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On some observable properties of longitudinal excitons
J. Phys. Chem. Solia3 Pergamon
Press 1960. Vol. 12. pp. 276-284.
ON SOME OBSERVABLE
PROPERTIES EXCITONS
J. J. HOPPIBLD Bell Telephone
Printed in Great Britain.
OF LONGITUDINAL
and D. G. THOMAS
Laboratories,
Murray Hill, New Jersey
(Received 6 July 1959)
Abstract-The continuum theory of the dielectric properties of direct excitons is briefly developed for uniaxial crystals (optically isotropic crystals are a special case). The finite energy difference between longitudinal and transverse excitons can quench the linear Zeeman effect in certain geometries in both isotropic and uniaxial crystals. In uniaxial crystals it should be possible to observe “longitudinal” excitons in optical absorption. These “longitudinal” excitons are expected to have energies and oscillator strengths which depend strongly on the direction of propagation of the exciton. “Longitudinal” excitons are not observable along the principal axes of a crystal. Optical absorption measurements carried out on hexagonal ZnO have demonstrated the existence of these excitons. Even for the rather complicated case of ZnO, in which two interacting longitudinal excitons are observed, the simple continuum theory is shown to agree quantitatively with experiment.
1. INTRODUCTION IT has long been recognized that in ionic cubic crystals there is a finite difference between the energy of a longitudinal optical phonon and the energy of a transverse optical phonon at k = 0. {Throughout the paper, “at k = 0" is to be understood to describe the limiting process in which first the crystal size becomes infinite and then k is allowed to approach zero.) This energy difference arises from the long-range coulomb interaction of the polarization field. For the same reason, an energy difference is to be expected between optically active longitudinal and transverse excitons at k = 0 in cubic crystals. The energy difference between longitudinal and transverse excitons in molecular crystals has been shown from a theoretical point of view by Fox and YATSIV~) and for excitons in cubic crystals by PEKAR(~)and HELLER and MARCUS(~). Infinitesimally close to k = 0,the excitons (or phonons) can be classified as purely longitudinal or purely transverse for any direction of k in cubic crystals. Light, being a purely transverse wave, interacts only with the transverse excitons (or phonons) in cubic crystals. The longitudinal excitons are therefore optically unobservable in cubic crystals. We shall show that longitudinal excitons and the nonanalytic nature of E(k) for
excitons at k = 0 are easily observed in hexagonal crystals. In Section 2, a continuum theory of the dielectric properties of excitons in a uniaxial crystal is developed. The theory is based upon replacing the crystal dielectric oscillators and their local field effects by a continuum of virtual oscillators. These virtual oscillators without local field effects have dielectric properties equivalent to the actual crystal oscillators with local field effects. The energies of optically active longitudinal and transverse excitons can then be obtained from the solution of a simple eigenvalue problem. A measurement of the optical absorption due to excitons in ZnO is described in Section 3. Two “longitudinal” excitons are observed. The energies and strengths of these lines can be calculated from measurements made on transverse excitons in ZnO and the theory of Section 2. The measured energies and strengths are both in agreement with the simple theory of Section 2. In Section 4, it is shown that there exist some observable effects of the energy difference between longitudinal and transverse excitons even in cubic excitons are not crystals, where longitudinal directly observable. In cubic crystals the energy difference between longitudinal and transverse excitons lifts a degeneracy which would otherwise
276
OBSERVABLE
PROPERTIES
OF
be present, and can thereby produce a quenching of the linear Zeeman effect in excitons. 2. THEORY It is most convenient to separate the radiation field from electrostatic interactions in a solid by use of the coulomb gauge. The coulomb interactions between charged particles in the crystal are then the instantaneous coulomb interactions; the radiation field then contains only transverse waves. In this form,’ the exciton is a particle which represents the quantization of a classical polarization field.@) The dipole distribution associated with a hydrogenic exciton cannot be calculated unless detailed wave functions of electrons and holes are known. In the absence of such knowledge, two simple approximations could be used; the continuum approximation (the dipole moment is assumed to be spread uniformly) and the point dipole approximation. The dipole moment distribution which is assumed to be a point dipole (in the point dipole approximation) or to be spread uniformly over an entire unit cell (in the continuum approximation) is the dipole moment distribution r%(r’-r)z4T(r’)t42(r’)dW. P(r) = 1 cell In this expression, for simple bands having k = 0 minima, 242is the periodic part of the Bloch function of the conduction band at k = 0,and ur is the periodic part of the valence-band Bloch function at k = 0.The spatial extent of the exciton does not matter. Neither of these approximations represents the actual state of affairs in a real crystal. To avoid this difficulty, we develop the continuum theory of virtual oscillators. It is well known that for long wavelengths, there exists a continuum virtual oscillator without local field effects which is equivalent to point dipole oscillators with local field effects.(s) We will make use of these equivalent continuum oscillators. The equivalent continuum model avoids the problem of relating actual observed excitons to the microscopic theory of the dielectric, and permits the calculation of relationships between observed parameters independent of the microscopic model. It is sufficient to treat the unquantized form of dielectric theory, with excitons replaced by
LONGITUDINAL
EXCITONS
277
polarization oscillators. In a uniaxial crystal there are two kinds of polarization oscillators. One kind is constrained to have its polarization in the ~,y plane (normal to the c-axis). The second set is constrained to have its polarization vector in the z-direction (along the c-axis). The first kind of polarization oscillator represents excitons belonging to the representation (~,y) in hexagonal crystals. These will be denoted by a subscript 1. The second kind of polarization oscillator represents excitons belonging to the representation (2) in hexagonal crystals. These will be denoted by a subscript 11.From this point of view, a cubic crystal is a special case of an hexagonal crystal in which, for each set of polarization oscillators of the first kind, there is a set of polarization oscillators of the second kind having the same strength and the same natural frequency. The constitutive equation can most simply be written by separating the longitudinal and the transverse parts of the electric field. The transverse part is to be thought of as representing the radiation field only, while the longitudinal part belongs to the bulk properties of the crystal in the coulomb gauge.* The constitutive relations then are
1 a2p,,t ___+ b,, = IS,,tex [(Etr&n8VerBe+ (w&2 at2 -I- Longitudinal) X
e]
= B,.E, -
1 2
w,, ,2
asp 1.t -+Pe,
(1)
, = B, ,te [@transverse
+
ata
-I- Longitudinal) .
= B,E,
e] (2)
* This separation appears highly artificial in classical theory. The origin of the separation is the use of the coulomb gauge. The use of this gauge is arbitrary, but extremelj convenient, because the quantum-mechanical solid is held together by coulomb forces. In the classical dielectric theory, however, one does not ask what holds the dielectric together on a microscopic level, and gauge becomes unimportant. In making the transition from the quantum-mechanical system to the classical continuum, one loses the apparent reason for preserving the coulomb gauge. It is preserved in orderto includethe staticcoulomb interaction in the crystal Hamiltonian. In the classical case, the sense of the physics is better preserved if the separation is made, in that the energy levels thus derived will then correspond to the dispersion frequencies of oscillators observed by means of a transverse electromagnetic wave.
J.
278
J.
HOPFIELD
In these equations, e is a unit vector in the direction of the hexagonal c-axis, here taken to be the x-direction. In general, there will be many different oscillators i, each having its own characteristic frequency. The numbers /3s are the zero-frequency polarizabilities of the oscillators i. It should be noted that in the equivalent continuum model, local field effects are taken into account in the frequencies and polarizabilities of the oscillators. The total polarization field P is given by the sum of the individual contributions P,i and Iir,+ The electric field can be separated into the longitudinal and transverse components in a plane wave representation. Let k be the propagation vector of a plane wave. The transverse component of the electromagnetic field is due to the incident radiation. The longitudinal component of the electric field is caused by the charge density of the longitudinal component of the polarization field, and is given (for propagation vector k) by
k
Equation (3) is equivalent to the condition of no “free charge. “(5) The exciton energies are determined by the disperson relation which can be obtained by combining equations (l), (2) and (3), at the same time setting Etram%erse equal to zero (i.e. no interaction with radiation is considered in finding the exciton energies). In order to gain a qualitative appreciation of the dispersion relation, we treat the special case in which one Ok,* has a frequency sufficiently removed from all other resonant frequencies that all other oscillators will contribute a constant amount to a background dielectric constant E. For convenience, this background term will be taken to be isotropic. In a cubic crystal, there would necessarily be an oscillator o,,,+ having the same frequency as aI,*. The idealized &axial crystal treated here deviates as far as possible from a cubic crystal, and has no 11oscillator corresponding to W&i.
and
D.
G.
THOMAS
Equations
(1) and (3) then become :
1 3P,
-
w12
-+P,
at2
brgitudinal
= pleX [&ongitudinalXe]
=
The contribution of all other oscillators far removed in frequency enters through E. The dispersion relation is then simply:
w2(k> = PL+~~L(ex(kxe)j(k.PL).(4) PL--cl? E There
are two solutions to equation (4):
P,.k = 0 : d(k) = wI
(transverse
mode)
P,.(kXe)=O:d(k)=wt (mixed ‘mode). The transverse mode has its polarization vector in the x-y plane and normal to k. The mixed mode has its polarization vector in the projection of k on
FIG. 1. E(k)for an isolated 1 exciton near k = 0 in the plane kv = 0. The energy surface ST belongs to the pure transverse exciton. The energy surface 5’~ beIongs to the exciton which is purely longitudinal for k parallel to the c-axis, and purely transverse for k perpendicular to the C-Z&.
the x-y plane. The transverse mode is purely transverse; the mixed mode is in general mixed, going from pure longitudinal for k perpendicular to the c-axis to pure transverse fork parallel to the c-axis. The energy surface for the two modes is sketched in Fig. 1 for the plane ink space defined
OBSERVABLE
PROPERTIES
by K, = 0. For k perpendicular modes are related by
OF
to the c-axis, the
the usual LYDDANE-SACKS-TELLER@) relation for cubic crystals. In order to observe the mixed modes in hexagonal crystals, it is necessary to use light whose propagation vector in the crystal makes an angle 4 # 90” or 0” with the c-axis. For light polarized in the plane containing both k and the c-axis, only the mixed mode would be observed. For light polarized perpendicular to the c-axis, only the transverse mode would be observed. For these two experiments, respectively one would obtain the results in Table 1.
Table 1 Frequency of observed line
Oscillator strength of absorption line (in proper units)
In cubic crystals, due to the presence of a 11oscillator, all excitons would be purely transverse or purely longitudinal.
Optical phonons The effects which have been described so far exist for both excitons and optical phonons in hexagonal crystals. In the case of optical phonons, the effects should be particularly small. For the optical phonons, wI and o,, are usually quite close. The effect of the polarization field interaction is to try to make all polarization normal modes either purely transverse or purely longitudinal. The strength of this electrostatic interaction is given by
For most hexagonal ionic crystals the strengthbf
LONGITUDINAL
EXCITONS
279
this electrostatic interaction is experimentally about 0~14. On the other hand, the crystal field splitting 1o L-w,ll in typical materials is roughly wJO0. (The experimental numbers used here are those appropriate to hexagonal SiC.0) Because the electrostatic energy is so much larger than the crystal field splitting, all modes will be very nearly pure longitudinal and pure transverse. In restrahl measurements, the longitudinal phonons will be only very weakly active for nonprincipal directions of k. In practice, these lines might be observed for certain polarizations of the incident light in nonprincipal directions. In crystals which deviate further from optical isotropy, the effects would be correspondingly larger.
Excitons In general, it would be rather difficult to investigate by optical means the “longitudinal” (mixed) phonons in hexagonal crystals due to the dominance of the electrostatic interaction over the crystal field splittings. For excitons in hexagonal crystals, the electrostatic interaction is no longer dominant (4+3 is very small for excitons of large radius), The absorption which one hopes to observe due to mixed excitons is qualitatively different from that which might have been off-hand expected. A few degrees from a principal direction, one expects to see new absorption lines which differ in energy from all ordinary (transverse) exciton lines seen in any principal direction. These absorption lines interact with all the light in a properly chosen geometry (as would not be the case for a slightly misaligned plane of polarization), and can easily be strong enough to lead to complete absorption in crystals a few microns thick for angles of deviation as small as 5” from principal directions. These absorption lines are due to “longitudinal” excitons (i.e. mixed excitons having chiefly a longitudinal component). In ZnO, a situation rather similar to that described in the text occurs for the excitons. There are two modes (excitons) P,I and P,z which have been identified by appropriate reflection measurements.@) The lines are well enough separated from all other reflection lines that E is approximately constant and isotropic. The parameters wr I, wsI, /3tL, j3sL and E have been determined by an analysis of the reflectivity measured with E and k perpendicular to the c-axis. The values of
280
J. f. HOPFIELD
the parameters are 01~ = 3_376seV, ~9~ = 01~ + 0*006s, pr I = 0*0074 f 15 per cent. 13s1 = 0.024 f 15 per cent and E = 5.0 f 10 per cent at 42°K. The energies wr I and wzI are slightly temperature-dependent, but their difference (which is the energy of importance for the calculations) is not. Two longitudinal excitons, corresponding to the two oscillators WI& and wsl are expected. The experiment is designed to observe these “longitudinal” excitons by virtue of their small transverse part (which does not exist for cubic crystals). The frequencies and strengths of the expected mixed excitons can be found by solving equations (1) and (3) for the case in which only two oscillators are present. The solution of the eigenvalne problem defined by (1) and (3) for the transverse mode energies is trivial. There are two “longitudinal” (mixed) solutions. One eigenstate has almost equal amounts of the two polarization fields w L1and o 19 oscillating 180” out of phase, with a resultant small energy shift and small strength. The higher-frequency eigenstate has almost equal amounts of the two polarization fields oscillating in phase, with a resultant large energy shift and large strength. Qualitatively, the lower-energy longitudinal line has a large effective E due to the upper line, so its shift is small. The higher-energy longitudinal line has a small effective e due to the lower line, so its shift is large. The two modes wrl and wsl are the lowestlying excitons of appreciable strength. An exciton of approximately the same strength lies approximately O-040 eV above these two excitons, out of the current range of interest. All excited states of these two low-lying excitons also lie more than 0*040 eV above these two strong lines. In addition, there exists a very weak (exciton) mode PI, having almost the same frequency as P,r. The optical activity of this weak line is associated with the small spin-orbit coupling which splits the excitons P,l and P L~.All other excitons falling in the region of experimental interest are known from group theory (confirmed by experiment) to be optically inactive.(9) 3. Expm An experiment was performed on ZnO to check the theory and to determine whether some absorption lines which had already been observed in ZnO
and
D.
G.
THOMAS
were due to “longitudinal” excitons. The absorption constant of ZnO was measured as a function of the angle of incidence. The geometry used is shown in Fig. 2. A ~ntinu~ source of illumination was used, and the transmitted light analyzed
FIG. 2. The experimental geometry of k,E and c-axis. All vectors drawn lie in the plane of the drawing.
by a Bausch and Lomb spectrograph. A focused beam of 3” full angle was used because of the small size of the ZnO crystals. Densitometer traces of the spectrograph photographs were analyzed to determine the absorption constant. Polarized light was used (as is indicated in the figure) in order to calibrate the transmission measurements. Small deviations of the plane of polarization from the nominal plane of polarization are unimpo~~t. Such an error would result in the presence of a small amount of light polarized perpendicular to the c-axis. For this component, the absorption constant is so great (about 105 cm-l) in the spectral region of interest that essentially none of this unwanted component can get through the 10,~ thick crystal. The qualitative results of the absorption experiment at 4.2”K are shown in Fig. 3. The energies wI1 and w,_9 are indicated by arrows. Line A is due to the previously mentioned weak exciton wll, (This exciton is “weak” compared with ~~1~ which, as a transverse exciton, has a peak absorption coefficient of about 10s cm-l.) The sloping background is due to the tail of a very strong exciton absorption line lying to higher energy. Lines B and C are the lines of interest. Within the accuracy of experimental measurement, these lines lie at the same energy for 8 = O”, 2”, 4” and 6”. Lines B and C have roughly the same strength at 2* as at O”, but increase in strength rapidly as 0
OBSERVABLE
PROPERTIES
OF
LONGITUDINAL
EXCITONS
281
background absorption) as a function of angle for line B at 77°K. The line width for line B was also measured at 77°K to provide a means of evaluating the oscillator strength. This width was independent of angle over the range of angles measured. The experimental points fall within experimental accuracy on a straight line.
i 3vJoO 8 400
200 00
e=40
2000
O,S>O
D A
5 L 1000 Y 6 400
100
X,B
-,a=70+15-482
C
150 a, CM-’
0
‘UlP
3.395
3.385
100
011
50
3.37 ENER GY
FIG. 3. The absorption spectrum of ZnOas afunctionof B at 4OK. Note that the absorption constant scale is nonlinear.
becomes larger than 4”. Line A, on the other hand, remains constant in strength. The rapidly increasing strengths of lines B and C is characteristic of excitons which are nearly completely longitudinal; the constant strength of line A indicates that this exciton is (as was assumed) chiefly transverse. The same absorption experiment was also carried out at 77°K to obtain more quantitative information about the line strength as a function of angle. At 77°K the exciton lines are broadened and peak absorption coefficients are therefore smaller. Unfortunately, due to the increased background absorption, it was possible to measure only the behaviour of line B. The results of this experiment are given in Fig. 4, which shows the peak absorption coefficient (after the subtraction of the
,,,I
345
I
I
6
8 7 e IN DEGREES
I
I
1
9
10
11
Fig. 4. The peak absorption coefficient of peak B at 77°K. as a function of 0. The angular scale is linear in ea. The circles are experimental points for B > 0 and the crosses for ~4< 0”.
From the theory of Section 2, the parameters E, PII, rB1s, w *I and w 1s given at the end of Section 2, and the measured width of line B at 77”K, the complete behaviour of lines B and C can be compputed. A comparison of theory (for small angles) and experiment is given below. The energy 0~1 is taken as a reference zero. A refractive correction is included in the calculations to relate the exterior angle 8 to the interior angle 4. The “Theory” column of Table 2 is computed from the reflection measurements alone; there are no free parameters to adjust.
Table 2
I Energy of line B Energy of line C Strength of line B
at 4.2”K
( Strength of line C Peak absorption coefficient of line B at 77°K
Theory
I
Experiment
0~0010 0.0158 0.035
0.0016 eV 0*0168 eV 0.05 + 100x--50%
c$$& = IO@
(“p& = 70+15*4L9s
282
J.
J.
H‘OPFIELD
and
Most of the experimental data agree with the theory within the error introduced into the theoretical calculation by the uncertainty of the parameters used. It should be noted that the ratio of strengths and the slope of the peak absorption are sensitive functions of 0s ,_ - wl I, changing about 30 per cent for a change of 0.001 eV in w2 I - w1 I. One point, however, is in definite disagreement with theory. The theory states unambiguously that the longitudinal lines should be unobservable for 0 = 0”, whereas Figs. 3 and 4 indicate* that the lines are small but present. We believe that their presence is due not to experimental error (the 0” residual absorption is an order of magnitude too large to be due to the 3” convergent beam), but is instead due to the failure of the infinite-wavelength theory used. Ordinarily, in treating optical processes in crystals, the effect of the finite wave vector of the light in the crystal is neglected. The error thus introduced is of the order (Ku)s, where K is the wave vector of the light in the crystal and a is a typical lattice dimension. In the case at hand, (lza)s M 10-d. F rom the known strengths of the lines w I~ and o 1s and the width of peak B, one could then expect a residual “error” for amax of the order of 100 cm-l. This is of the same order of magnitude as the observed residual absorption. The effect can be regarded as due to the small but finite photon momentum or, equivalently, to the fact that optical transitions are not precisely vertical. Such effects are not unkown in cubic crystals, and must be invoked, for example, to explain the existence of the “n = 1” line in the exciton absorption spectrum of CusO.(lo) The observed widths of the two longitudinal exciton lines differ considerably, as can be seen in Fig. 3. The decreased lifetime of the higherenergy longitudinal exciton may simply be due to the fact that the higher-energy exciton of small k may be scattered by impurities or phonons into states of the lower-energy exciton of higher k. NO corresponding states exist for scattering of the longitudinal exciton. A second lower-energy possibility is that the scattering is related to the size of the oscillating dipole, and that the large dipole matrix elements of the higher * The mixed mode at 0” angle of incidence was below noise level for the experiment at 77°K. Quantitative measurements could be made at 77°K only for external angles of incidence greater than l-5”.
D.
G.
THOMAS
exciton lead to strong interactions and rapid scattering. A few qualitative points should be emphasized concerning the “longitudinal” effects in hexagonal crystals. First, as is well known in the case of cubic crystals, group theory can be misleading with reference to exciton lines. The usual analysis of the group theory of exciton lines applies at k = 0.To apply this result to excitons of small but finite k is not strictly permissible, because of the nonanalytic nature of the exciton energy surface. In cubic crystals the errors which are made by extending the results of group theory at k = 0 to finite k are not important in determining selection rules, since the chief error is the energy difference between the longitudinal (unobservable) exciton and the transverse excitons. In hexagonal crystals, on the other hand, because the “longitudinal” (mixed) modes are optically observable, it is necessary to understand the nature of the singularity in E(k) in order to understand quantitatively the positions and oscillator strengths of observed exciton absorption lines. Second, it is essential to use single crystals in investigating the exciton states in hexagonal crystals. In the case of ZnO, for instance, the use of polycrystalline films would have produced a confusing jumble of experimental results due to the varying energy position of the “longitudinal” exciton line. Finally, the finite angular aperture of an optical system can contribute strongly to observed effects. For the small-aperture Bausch and Lomb spectrometer used in this study, the correction due to the finite aperture was about 1.5 per cent of the observed residual absorption at 0”. 4. OTHER EFFECTS IN CUBIC CRYSTALS
The linear Zeeman effect in certain crystal geometries can be quenched by the energy difference between the longitudinal and transverse excitons. As an example of this, we consider the case of the CusO yellow exciton series which has been studied extensively by GROSS and co-workers.(ls) In CusO the direct transition is forbidden, and excitons are observed in p-like orbital states. (The n = 1 line, though present, is very weak.) The electron spin effectively does not enter the problem. Let the magnetic field be along the z-direction, let y be the direction of the incident light and let z be the direction of polarization of the incident light-
OBSERVABLE
PROPERTIES
OF
At first sight, one would have expected a linear Zeeman effect to be observed in this geometry. However, because of the direction of propagation of the incident light, the energy difference between the longitudin~ and transverse excitons is important. The Hamiltonian matrix for the linear Zeeman effect in this geometry is :
LONGITUDINAL
checked. If the light is instead sent aEongthe magnetic field, the Hamiltonian becomes :
P, H=P, ps i
where
S = E[&+y)-I]. E is the energy of the exciton and @ and z are as defined earlier in the text. The existence of the term 6 will quench the linear Zeeman effect if 6 2 $3. For the case of CusO, in the largest fields used by GROSS,the effect would be quenched if eirli3 -~lO-~ i rf the electron and hole masses differ by a factor of 2. For comparison, in ZnO h/3,/~M 4~ 10-s. The CusO excitons are “forbidden,” so &r/3/~ would be less by a factor of exciton radius s ( lattice constant 1 than for ZnO. On the other hand, the excitons are more tightly bound in CusO, which would tend to make ~T$/E larger. Unfortunately, ~+I/E is not known from experiments in CusO. GROSS has observed the absence of a linear Zeeman effect in CusO, and has attributed this absence to the equality of the electron and hole masses (f~ = 0). It would appear that quenching due to longitudinal excitons can equally well be the cause of the absence of the linear Zeeman effect in the low-lying exciton lines of CusO.* The quenching hypothesis can be rather easily * NOM added in grooj: One of us (D.G.T.) has since made measurements which indicate that 4?rfl/c is too small to quench the Zeeman effect in CusO.
283
EXCITONS
PS
P,
PZ
E
;EtH
0
-+H
E
0
0
0
ES-8 f
and a linear Zeeman effect will be obtained if p is not zero. This quenching is to be predicted for all direct exciton Zeeman effects in which the Zeeman splitting is expected to arise from the splitting of an exciton which belongs to a three-dimensional irreducible representation. The stronger the exciton line, the more important this quenching becomes. Conversely, the longitudinal-transverse energy difference goes to zero faster than the exciton binding energy for highly excited exciton states, and so rapidly becomes unimportant for excited states. There is no effect for continuum states. Linear Zeeman effects in excitons can sometimes arise from a near degeneracy of exciton levels of different symmetry which are mixed by a magnetic field (such a case has been observed by THOMAS in GO). Accidental degeneracies can easily arise in hydrogenic excitons, for spin-spin interactions between electrons and holes are very short-range forces. If the continuum approximation is not valid, it is possible for the long-range coulomb effects to shift the transverse exciton states as well as the longitudinal states. This shift could remove the accidental degeneracy which is responsible for the linear Zeeman effect in the case of near degeneracy that might otherwise have been expected, and lead to a quenching of this Zeeman effect also. All the effects treated so far have referred to direct excitons. The effect of the long-range electrostatic interaction on indirect excitons is a much more difftcult problem, for deviations from the continuum approximations must necessarily occur for excitons of large k. (Such deviations are necessary to preserve the periodicity of the exciton in the Brillouin zone.) The effects will probably be far less interesting. It is clear, however, that the longrange electrostatic interaction should give rise to &-rite energy corrections and configuration mixings which do not appear explicitly in the effective mass
284
J.
J.
HOPFIELD
approximation. Suppose, for example, the holes were extremely heavy compared to the electrons. The usual effective mass problem would yield an exciton binding energy which would be the same as the binding energy of a donor. The electrostatic effects, however, could produce an energy shift from this calculated energy which could be as large as 20 per cent of the exciton binding energy for some states. This effect is not due to polaron effects, but is caused by the same electrostatic interactions which lead to the finite energy difference between longitudinal and transverse excitons near k = 0.
REFERENCES 1. Fox
D and YATSIVS., Phys. Rev. 108, 938 (1957).
and
D.
G.
THOMAS
2. PBWR S. I., Zh. eksp. teor.jk 8, 360 (1958). 3. HELLW W. R. and MARCUSA., Phys. Rev. 84,809 (1951). 4. HOPFIELDJ. J., Phys. Rev. 112, 1555 (1958). 5. BORN M. and HUANG I-C., Dynamical Theory of Cvystal Lattices, Section 7. Oxford University Press (1954). 6. LYI~DANB R. H., SACHSR. G. and TELLERE., Phys. Rev. 59, 673 (1941). 7. SPIKER W. G., KLEINMAND. and WALSHD., Phys. Rev. 113,127 (1959). 8. THOMASD. G., Bull. Amer. Phys. Sot. 4,154(1959). 9. HOPFIELDI. J.. Bull. Amer. Phvs. Sot. 4.154 (1959). 10. GROSSE. ‘F: hd ZAKI-LUKHJ&A B. $. , J.. Tech. Phys. (U.S.S.R.) 2, 1808 (1957). GROSSE. F., ZAKHARCHENYA B. P. and PAVENSKII P. P.. Zh. tekh. fiz. 2.2018 (19571. GROSSE. F. and ZA&ARC&YA b. P., J. Phys. Radium 18,68 (1957).
Press 1960. Vol. 12. pp. 276-284.
ON SOME OBSERVABLE
PROPERTIES EXCITONS
J. J. HOPPIBLD Bell Telephone
Printed in Great Britain.
OF LONGITUDINAL
and D. G. THOMAS
Laboratories,
Murray Hill, New Jersey
(Received 6 July 1959)
Abstract-The continuum theory of the dielectric properties of direct excitons is briefly developed for uniaxial crystals (optically isotropic crystals are a special case). The finite energy difference between longitudinal and transverse excitons can quench the linear Zeeman effect in certain geometries in both isotropic and uniaxial crystals. In uniaxial crystals it should be possible to observe “longitudinal” excitons in optical absorption. These “longitudinal” excitons are expected to have energies and oscillator strengths which depend strongly on the direction of propagation of the exciton. “Longitudinal” excitons are not observable along the principal axes of a crystal. Optical absorption measurements carried out on hexagonal ZnO have demonstrated the existence of these excitons. Even for the rather complicated case of ZnO, in which two interacting longitudinal excitons are observed, the simple continuum theory is shown to agree quantitatively with experiment.
1. INTRODUCTION IT has long been recognized that in ionic cubic crystals there is a finite difference between the energy of a longitudinal optical phonon and the energy of a transverse optical phonon at k = 0. {Throughout the paper, “at k = 0" is to be understood to describe the limiting process in which first the crystal size becomes infinite and then k is allowed to approach zero.) This energy difference arises from the long-range coulomb interaction of the polarization field. For the same reason, an energy difference is to be expected between optically active longitudinal and transverse excitons at k = 0 in cubic crystals. The energy difference between longitudinal and transverse excitons in molecular crystals has been shown from a theoretical point of view by Fox and YATSIV~) and for excitons in cubic crystals by PEKAR(~)and HELLER and MARCUS(~). Infinitesimally close to k = 0,the excitons (or phonons) can be classified as purely longitudinal or purely transverse for any direction of k in cubic crystals. Light, being a purely transverse wave, interacts only with the transverse excitons (or phonons) in cubic crystals. The longitudinal excitons are therefore optically unobservable in cubic crystals. We shall show that longitudinal excitons and the nonanalytic nature of E(k) for
excitons at k = 0 are easily observed in hexagonal crystals. In Section 2, a continuum theory of the dielectric properties of excitons in a uniaxial crystal is developed. The theory is based upon replacing the crystal dielectric oscillators and their local field effects by a continuum of virtual oscillators. These virtual oscillators without local field effects have dielectric properties equivalent to the actual crystal oscillators with local field effects. The energies of optically active longitudinal and transverse excitons can then be obtained from the solution of a simple eigenvalue problem. A measurement of the optical absorption due to excitons in ZnO is described in Section 3. Two “longitudinal” excitons are observed. The energies and strengths of these lines can be calculated from measurements made on transverse excitons in ZnO and the theory of Section 2. The measured energies and strengths are both in agreement with the simple theory of Section 2. In Section 4, it is shown that there exist some observable effects of the energy difference between longitudinal and transverse excitons even in cubic excitons are not crystals, where longitudinal directly observable. In cubic crystals the energy difference between longitudinal and transverse excitons lifts a degeneracy which would otherwise
276
OBSERVABLE
PROPERTIES
OF
be present, and can thereby produce a quenching of the linear Zeeman effect in excitons. 2. THEORY It is most convenient to separate the radiation field from electrostatic interactions in a solid by use of the coulomb gauge. The coulomb interactions between charged particles in the crystal are then the instantaneous coulomb interactions; the radiation field then contains only transverse waves. In this form,’ the exciton is a particle which represents the quantization of a classical polarization field.@) The dipole distribution associated with a hydrogenic exciton cannot be calculated unless detailed wave functions of electrons and holes are known. In the absence of such knowledge, two simple approximations could be used; the continuum approximation (the dipole moment is assumed to be spread uniformly) and the point dipole approximation. The dipole moment distribution which is assumed to be a point dipole (in the point dipole approximation) or to be spread uniformly over an entire unit cell (in the continuum approximation) is the dipole moment distribution r%(r’-r)z4T(r’)t42(r’)dW. P(r) = 1 cell In this expression, for simple bands having k = 0 minima, 242is the periodic part of the Bloch function of the conduction band at k = 0,and ur is the periodic part of the valence-band Bloch function at k = 0.The spatial extent of the exciton does not matter. Neither of these approximations represents the actual state of affairs in a real crystal. To avoid this difficulty, we develop the continuum theory of virtual oscillators. It is well known that for long wavelengths, there exists a continuum virtual oscillator without local field effects which is equivalent to point dipole oscillators with local field effects.(s) We will make use of these equivalent continuum oscillators. The equivalent continuum model avoids the problem of relating actual observed excitons to the microscopic theory of the dielectric, and permits the calculation of relationships between observed parameters independent of the microscopic model. It is sufficient to treat the unquantized form of dielectric theory, with excitons replaced by
LONGITUDINAL
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277
polarization oscillators. In a uniaxial crystal there are two kinds of polarization oscillators. One kind is constrained to have its polarization in the ~,y plane (normal to the c-axis). The second set is constrained to have its polarization vector in the z-direction (along the c-axis). The first kind of polarization oscillator represents excitons belonging to the representation (~,y) in hexagonal crystals. These will be denoted by a subscript 1. The second kind of polarization oscillator represents excitons belonging to the representation (2) in hexagonal crystals. These will be denoted by a subscript 11.From this point of view, a cubic crystal is a special case of an hexagonal crystal in which, for each set of polarization oscillators of the first kind, there is a set of polarization oscillators of the second kind having the same strength and the same natural frequency. The constitutive equation can most simply be written by separating the longitudinal and the transverse parts of the electric field. The transverse part is to be thought of as representing the radiation field only, while the longitudinal part belongs to the bulk properties of the crystal in the coulomb gauge.* The constitutive relations then are
1 a2p,,t ___+ b,, = IS,,tex [(Etr&n8VerBe+ (w&2 at2 -I- Longitudinal) X
e]
= B,.E, -
1 2
w,, ,2
asp 1.t -+Pe,
(1)
, = B, ,te [@transverse
+
ata
-I- Longitudinal) .
= B,E,
e] (2)
* This separation appears highly artificial in classical theory. The origin of the separation is the use of the coulomb gauge. The use of this gauge is arbitrary, but extremelj convenient, because the quantum-mechanical solid is held together by coulomb forces. In the classical dielectric theory, however, one does not ask what holds the dielectric together on a microscopic level, and gauge becomes unimportant. In making the transition from the quantum-mechanical system to the classical continuum, one loses the apparent reason for preserving the coulomb gauge. It is preserved in orderto includethe staticcoulomb interaction in the crystal Hamiltonian. In the classical case, the sense of the physics is better preserved if the separation is made, in that the energy levels thus derived will then correspond to the dispersion frequencies of oscillators observed by means of a transverse electromagnetic wave.
J.
278
J.
HOPFIELD
In these equations, e is a unit vector in the direction of the hexagonal c-axis, here taken to be the x-direction. In general, there will be many different oscillators i, each having its own characteristic frequency. The numbers /3s are the zero-frequency polarizabilities of the oscillators i. It should be noted that in the equivalent continuum model, local field effects are taken into account in the frequencies and polarizabilities of the oscillators. The total polarization field P is given by the sum of the individual contributions P,i and Iir,+ The electric field can be separated into the longitudinal and transverse components in a plane wave representation. Let k be the propagation vector of a plane wave. The transverse component of the electromagnetic field is due to the incident radiation. The longitudinal component of the electric field is caused by the charge density of the longitudinal component of the polarization field, and is given (for propagation vector k) by
k
Equation (3) is equivalent to the condition of no “free charge. “(5) The exciton energies are determined by the disperson relation which can be obtained by combining equations (l), (2) and (3), at the same time setting Etram%erse equal to zero (i.e. no interaction with radiation is considered in finding the exciton energies). In order to gain a qualitative appreciation of the dispersion relation, we treat the special case in which one Ok,* has a frequency sufficiently removed from all other resonant frequencies that all other oscillators will contribute a constant amount to a background dielectric constant E. For convenience, this background term will be taken to be isotropic. In a cubic crystal, there would necessarily be an oscillator o,,,+ having the same frequency as aI,*. The idealized &axial crystal treated here deviates as far as possible from a cubic crystal, and has no 11oscillator corresponding to W&i.
and
D.
G.
THOMAS
Equations
(1) and (3) then become :
1 3P,
-
w12
-+P,
at2
brgitudinal
= pleX [&ongitudinalXe]
=
The contribution of all other oscillators far removed in frequency enters through E. The dispersion relation is then simply:
w2(k> = PL+~~L(ex(kxe)j(k.PL).(4) PL--cl? E There
are two solutions to equation (4):
P,.k = 0 : d(k) = wI
(transverse
mode)
P,.(kXe)=O:d(k)=wt (mixed ‘mode). The transverse mode has its polarization vector in the x-y plane and normal to k. The mixed mode has its polarization vector in the projection of k on
FIG. 1. E(k)for an isolated 1 exciton near k = 0 in the plane kv = 0. The energy surface ST belongs to the pure transverse exciton. The energy surface 5’~ beIongs to the exciton which is purely longitudinal for k parallel to the c-axis, and purely transverse for k perpendicular to the C-Z&.
the x-y plane. The transverse mode is purely transverse; the mixed mode is in general mixed, going from pure longitudinal for k perpendicular to the c-axis to pure transverse fork parallel to the c-axis. The energy surface for the two modes is sketched in Fig. 1 for the plane ink space defined
OBSERVABLE
PROPERTIES
by K, = 0. For k perpendicular modes are related by
OF
to the c-axis, the
the usual LYDDANE-SACKS-TELLER@) relation for cubic crystals. In order to observe the mixed modes in hexagonal crystals, it is necessary to use light whose propagation vector in the crystal makes an angle 4 # 90” or 0” with the c-axis. For light polarized in the plane containing both k and the c-axis, only the mixed mode would be observed. For light polarized perpendicular to the c-axis, only the transverse mode would be observed. For these two experiments, respectively one would obtain the results in Table 1.
Table 1 Frequency of observed line
Oscillator strength of absorption line (in proper units)
In cubic crystals, due to the presence of a 11oscillator, all excitons would be purely transverse or purely longitudinal.
Optical phonons The effects which have been described so far exist for both excitons and optical phonons in hexagonal crystals. In the case of optical phonons, the effects should be particularly small. For the optical phonons, wI and o,, are usually quite close. The effect of the polarization field interaction is to try to make all polarization normal modes either purely transverse or purely longitudinal. The strength of this electrostatic interaction is given by
For most hexagonal ionic crystals the strengthbf
LONGITUDINAL
EXCITONS
279
this electrostatic interaction is experimentally about 0~14. On the other hand, the crystal field splitting 1o L-w,ll in typical materials is roughly wJO0. (The experimental numbers used here are those appropriate to hexagonal SiC.0) Because the electrostatic energy is so much larger than the crystal field splitting, all modes will be very nearly pure longitudinal and pure transverse. In restrahl measurements, the longitudinal phonons will be only very weakly active for nonprincipal directions of k. In practice, these lines might be observed for certain polarizations of the incident light in nonprincipal directions. In crystals which deviate further from optical isotropy, the effects would be correspondingly larger.
Excitons In general, it would be rather difficult to investigate by optical means the “longitudinal” (mixed) phonons in hexagonal crystals due to the dominance of the electrostatic interaction over the crystal field splittings. For excitons in hexagonal crystals, the electrostatic interaction is no longer dominant (4+3 is very small for excitons of large radius), The absorption which one hopes to observe due to mixed excitons is qualitatively different from that which might have been off-hand expected. A few degrees from a principal direction, one expects to see new absorption lines which differ in energy from all ordinary (transverse) exciton lines seen in any principal direction. These absorption lines interact with all the light in a properly chosen geometry (as would not be the case for a slightly misaligned plane of polarization), and can easily be strong enough to lead to complete absorption in crystals a few microns thick for angles of deviation as small as 5” from principal directions. These absorption lines are due to “longitudinal” excitons (i.e. mixed excitons having chiefly a longitudinal component). In ZnO, a situation rather similar to that described in the text occurs for the excitons. There are two modes (excitons) P,I and P,z which have been identified by appropriate reflection measurements.@) The lines are well enough separated from all other reflection lines that E is approximately constant and isotropic. The parameters wr I, wsI, /3tL, j3sL and E have been determined by an analysis of the reflectivity measured with E and k perpendicular to the c-axis. The values of
280
J. f. HOPFIELD
the parameters are 01~ = 3_376seV, ~9~ = 01~ + 0*006s, pr I = 0*0074 f 15 per cent. 13s1 = 0.024 f 15 per cent and E = 5.0 f 10 per cent at 42°K. The energies wr I and wzI are slightly temperature-dependent, but their difference (which is the energy of importance for the calculations) is not. Two longitudinal excitons, corresponding to the two oscillators WI& and wsl are expected. The experiment is designed to observe these “longitudinal” excitons by virtue of their small transverse part (which does not exist for cubic crystals). The frequencies and strengths of the expected mixed excitons can be found by solving equations (1) and (3) for the case in which only two oscillators are present. The solution of the eigenvalne problem defined by (1) and (3) for the transverse mode energies is trivial. There are two “longitudinal” (mixed) solutions. One eigenstate has almost equal amounts of the two polarization fields w L1and o 19 oscillating 180” out of phase, with a resultant small energy shift and small strength. The higher-frequency eigenstate has almost equal amounts of the two polarization fields oscillating in phase, with a resultant large energy shift and large strength. Qualitatively, the lower-energy longitudinal line has a large effective E due to the upper line, so its shift is small. The higher-energy longitudinal line has a small effective e due to the lower line, so its shift is large. The two modes wrl and wsl are the lowestlying excitons of appreciable strength. An exciton of approximately the same strength lies approximately O-040 eV above these two excitons, out of the current range of interest. All excited states of these two low-lying excitons also lie more than 0*040 eV above these two strong lines. In addition, there exists a very weak (exciton) mode PI, having almost the same frequency as P,r. The optical activity of this weak line is associated with the small spin-orbit coupling which splits the excitons P,l and P L~.All other excitons falling in the region of experimental interest are known from group theory (confirmed by experiment) to be optically inactive.(9) 3. Expm An experiment was performed on ZnO to check the theory and to determine whether some absorption lines which had already been observed in ZnO
and
D.
G.
THOMAS
were due to “longitudinal” excitons. The absorption constant of ZnO was measured as a function of the angle of incidence. The geometry used is shown in Fig. 2. A ~ntinu~ source of illumination was used, and the transmitted light analyzed
FIG. 2. The experimental geometry of k,E and c-axis. All vectors drawn lie in the plane of the drawing.
by a Bausch and Lomb spectrograph. A focused beam of 3” full angle was used because of the small size of the ZnO crystals. Densitometer traces of the spectrograph photographs were analyzed to determine the absorption constant. Polarized light was used (as is indicated in the figure) in order to calibrate the transmission measurements. Small deviations of the plane of polarization from the nominal plane of polarization are unimpo~~t. Such an error would result in the presence of a small amount of light polarized perpendicular to the c-axis. For this component, the absorption constant is so great (about 105 cm-l) in the spectral region of interest that essentially none of this unwanted component can get through the 10,~ thick crystal. The qualitative results of the absorption experiment at 4.2”K are shown in Fig. 3. The energies wI1 and w,_9 are indicated by arrows. Line A is due to the previously mentioned weak exciton wll, (This exciton is “weak” compared with ~~1~ which, as a transverse exciton, has a peak absorption coefficient of about 10s cm-l.) The sloping background is due to the tail of a very strong exciton absorption line lying to higher energy. Lines B and C are the lines of interest. Within the accuracy of experimental measurement, these lines lie at the same energy for 8 = O”, 2”, 4” and 6”. Lines B and C have roughly the same strength at 2* as at O”, but increase in strength rapidly as 0
OBSERVABLE
PROPERTIES
OF
LONGITUDINAL
EXCITONS
281
background absorption) as a function of angle for line B at 77°K. The line width for line B was also measured at 77°K to provide a means of evaluating the oscillator strength. This width was independent of angle over the range of angles measured. The experimental points fall within experimental accuracy on a straight line.
i 3vJoO 8 400
200 00
e=40
2000
O,S>O
D A
5 L 1000 Y 6 400
100
X,B
-,a=70+15-482
C
150 a, CM-’
0
‘UlP
3.395
3.385
100
011
50
3.37 ENER GY
FIG. 3. The absorption spectrum of ZnOas afunctionof B at 4OK. Note that the absorption constant scale is nonlinear.
becomes larger than 4”. Line A, on the other hand, remains constant in strength. The rapidly increasing strengths of lines B and C is characteristic of excitons which are nearly completely longitudinal; the constant strength of line A indicates that this exciton is (as was assumed) chiefly transverse. The same absorption experiment was also carried out at 77°K to obtain more quantitative information about the line strength as a function of angle. At 77°K the exciton lines are broadened and peak absorption coefficients are therefore smaller. Unfortunately, due to the increased background absorption, it was possible to measure only the behaviour of line B. The results of this experiment are given in Fig. 4, which shows the peak absorption coefficient (after the subtraction of the
,,,I
345
I
I
6
8 7 e IN DEGREES
I
I
1
9
10
11
Fig. 4. The peak absorption coefficient of peak B at 77°K. as a function of 0. The angular scale is linear in ea. The circles are experimental points for B > 0 and the crosses for ~4< 0”.
From the theory of Section 2, the parameters E, PII, rB1s, w *I and w 1s given at the end of Section 2, and the measured width of line B at 77”K, the complete behaviour of lines B and C can be compputed. A comparison of theory (for small angles) and experiment is given below. The energy 0~1 is taken as a reference zero. A refractive correction is included in the calculations to relate the exterior angle 8 to the interior angle 4. The “Theory” column of Table 2 is computed from the reflection measurements alone; there are no free parameters to adjust.
Table 2
I Energy of line B Energy of line C Strength of line B
at 4.2”K
( Strength of line C Peak absorption coefficient of line B at 77°K
Theory
I
Experiment
0~0010 0.0158 0.035
0.0016 eV 0*0168 eV 0.05 + 100x--50%
c$$& = IO@
(“p& = 70+15*4L9s
282
J.
J.
H‘OPFIELD
and
Most of the experimental data agree with the theory within the error introduced into the theoretical calculation by the uncertainty of the parameters used. It should be noted that the ratio of strengths and the slope of the peak absorption are sensitive functions of 0s ,_ - wl I, changing about 30 per cent for a change of 0.001 eV in w2 I - w1 I. One point, however, is in definite disagreement with theory. The theory states unambiguously that the longitudinal lines should be unobservable for 0 = 0”, whereas Figs. 3 and 4 indicate* that the lines are small but present. We believe that their presence is due not to experimental error (the 0” residual absorption is an order of magnitude too large to be due to the 3” convergent beam), but is instead due to the failure of the infinite-wavelength theory used. Ordinarily, in treating optical processes in crystals, the effect of the finite wave vector of the light in the crystal is neglected. The error thus introduced is of the order (Ku)s, where K is the wave vector of the light in the crystal and a is a typical lattice dimension. In the case at hand, (lza)s M 10-d. F rom the known strengths of the lines w I~ and o 1s and the width of peak B, one could then expect a residual “error” for amax of the order of 100 cm-l. This is of the same order of magnitude as the observed residual absorption. The effect can be regarded as due to the small but finite photon momentum or, equivalently, to the fact that optical transitions are not precisely vertical. Such effects are not unkown in cubic crystals, and must be invoked, for example, to explain the existence of the “n = 1” line in the exciton absorption spectrum of CusO.(lo) The observed widths of the two longitudinal exciton lines differ considerably, as can be seen in Fig. 3. The decreased lifetime of the higherenergy longitudinal exciton may simply be due to the fact that the higher-energy exciton of small k may be scattered by impurities or phonons into states of the lower-energy exciton of higher k. NO corresponding states exist for scattering of the longitudinal exciton. A second lower-energy possibility is that the scattering is related to the size of the oscillating dipole, and that the large dipole matrix elements of the higher * The mixed mode at 0” angle of incidence was below noise level for the experiment at 77°K. Quantitative measurements could be made at 77°K only for external angles of incidence greater than l-5”.
D.
G.
THOMAS
exciton lead to strong interactions and rapid scattering. A few qualitative points should be emphasized concerning the “longitudinal” effects in hexagonal crystals. First, as is well known in the case of cubic crystals, group theory can be misleading with reference to exciton lines. The usual analysis of the group theory of exciton lines applies at k = 0.To apply this result to excitons of small but finite k is not strictly permissible, because of the nonanalytic nature of the exciton energy surface. In cubic crystals the errors which are made by extending the results of group theory at k = 0 to finite k are not important in determining selection rules, since the chief error is the energy difference between the longitudinal (unobservable) exciton and the transverse excitons. In hexagonal crystals, on the other hand, because the “longitudinal” (mixed) modes are optically observable, it is necessary to understand the nature of the singularity in E(k) in order to understand quantitatively the positions and oscillator strengths of observed exciton absorption lines. Second, it is essential to use single crystals in investigating the exciton states in hexagonal crystals. In the case of ZnO, for instance, the use of polycrystalline films would have produced a confusing jumble of experimental results due to the varying energy position of the “longitudinal” exciton line. Finally, the finite angular aperture of an optical system can contribute strongly to observed effects. For the small-aperture Bausch and Lomb spectrometer used in this study, the correction due to the finite aperture was about 1.5 per cent of the observed residual absorption at 0”. 4. OTHER EFFECTS IN CUBIC CRYSTALS
The linear Zeeman effect in certain crystal geometries can be quenched by the energy difference between the longitudinal and transverse excitons. As an example of this, we consider the case of the CusO yellow exciton series which has been studied extensively by GROSS and co-workers.(ls) In CusO the direct transition is forbidden, and excitons are observed in p-like orbital states. (The n = 1 line, though present, is very weak.) The electron spin effectively does not enter the problem. Let the magnetic field be along the z-direction, let y be the direction of the incident light and let z be the direction of polarization of the incident light-
OBSERVABLE
PROPERTIES
OF
At first sight, one would have expected a linear Zeeman effect to be observed in this geometry. However, because of the direction of propagation of the incident light, the energy difference between the longitudin~ and transverse excitons is important. The Hamiltonian matrix for the linear Zeeman effect in this geometry is :
LONGITUDINAL
checked. If the light is instead sent aEongthe magnetic field, the Hamiltonian becomes :
P, H=P, ps i
where
S = E[&+y)-I]. E is the energy of the exciton and @ and z are as defined earlier in the text. The existence of the term 6 will quench the linear Zeeman effect if 6 2 $3. For the case of CusO, in the largest fields used by GROSS,the effect would be quenched if eirli3 -~lO-~ i rf the electron and hole masses differ by a factor of 2. For comparison, in ZnO h/3,/~M 4~ 10-s. The CusO excitons are “forbidden,” so &r/3/~ would be less by a factor of exciton radius s ( lattice constant 1 than for ZnO. On the other hand, the excitons are more tightly bound in CusO, which would tend to make ~T$/E larger. Unfortunately, ~+I/E is not known from experiments in CusO. GROSS has observed the absence of a linear Zeeman effect in CusO, and has attributed this absence to the equality of the electron and hole masses (f~ = 0). It would appear that quenching due to longitudinal excitons can equally well be the cause of the absence of the linear Zeeman effect in the low-lying exciton lines of CusO.* The quenching hypothesis can be rather easily * NOM added in grooj: One of us (D.G.T.) has since made measurements which indicate that 4?rfl/c is too small to quench the Zeeman effect in CusO.
283
EXCITONS
PS
P,
PZ
E
;EtH
0
-+H
E
0
0
0
ES-8 f
and a linear Zeeman effect will be obtained if p is not zero. This quenching is to be predicted for all direct exciton Zeeman effects in which the Zeeman splitting is expected to arise from the splitting of an exciton which belongs to a three-dimensional irreducible representation. The stronger the exciton line, the more important this quenching becomes. Conversely, the longitudinal-transverse energy difference goes to zero faster than the exciton binding energy for highly excited exciton states, and so rapidly becomes unimportant for excited states. There is no effect for continuum states. Linear Zeeman effects in excitons can sometimes arise from a near degeneracy of exciton levels of different symmetry which are mixed by a magnetic field (such a case has been observed by THOMAS in GO). Accidental degeneracies can easily arise in hydrogenic excitons, for spin-spin interactions between electrons and holes are very short-range forces. If the continuum approximation is not valid, it is possible for the long-range coulomb effects to shift the transverse exciton states as well as the longitudinal states. This shift could remove the accidental degeneracy which is responsible for the linear Zeeman effect in the case of near degeneracy that might otherwise have been expected, and lead to a quenching of this Zeeman effect also. All the effects treated so far have referred to direct excitons. The effect of the long-range electrostatic interaction on indirect excitons is a much more difftcult problem, for deviations from the continuum approximations must necessarily occur for excitons of large k. (Such deviations are necessary to preserve the periodicity of the exciton in the Brillouin zone.) The effects will probably be far less interesting. It is clear, however, that the longrange electrostatic interaction should give rise to &-rite energy corrections and configuration mixings which do not appear explicitly in the effective mass
284
J.
J.
HOPFIELD
approximation. Suppose, for example, the holes were extremely heavy compared to the electrons. The usual effective mass problem would yield an exciton binding energy which would be the same as the binding energy of a donor. The electrostatic effects, however, could produce an energy shift from this calculated energy which could be as large as 20 per cent of the exciton binding energy for some states. This effect is not due to polaron effects, but is caused by the same electrostatic interactions which lead to the finite energy difference between longitudinal and transverse excitons near k = 0.
REFERENCES 1. Fox
D and YATSIVS., Phys. Rev. 108, 938 (1957).
and
D.
G.
THOMAS
2. PBWR S. I., Zh. eksp. teor.jk 8, 360 (1958). 3. HELLW W. R. and MARCUSA., Phys. Rev. 84,809 (1951). 4. HOPFIELDJ. J., Phys. Rev. 112, 1555 (1958). 5. BORN M. and HUANG I-C., Dynamical Theory of Cvystal Lattices, Section 7. Oxford University Press (1954). 6. LYI~DANB R. H., SACHSR. G. and TELLERE., Phys. Rev. 59, 673 (1941). 7. SPIKER W. G., KLEINMAND. and WALSHD., Phys. Rev. 113,127 (1959). 8. THOMASD. G., Bull. Amer. Phys. Sot. 4,154(1959). 9. HOPFIELDI. J.. Bull. Amer. Phvs. Sot. 4.154 (1959). 10. GROSSE. ‘F: hd ZAKI-LUKHJ&A B. $. , J.. Tech. Phys. (U.S.S.R.) 2, 1808 (1957). GROSSE. F., ZAKHARCHENYA B. P. and PAVENSKII P. P.. Zh. tekh. fiz. 2.2018 (19571. GROSSE. F. and ZA&ARC&YA b. P., J. Phys. Radium 18,68 (1957).