLIOTT@*), and we shall also make use of the tables in these two articles. We note that the product of Bloch functions in (2.1) transforms like rixr; = I’;, which is a four-dimensional irreducible representation. To obtain the representation by which the total exciton wave-function transforms, I?; must be multiplied by the appropriate representation for F,,(r), whose nature we consider in Section 3. (b) The indirect exciton in germanium The conduction band in germanium has minima at the zone edges in the (111) and similar directions. The indirect exciton consists of an electron in the conduction band at one of these minima and a hole in the valence band at k = 0. The constant-energy surfaces close to the (111) direction minimum are ellipsoids of revolution about the (111) direction, and it is convenient to take a Cartesian co-ordinate system (1, 2, 3) in which the 3-axis points along this direction, For k close to ke, the conductionband energy is then :
where ml = 160m and mt = 0.0813m are known from cyclotron-resonance measurements and Ak= k-kc; EC is the minimum energy gap between the and conduction bands-the indirect valence energy gap. For Hv(k) we can again neglect the contribution to the exciton wave-function from the twofold degenerate valence bands and take just the upper left-hand 4 x 4 block of the complete valence-band matrix. The co-ordinate axes (x, y, z) for Hi(k) are normally taken in the crystal directions (loo), (010) and (001). For the indirect exciton Hamiltonian it is more convenient to apply to q(k) a unitary transformation which alters the reference axes to (1,2,3). We denote this transformed matrix by Z?;(k). Substitution of the above valence- and conduction-band Hamiltonians into (2.3) gives the exciton Hamiltonian. In state TJthe exciton binding energy is E~-Enko where the Enac are the eigenvalues of this Hamiltonian. The energy levels of
T.
4
P.
MCLEAN
the excitor-is with wave-vectors corresponding to the other conduction-band minima are degenerate with the &,. The Bloch functions at ke form bases for the irreducible representations of the symmetry group L, which is a subgroup of the full cubic group I’. The L representations corresponding to k = 0 and k = kc are different, and we must therefore distinguish between them by quoting the k-vector. The compatibility relation between I’: and L representations is :
L:(O) is two-dimensional
and L:(O) and L:(O) are both one-dimensional, being degenerate by time reversal. The conduction-band Bloch function at kc transforms as the scalar representation Ll(kc), so that the product of Bloch functions in (2.1) transforms either as L,+(kc)+L$(kc) or as Li(.kc). Apart from accidental degeneracy, the exciton states in this case cannot have a degeneracy greater than twofold. (c) The indirect exciton in silicon In silicon the experimentally determined binding energy of the indirect exciton is about 0.01 eV, which is about one-third of the estimated spinorbit splitting of the valence bands at k = 0. The twofold degenerate valence bands may therefore make a significant contribution to the exciton state. On account of this one should use the complete 6x 6 valence-band matrix P(k) in forming the exciton Hamiltonian. However, this leads to rather and we have therefore lengthy computations, carried out calculations in two extreme limits, firstly proceeding as in the case of germanium and considering only contributions from the top four valence-band states (infinite spin-orbit splitting), and secondly taking the spin-orbit splitting to be zero, which corresponds to neglecting the valenceband spin. The conduction-band minima in silicon lie in the (100) and similar directions inside the first zone. Near the minimum in the (001) direction situated at k = kc, the surfaces of constant energy are ellipsoids of revolution about the (001) direction, the energy being given by:
Hc(k-kc)
= &+g(Ak;+Ak;)+2k: t
2
(2.7)
and
R.
LOUDON
where ml = 0.9% and mt = 0.19m are known from cyclotron-resonance measurements, and the co-ordinate axes (x, y, Z) are defined in Section 2(b). Assuming infinite spin-orbit splitting, we take for the valence-band matrix in (2.3) the matrix Hi(k) defined above. The Bloch functions at kc now form bases for irreducible representations of the A subgroup of the full cubic group, the conduction-band Bloch function transforming as AI( The A representations corresponding to k = 0 and k = kc are not identical, so we display the k-vectors of the representations in parenthesis. The compatibility relation between I’: and A group representations is :
~;+&(o)+&(o). If we assume zero spin-orbit splitting, we must take for the valence-band matrix in (2.3) the 3 x 3 matrix which describes the valence-band edge in the absence of spin-orbit coupling(r5) and which we shall denote by Hi(k). In this case the three degenerate valence-band Bloch functions at k = 0 form a basis for the full cubic group representation r&. When the symmetry group is restricted to A, this representation decomposes as follows :
Since the conduction-band Bloch function transforms as the scalar representation, the symmetry character of the product of Bloch functions in (2.1) is the same as that of the valence-band Bloch functions alone, except that the k-vector of the representation is changed from zero to kc. The exciton states again cannot have degeneracy greater than twofold. 3. THE VARIATIONAL
PROCEDURE FUNCTIONS
AND
TRIAL
In none of the three cases is it possible to solve equations (2.4) exactly for the F&(r), and we therefore adopt a variational procedure taking trial functions for the Ft. Much labour can be avoided by a suitable choice of trial functions, and we construct our trial functions by using the method of SCHECHTER(~). We write the FL&r) in a column matrix which we note by Fnkc(r), and we take as trial function for this matrix a linear sum of column matrices, the coefficients in the sum being
EXCITON
LEVELS
IN
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variable parameters. This is the Ritz-Rayleigh variational procedure. We are most interested in the ground state of the exciton for each symmetry type, and we expect that the functions Fnk,(r) will be dominated by a term of s-like symmetry for such states. However, the form of the exciton Hamiltonian suggests that there should be some admixture of higher angular-momentum states. For example, without these higher states one effectively neglects the fluted nature of the valence bands. Now the exciton Hamiltonians are even under inversion, so that they do not mix states of different parity. Hence the functions included in the ground-state trial function must all be even under inversion. After the s-term we expect that terms of d-like symmetry will be the most important, and in fact we shall not include in the sum matrices of higher angular momentum than d. Taking only the s-term in the trial function, the Fnkc(r) will transform like scalars, and hence the total exciton wave-function as given by (2.1) will transform in the same way as the product of valenceand conduction-band Bloch functions. For example, in the case of the direct exciton in germanium Y&re, rh), where n refers to the ground state, must transform like IWhen matrices of d-like symmetry are included’fn the trial function, these matrices must be such that Y&r,, r-h) still transforms as Is. This fact enables us to restrict the number of d-terms which need be included in the trial function. The number of such d-terms may be determined as follows, still taking the direct exciton in germanium as an example. Under the operations of the cubic point-group, the five
1
AND
SILICON
angular-momentum d-functions form a basis for the point-group irreducible representations corresponding to rre and I&. Hence, using terms of dsymmetry in the trial function, the FAo(r) form a basis for the irreducible representations I?12 and r' of the full cubic space-group. We emphasize 25. again that the transformations of this group are to be applied simultaneously to re and rh, not to r. It follows that the products of three functions appearing in the exciton wave-function (2.1) form bases for the following representations :
G+r&)xrg
= 2(r,+r;)+3ri.
The representation Is occurs three times in the reduction of the direct product. This shows that there are three different matrices of d-symmetry which can be included in the trial function for Fno(r) such that the total exciton wave-function has l?, symmetry. The actual determination of the d-type matrices can be carried out by standard group theoretical methods involving the use of projection operators, or by a simpler method described by SCHECHTER(~). For the radial parts of the trial functions we take exp(--v/rr) for the s-terms and exp(-r/ra) for the d-terms. We consider the algebraic form of the trial functions separately for the three cases in which we are interested. (a) The direct exciton in germanium The number of d-terms in the trial function has been determined above, and the complete function is found to be :
_ _ 1 0 Cl
0
0 _ _
0
iz(x+iy) +c3
z2--_:(x2+y2) 0
exp(--/yl)+cz
ixy 0
ti(3/2)(x2-Y2)
I l-
5
O
-I
I 1 0
iz(x+
iy)
- 2ixy
id( 3)z(x - iy)
exp(-y/y&
6
T.
P.
MCLEAN
and
Here the four c’s are variational parameters and also rr and rs. The last matrix in the sum is not coupled to the s-matrix by the exciton Hamiltonian, but only to the other d-matrices, so that its effect on the exciton binding energy is small. There are, of course, three other functions similar to the above, but since they are degenerate with it they need not be considered. We note that an exciton state of this type can be produced optically, for the matrix element governing such transitions contains the ground-state wave-function of symmetry Fr and the gradient operator transforming as I’rb, so that the final state wave-function must transform as Fr X Frs = r15 for the matrix element to be non-zero. The electron spin function must now be included and transforms like 01,s of the rotation group, so that the total exciton wave-function has symmetry F, x D1,2 = I?;,+r,5+r20. This contains F1a, SO that optical transitions into this exciton state are allowed. (b) The indirect exciton in germanium We have shown above that the products of valence- and conduction-band Bloch functions in this case form a set of basis functions for Ll(k,)+ Ll(kc)+Ll(kc). The angular-momentum d-functions can be used to form a basis for the irreducible representations &(0)+2&(O). The products of these two sets of functions therefore form bases for:
[-b(o)+2h(o)]
X [L~(k,)+L:(k,)+L~(k,)]
= 3[L~(k,)+L:(k,)l+7L~(k,). There are therefore three matrices of d-symmetry which can be included in the trial function for the Li(k,)+Li(k,) exciton and seven such matrices for the case of the Lz(kc) exciton. We shall not exhibit here the actual form of the trial functions, but will confine ourselves to giving the result of the variational calculation below. It is easily verified that both these exciton states can be excited by indirect transitions involving either of the two acoustical phonons at the L symmetry point, but not the optical phonons. (c) The indirect exciton in silicon (i) Infinite spin-orbit splitting. There are two twofold degenerate exciton levels having symmetries As(k,) and A,(ke). The angular momentum
R.
LOUDON
d-functions can be used to form a basis for Ar(O)+Az(O)+AL(O)+As(O). We have:
[~~(~)+~z(O)+~~(~)+~~(~>l x Ps(kc)+&(ke)l = %(kc)+5&(kc) SO that for each exciton there having the correct symmetry clusion in the trial function. types of exciton level can be transitions involving any type symmetry direction.
are five d-matrices properties for inEither of the two excited by indirect of phonon in the A
(ii) Zero spin-orbit splitting. The two exciton levels are a singlet Ak(kC) and a doublet As(kc). Now: [Al(O)+As(O)+A;(O)+Aa(O)]
x [A;(k,)+As(ke)]
= 2Al(k,)+2A;(k,)+Az(ke)+2A;(ke)+4As(ke) so there are two d-type matrices for the Ag(k,) exciton trial function and four such matrices for the A5(ke) exciton. Either exciton level can be excited by indirect transitions involving any type of phonon in the A symmetry direction, with the one exception that the Ab(kJ level cannot be excited by a transition involving the longitudinal acoustic phonon. 4. RESULTS Minimizing the energies, corresponding to the various trial functions, with respect to the parameters cd, i = 1,2 . . . n, ~1 and rs involves finding the minimum latent root of a certain nxn determinant (whose elements are functions of rr and rs) which can be produced by variation of ~1 and 7s. This minimization procedure was carried out numerically in each case on the R.R.E. electronic computer. The results are shown in Table 1 for germanium and Table 2 for silicon. For the direct exciton in germanium the calculated binding energy of 0.0014 eV is only about 10 per cent larger than the value obtained by taking only the s-part of the trial function, so that one would expect the inclusion of higher angular-momentum states than d-states to produce a negligible increase in the binding energy. This value agrees reasonably with the R.R.E. experimental value of O*OOll& 0.0001 eV, but is in disagreement with the Lincoln Laboratory value of 0*0025&0.0004 eV. However, both of these values may be in error; the R.R.E.
EXCITON
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Table 1. The binding energies of excitons in germanium, using both -s-type and- (s+d)-type trial functions and band parameters quoted by KoHN(@, and the corresponding values of the variational parameters r1 and r2.
= Binding energy (eV)
__ Direct exciton-s-type function Direct exciton-sand dtype functions
(2) .-
0.00127
350
-
0.00138
320
220
0.00244
180
-
0.00347
120
87
0.00288
150
85
._
Indirect exciton-s-type function Indirect exciton-sand d-type functions w:+q Indirect exciton--sand d-type functions
L$
=I
AND
7
SILICON
value was obtained from an analysis of the absorp-
tion spectrum appropriate to a material having simple band edges, the error quoted here merely being due to a scatter in the values obtained at different temperatures; the Lincoln Laboratory value was obtained more directly, but from a specimen in a state of strain which has been shewn to influence considerably the value of the energy gap.(le) This strain resulted from the difference in thermal contraction of the thin slab of germanium and the glass mounting to which it was cemented. On cooling to liquid-helium temperatures, a uniform isotropic strain in the plane of the germanium slab is thus set up. In the presence of a strain of this type, the valence-band degeneracy at k = 0 is lifted, giving two valence-band edges and correspondingly two groups of exciton states. At sufficiently small values of k, the surfaces of constant energy for the two valence bands are ellipsoids of revolution about the direction of the slab normal, being prolate ellipsoids for one band and oblate ellipsoids for the other. If we assume that
Table 2. The binding energies of the indirect excitons in silicon, using both s and (s+d)-type trial functions and band parameters quoted by KOHN@), and the corresponding values of the variational parameters rl and r2 for both zero and infinitely large spin-orbit coupling and for both positive and negative values of the eflective mass parameter (L-M).(15)
_
s-type function
(L-W><0
s- and d-type functions
(L--M)<0
Binding energy (eV)
2,
0.0120
50
-
A;
0.0132
44
33
A5
0.0137
42
30
0.0145
39
28
0.0133
44
30
j
(%
s- and d-type functions
s-type function
(L-M)><0
s- and d-type functions
(L-M)<0
s- and d-type functions
(L-W>0
0.0120
50
i -
A6
0.0130
44
31
A7
1 0.0124
47
33
A6
0.0124
47
33
A7
0.0130
44
31
8
T.
P.
MCLEAN
the splitting between the bands at k = 0 is large compared with the exciton binding energy, then it is this region of ellipsoidal energy-dependence which is important in determining the hole effective mass for use in the exciton wave-equation. Taking the conduction-band effective mass at k = 0 to be unaltered by the strain, the exciton functions F&r) then satisfy an equation similar to that for a donor state in germanium or silicon, and the ground-state binding energies can be calculated by the same method as has been used for the donor problem.(l7) It is found that the excitons due to both bands have approximately the same binding energy of about 0.0013 eV. Since the measured value of the valence-band splitting in the Lincoln Laboratory experiments was 0.005 eV,(ls) the exciton binding energy is as large as one-quarter of the splitting, so that the assumption on which the above estimate is based is not very well satisfied. However, the calculation indicates that the difference between the binding energies observed by the two experimental groups is not likely to be entirely accounted for by the strain in the Lincoln Laboratory specimen. As for the accuracy of the calculated binding energy, we may note that a similar theory applied to the ground states of acceptors in germanium(T) gives a binding energy which is about 20 per cent smaller than that measured experimentally, and the discrepancy is thought to be largely due to breakdown of effective-mass theory in the immediate vicinity of the impurity atom. However, the mean radius of the direct exciton wave-function in germanium is almost ten times that of the groundstate acceptor wave-function, so that effects due to the breakdown of effective-mass theory when the electron and hole are close together would be expected to be correspondingly smaller. Further, for the exciton there is no distortion of the lattice analogous to that introduced by the inclusion of the impurity atom which renders the effectivemass theory less accurate for the case of an acceptor. A better agreement between the calculated and experimental results for the direct exciton in germanium might therefore be expected. For the indirect exciton in germanium, the calculated ground-state binding energy is 0.0035 eV, and an excited state lies above this by an energy 0.0006 eV. The two binding energies represent substantial increases over the values obtained by
and
R.
LOUDON
using only an s-type trial function. Experimentally, the R.R.E. value for the ground-state binding energy is 0~0027f0~0004 eV, with an excited state at 0~0010~0~0001 eV above this, while for the same quantities Lincoln Laboratory measures 0.0033 f 0.0004 and 0.0010 eV, respectively. The reported ground-state binding energies are therefore consistent with a value of about 0.0030 eV, and the splittings between the two levels agree at 0.0010 eV. There is fair agreement between experimental and theoretical values for the ground-state binding energy, but for the splitting between the two levels the calculated value is significantly lower than that measured experimentally. We note that the energies calculated with the present trial functions are larger than the values of 0.0025 and 0.0033 eV calculated by the Lincoln Laboratory,@, 10) using the trial function exp{ - [u?za+ba(~a+ys)]}. For the various cases considered for the indirect exciton in silicon, the inclusion of the d-states in the trial function produces at most an increase of about 20 per cent over the binding energy found by using only an s-type trial function. The calculated ground-state binding energy is about 0.0120.014 eV, and the excited state lies above this by about 0~0005-0~0012 eV. These results are to be compared with the values measured at R.R.E. of about 0.01 eV for the binding energy, and 0.0055 eV for the splitting. The agreement between the calculated and observed ground-state binding energies is fair considering the uncertainty in the latter, but the agreement between the splittings is so bad as to suggest that the excited state observed experimentally is not that whose energy we have calculated. The mean radii of the indirect exciton wavefunctions are smaller, particularly in silicon, than that of the direct exciton in germanium, and, on the basis of the above remarks concerning the breakdown of effective-mass theory when the electron and hole are close together, we should expect the calculated values for these indirect exciton binding energies to be less accurate than the calculated binding energy of the direct exciton in germanium. However, the mean indirect exciton radii are larger than the ground acceptor state radii in the same semiconductor by a factor of about three, so that the accuracy of the exciton calculations would be expected to be correspondingly better.
EXCITON
LEVELS
IN
GERMANIUM
AND
SILICON
9
2. ZWERDLINGS., ROTH L. M. and LAX B., Phys. Rev. 109, 2207 (1958); J. Phys. Chem. Solids 8, 397 (1959). BUTTON K. J., ROTH L. M., KLEINER W. H., ZWERDLING S. and LAX B., Phys. Rev. Letters 2, 161 (1959). 3. DRE~SELHAUS G., J. Phys. Chem. Solids 1, 14 (1956). 4. WANNIER G. H., Phys. Rev. 52, 191 (1937). 5. KOHN W., Solid State Physics 5, 257 (1957). 6. KOHN W. and SCHECHTERD., Phys. Rev. 99, 1903 (1955). 7. SCHECHTERD., Thesis, Carnegie Institute of TechAcknowledgements-The authors wish to thank the nology (1958). Lincoln Laboratory for pre-publication copies of their 8. KOSTER G. F., Solid State Physics, 5, 173 (1957). results. Their thanks are also due to Dr. D. P. JENKINS 9. KAHN A. H., Phys. Rev. 97, 1647 (1955). for help in programming some parts of the calculation 10. ROTH L. M., LAX B. and ZWERDLINGS., Phys. Rev for the computer. to be published. This work was principally carried out in the summer 11. ELLIOTT R. J., Phys. Rev. 96, 266 (1954). of 1958, when one of us (R.L.) was a Vacation Student 12. HERMAN F., Phys. Rev. 95. 847 (1954). at the Royal Radar Establishment. He would like to take 13. BOUCKAERTL. P., SMOLU&OW&U g. and WIGNER this opportunity to thank Dr. G. G. MACFARLANEand E.. Phvs. Rev. 50. 58 (1936). his group for a pleasant and stimulating time. 14. ELLIOTT-R. J., Phyi. Rev. 96: 280 (1954). This paper is published by permission of the ConG., KIP A. F. and KITTEL C., Phys. 15. DRESSELHAUS troller, Her Britannic Majesty’s Stationery Office. Rev. 98, 368 (1955). 16. MACFARLANEG. G., MCLEAN T. P., QUARRINGTON REFERENCES J. E. and ROBERTSV., Phys. Rev. Letters 2, 252 (1959). 1. MACFARLANEG. G., MCLEAN I?. P., QUAFUUNGTON 17. KOHN W. and LUTTINGERJ. M., Phys. Rev. 98, 915 J, E. and ROBERTSV., Phys. Rev. 108, 1377 (1957); Proc. Phys. Sot. Lond. 71, 863 (1958); Phys. Rev. (1955). 18. KLEIN& W. H. and ROTH L. M., Phys. Rev. Letters 111, 1245 (1958); J. Phys. Chem. Solids 8, 388 2, 334 (1959). (1959).
For each of the three types of exciton there is fair agreement between the measured and calculated values of the ground-state binding energies. A more detailed comparison of theory with experiment must await more accurate experiments on silicon and a reconciliation of the two experimental groups whose differing observations of the direct exciton in germanium we have discussed.