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Second harmonic generation in solids
J. Phys. Chem. Solids
Pergamon Press 1963. Vol. 24, pp. 1113-1119.
SECOND
HARMONIC
Printed in Great Britain.
GENERATION
IN SOLIDS
P. L. KELLEY Lincoln Laboratory,* Massachusetts Institute of Technology, Lexington 73, Massachusetts (Received 18 December 1962; revised 12 April 1963)
Abstract-This paper is concerned with the relation in solids between second harmonic generation and electronic band structure. The second order response function is evaluated in the dipole approximation for two- and three-band transition schemes in a system with an idealized band structure. In addition, an estimate is made of the multipole terms which are of one order higher than the dipolar.
1. INTRODUCTION A NUMBER of fruitful investigation&-@ of the harmonic generation of light due to the interaction of solids with optical maser beams have been made recently. Early theories of harmonic generation dealt with: (1) a phenomenological description of electrons interacting with radiation,@) and (2) localized electrons interacting with radiation using semiclassical radiation theory.@) In a previous paper by the author,(7) that part of the electronic response of a solid which is proportional to the square of the classical electromagnetic field was derived using operator perturbation theory. The results were obtained in a form which was applicable to non-localized electrons such as Bloch electrons. The spatial dependence of the second harmonic wave within the solid was found. In addition, a three-band transition scheme was used to estimate the output of second harmonic power in the limit where both the first and second harmonic energies are near band gaps. This limit would most likely imply that the medium is absorbing to the second harmonic. The present work is concerned primarily with describing the effect of electronic band structure on second harmonic generation. Direct transitions (those not involving phonons) which are allowed in the dipole approximation are described. It is found that two- and three-band schemes are
* Operated with support from the U.S. Army,
Navy
and Air Force. 1113
allowed, with the two-band scheme including an intraband transition. The response function is then calculated for the two- and three-band transition schemes. In order to facilitate the calculations, a band structure is assumed in which the bands are parabolic and widely separated. In this idealized calculation, the two-band scheme does not exhibit any singular behavior near the band gap, while the three-band scheme may show for certain choices of the band parameters singular behavior but not necessarily at the band gap. Finally, an estimate is made of the term in the response function one order higher than the dipolar.
NONLINEAR RESPONSE FUNCTION The quantity of interest in this paper is that part of the current induced in a solid which is proportional to the square of the external field. The second order nonlinear response function, Q, relates the space and time Fourier transform of the current J”(q’w) to the product of the transformed vector potentials Afi (Q’w’) and Av (Q”, w”) in the following way: 2. THE
Je(q,
W) = lssj
dw’dw” dq’dq”
x aW (q, q’, Q”, w’f
con, w’)
x Ar(q’, w’) Av(q”, co”) S(w’+ w”The
response
function,
obtained
w).
(1)
in an earlier
1114
P. L. KELLEY
paper,(‘) is given by Q”PY(Q,q’, q”, w’+d,
The problem to be considered in the following calculations is that of frequency doubling. In this WI and w’+ w”= wz=2wl. case, w’ = w”= Here, wl is the first harmonic frequency and ~2 is the second harmonic frequency.
cd)
=& 1 c
exp[i(q+q’+q”)
x
rw
. %]
(q, q’, q”, cd+ co”, co’),
I
3. THE DIPOLE APPROXIMATION
(2)
where WCis the volume of a unit cell, R, is a lattice vector, and r =fi”(q, q’, q”, w’+ con,w’)
~x,_q-*‘~ii(Q'+Q")I$oJ~
+247vo+ w”) @z-w))>L-4, P (@)Is,, +;
[(7(9o+w’)
&,-,*$a
[(7(9o+w’+J>P
(q)),
(3)
(@)I), %, n*,jY (@?I>*
In the above equation, p(ts) is the equilibrium one electron distribution operator, j(q) is the cellular transform of the momentum density operator $+I)
= N
j
dr exp(GI - r> p(r),
(4) rapv
cell
and (Fq) is the transform of the mass density operator. 7(w) is given by l/w+ A where l --f O+. The Liouville operator, 2’0, has been introduced and is defined by 2z,A
E l/ti [#I), A] ;
in addition, its analog operator, is defined by
K, for the
wave
vector
(2~1,
wl)
Here, k is the wave vector operator terms of Bloch states {n’k’) by =
k’ln’k’
>.
defined
in
(7)
The response function may now be evaluated by writing the trace as a sum over the Bloch states of the solid.
-sv
2
f( w&))
1
1 X-
1.wnn,(k)+2wl+ie
wnne(k)+wl+ie
X &n*&,n*&n(k)
1
-_ wn,n-(k)
(6)
=
k=k’zk” n,n’,n”
(5)
KA = [&A].
kln’k’ )
In the preceding Section, the second order nonlinear response function has been given as a trace over the one-electron states of the solid. In this Section, the dipole approximation will be made and the response function will be evaluated for a simple model band structure. The aim here is to obtain a picture of the dipole transition schemes and give an order of magnitude estimate of the response function for the various processes. The dipole approximation consists of making the approximation exp(iq - t) N 1 within the cellular integrals, under the assumption that Iq - (11-g 1 (where a is a lattice spacing). Furthermore, the q’s in the Kronecker deltas in equation (3) are set equal to zero. The last approximation constrains the interband transitions to be vertical. Although the transitions are not quite vertical, the error made in the vertical approximation is also of the order of ]q * al. With these approximations in mind, equation (3) can be written explicitly in terms of matrix elements between Bloch states {nk}, {n’K’} and (0”) as
+2w1+
1
ic
( wnn-(k)
X PL*&,n*&ta(k)
+ ~1 + ie
SECOND
HARMONIC
GENERATION
where tiw,,&) = (nkj.%‘slnk)(n’Kf&‘eln’~>. Note that the solid considered for the dipole process cannot have inversion symmetry, since the states of such a system can be chosen to have definite parity (these are not the Bloch states) and the product of three dipole operators cannot connect states of the same parity. (See Appendix.) The results can now be discussed in terms of transitions between energy bands. The three states {nR}, (n’k’), and (n”K”) may correspond to states in one, two, or three bands. If only one band were involved, expression (8) would vanish, since the four terms in (8) would cancel. Thus, the states must involve either two or three bands. The two-band scheme consists of two interband transitions and one intraband transition; the threeband scheme consists of three interband transitions. The magnitude of I’ will be calculated for the two and three band schemes. The energy bands will be taken to be parabolic about k = 0 in wave vector space. Figure I represents the band structure of the idealized solid. The valence band (band 1) has energy versus wave vector dependence (lkfS’ollk>
lizkz = El(k) = --=
(9)
1
while the conduction bands (bands 2 and 3) have the dependence <2klA=‘o\%) = Ez(k) = EgeZptt
(10) 2
SOLIDS
Pk2 2m
(11)
3
Band 1 will be considered completely filled, while bands 2 and 3 are empty. The, transitions will be taken to be virtual (Eg, Eg > 284; in other words, none of the denominators in (8) are allowed to vanish. This implies that the energy of the radiation field is conserved but that there is mixing of the modes of the electromagnetic field by the solid.
1115
for a Bloch state @, k) is given by
(12)
and is, therefore, an odd function of k. On converting the sum over k in equation (8) into an integral, it is evident that only those terms in the remainder of the integrand (i.e. apart from the
3
----- i/
mm--
-
i
_f
I
E!J
Et3
I
k--l 1
e-m---
FIG. 1.
4
Three-band model.
intraband matrix element) which are odd in K will contribute. Therefore, if the product of the two interband momentum matrix elements is expanded as a series in k, only the odd powers in k give a nonvanishing contribution. This expansion will be accomplished by means of a k rp perturbation calculation, where, for the sake of simplicity, only the term linear in K will be kept. This approximation is valid in the limit where the bands are far apart in energy. k lp perturbation theory contains l
The two-band calculation First, the two-band scheme is considered. The interband transitions are taken between bands 1 and 2. The intraband momentum matrix element
mm-
_u_ -I--_
and <3k~&‘o~3k> = l&(k) = E;+
IN
1116
P. L. KELLEY
energy denominators and interband momentum matrix elements. The assumption made here is that the bands are far enough apart so that the denominators are large enough to make the series convergent. The linear part of the product of the interband matrix elements is found to be (&(+&k))l’““a’
= - (&(k)$&(k>)““ear
The behavior of (15), as a function of WI, is not too transparent. However, it can be pointed out that (15) does not exhibit any strong singularities. If one goes to the limit of narrow bands, with 2Awr near enough to the band gap so that Eg-2Awl is small compared to the band width (in other words, a- Q & < a+, b*), the inverse tangent functions can be expanded to obtain
(13) where
3
___-
x (A;;Y+A;;P)
Pz’l
(14) In the last expression, the matrix elements of p refer to the center of the zone. The sum over j is taken so that the wu’s do not vanish. Note that if we were to include only bands 1 and 2 in the sum, then A would vanish. Other bands must be involved in the calculation of A. Substituting (9), (lo), (12) and (13) into (8), the following is obtained in the spherical zone approximation:
+
(a+)3j2
tan-l
tan-l --@&-I,
E-(b+)3/2
@+)1/s
(15) wherepm = hkM, and KM is the wave vector at the spherical Brillouin zone boundary. The a’s and b’s are defined by a* 3 2/.q2(E,+2fiwr),
(16)
b* = 2p12(Eg f
(17)
and The interband fined by
fiw1).
mass CL, introduced
pip =
mtmj -* mz+mj
above, is de-
(18)
PM
3
(19)
It should be mentioned that the two-band model places a restriction on the response tensor. PRICE(*) has shown that the third rank response tensor for a two-band process must be antisymmetric under the interchange of the two indices corresponding to the interband dipole matrix elements. This result may be obtained by applying Kramer’s theorem to the interband elements. However, those crystals which have a nonvanishing third rank tensor are not necessarily constrained to be symmetric under the interchange of two arbitrary indices. Therefore, the two-band scheme will not be forbidden by crystal symmetry. The three-band calculation Next, the three-band calculation where all the momentum matrix elements are interband is treated. Since these matrix elements are complicated functions of k, it is advisable to make a simplifying assumption and replace the matrix by the band edge matrix element; in other words, (nkljwjnk)
1: (nOJp+‘O)
= p&r.
(20)
Here, it is assumed that symmetry will allow transitions between the band edges. For a particular symmetry type, this assumption must be tested by studying the transformation properties of the wave functions at the band edge. Substituting (9), (lo), (11) and (16) into (8), the leading term in the three-band process gives: l?“P (2w1, WI) = 2
(2~12) (2~1s)
SECOND x
HARMONIC
-.L_{(a-ytan-l -&(c)i’s
GENERATION
tan-l
(21) where c- = 24E;-Awl).
(22)
Again assuming narrow bands, however, with 2tiwl near Eg (i.e. a- 4 p& Q c-), the inverse tangent functions can be expanded to obtain rep” (2Wl, Wl) = --&
(24
c&3)
x P”l2&&
PM
X
1113(E;,-Awl)--~~12(E~-2Aw~)
’
r2 Band _ r3 Band
N
2p12 -* 4J
(24)
Thus, it is seen that, under the assumption of bands narrow compared to the gap, the three band process will be dominant. It should be remembered, in considering the above results, that the amplitude of the second harmonic is proportional to l? and local field corrections. These local field corrections are dependent on the dielectric tensor which usually shows a strong singularity near the band gap. Thus, it may well be that any dependence of the second harmonic on the difference between the band edge and the second harmonic frequency is due, in large measure, to the linear polarizability of the medium at the
1117
SOLIDS
second harmonic frequency. The increased difficulty in phase matching will play an important role also, as the second harmonic approaches the band edge. 4. HIGHER ORDER EFFECTS In solids with inversion symmetry the dipolar process vanishes and higher order processes must be considered. An order of magnitude estimate will be made of the term in I’ which is one order higher than the dipolar; this is that part of I’ which is linear in Q. The linear term will then be compared with the dipolar term. Because of the complicated nature of the higher order processes only the first commutator in (3) will be evaluated here. The aim is to explicitly show for the simplified model used here that a typical term linear in q is of order pa or q/KM smaller than the dipolar. The other terms in (3) should also be of order q/kM smaller.
(23)
Whether I’ has a singularity, and the position of this singularity, is determined by the gaps Eg and Eg and by the interband masses ~12 and ~1s. The limit taken above constrains the second harmonic to be below both band gaps. This is in contrast t,o the previou0 three-band model where Eg > 2fiwl > Eg, in which case the medium will probably be absorbing to second harmonic radiation. The ratio of the two-band to the three-band process can now be estimated as
IN
{r 6pv (2wl, wI))(linear i* 4) =
c 8
- e3(27r)3
2m3c2Vh
k-k’
n,n’
XL1
wnn,(k) Iarm(k)
l +
2wl
1 -t wn,n(k) + 2USJ%a*
Pi&~$44 Ptw4 ]s
(25)
where q1 is the wave vector of the first harmonic. The matrix elements of the momentum operators will be taken between band 1 and band 2. As in (20), these matrix elements will be replaced by those at the band edge. Converting the sum in (25) to an integral, and carrying out the integration, the following is obtained 239 {r”pY (2wl, wl)}(ltnear in e) = m%%i2 8
c
x8 q1 $v P:2 p21 x
E
((a+)112 tan &+(+I2
x-
PM
(a-)1/2
- 2(a)l’a tan-1 -
tan-l
1118
P.
L.
Here, a* is defined by (16), and a = 2plzEg. Again going to the limit of narrow bands, but with 2Awl close to Eg (i.e. a-< p& < a, a+), equation (26) becomes {r EFV(2wl, wl)}(linear fn q) =
x
4!L
1112 T--
Pf2 p;2 PM.
It will now be shown that (3) vanishes for systems with inversion symmetry in the dipole approximation. This is most readily shown by introducing a complete orthonormal set of states defined in the following way:
(27)
Comparing this result with the dipolar term, it is found that the ratio is approximately Jhlear
in g
rdipolar
m ql/k&f w 10-d.
6. ARMSTRONGJ. A., BLOEMBERGBN N., DUCXIINGJ. and PERSHAN P. S., Pkys. Rew. 127, 1918 (1962). 7. KELLEYP. L., J. Phys. Chem. Solids 26,607 (1963). 8. PRICEP. J., Private communication.
APPENDIX
27X+3 _ mscatia
c 8
KELLEY
(28)
Thus, the second harmonic intensity, which is proportional to the square of r, is M 10ms smaller in crystals with inversion symmetry than in crystals without inversion symmetry. 5. CONCLUSIONS
The preceding has given an evaluation of the second harmonic response function for two- and three-band transition schemes in the dipole approximation, as well as an estimate of the response function for a two-band scheme for electromagnetic processes which are of higher order than the dipolar. The calculation has been carried out for a solid with highly idealized bands. This idealized model may be recognized as a first step toward the methods to be used in the calculation of the nonlinear response of an actual solid. The details of the calculation for a real solid would necessarily involve its symmetry properties and actual energy surfaces.
Ink +
> =42-&,
InO +
> =
+k)
*
1% -k)),
k,
< 0
k # OX
InK + )
k=O
jn,O>
= In,K>
(29)
k = K
{n,O) and {n,K}, respectively, indicate states at k = 0 and symmetry points on the zone face. As is well known, these are energy eigenstates 3Eaoltl, k +
> = &(k)ln,k
+
>.
(30)
It is easily shown for k # 0 that these are states of definite parity. If #z (k, r) = exp(ik
- r) unk(r) = (r[nk + ),
and the choice of phase then it can be shown that #‘,k: (4
U”L(-r)
=
= t&r)
is
(31) made,
(32)
+ $q&>*
The states at k = O,K, namely {n,O} and {n,K}, can also be chosen to have definite parity in systems with inversion symmetry; however, the + at k = 0,K does not necessarily indicate parity + . In taking the trace in (3) if the sum is carried out over the states defmed by (29), the following is obtained in the dipole approximation
Acknowledgements-The author would like to thank Dr. W. H. KLEINER and Dr. A. L. MCWHORTER for their critical reading of this manuscript. REFERENCES 1. FRANKENP. A., HILL A. E., PETERSC. W. and WEINFUXICH G., Phys. Rev. Letters 7, 118 (1961). 2. BASS M., FRANKEN P. A., HILL A. E., PETERS C. W. and WEINREICHG., Phys. Rev. Letters 8, 18 (1962). 3. GIORDMAINE J. A., Phys. Reo. Letters 8, 19 (1962). 4. MAKER P. D., TEFCHUNER. W., NISENOFF M. and SAVAGEC. M., Pkys. Rev. Letters 8, 21 (1962). 5. LAX B., MAVROIDES J. G. and EDWARDSD. F., Pkys. Rw. Letters 8, 166 (1962).
(nk f
Ip+z’k
+ ) (n’k If: Jp+“k
x
?
+ >
>
1 w,*,*(k) + 2~l+
in
1 w,tnn(k) + WI + k x (n’k f
jj+z”k
{nk + Jj+z’k f
) (d’k
+ )
k If+k
+ >
SECOND
+
1 w,e,( k) + WI+ ic
1
GENERATION
(nk f jjh’jn’k & )
x (n’k + Jj+“k
+
HARMONIC
f
\ ) (d’k + Jj+zk + )) 1
awn(k) +2w + ic own(k) + WI+ ic
IN
x (nk + jj+t’k
1119
SOLIDS
f
) (n’k + jj+“k
x (d’k + l$+zk + >I. I
+ ) (33)
The operator 5 is nonvanishinn only for states of opposite parity; hen& an odd product of p’s is nonvanishing only for states of opposite parity. Since the products of the p matrix elements have the same final as initial state, the product must vanish.
Pergamon Press 1963. Vol. 24, pp. 1113-1119.
SECOND
HARMONIC
Printed in Great Britain.
GENERATION
IN SOLIDS
P. L. KELLEY Lincoln Laboratory,* Massachusetts Institute of Technology, Lexington 73, Massachusetts (Received 18 December 1962; revised 12 April 1963)
Abstract-This paper is concerned with the relation in solids between second harmonic generation and electronic band structure. The second order response function is evaluated in the dipole approximation for two- and three-band transition schemes in a system with an idealized band structure. In addition, an estimate is made of the multipole terms which are of one order higher than the dipolar.
1. INTRODUCTION A NUMBER of fruitful investigation&-@ of the harmonic generation of light due to the interaction of solids with optical maser beams have been made recently. Early theories of harmonic generation dealt with: (1) a phenomenological description of electrons interacting with radiation,@) and (2) localized electrons interacting with radiation using semiclassical radiation theory.@) In a previous paper by the author,(7) that part of the electronic response of a solid which is proportional to the square of the classical electromagnetic field was derived using operator perturbation theory. The results were obtained in a form which was applicable to non-localized electrons such as Bloch electrons. The spatial dependence of the second harmonic wave within the solid was found. In addition, a three-band transition scheme was used to estimate the output of second harmonic power in the limit where both the first and second harmonic energies are near band gaps. This limit would most likely imply that the medium is absorbing to the second harmonic. The present work is concerned primarily with describing the effect of electronic band structure on second harmonic generation. Direct transitions (those not involving phonons) which are allowed in the dipole approximation are described. It is found that two- and three-band schemes are
* Operated with support from the U.S. Army,
Navy
and Air Force. 1113
allowed, with the two-band scheme including an intraband transition. The response function is then calculated for the two- and three-band transition schemes. In order to facilitate the calculations, a band structure is assumed in which the bands are parabolic and widely separated. In this idealized calculation, the two-band scheme does not exhibit any singular behavior near the band gap, while the three-band scheme may show for certain choices of the band parameters singular behavior but not necessarily at the band gap. Finally, an estimate is made of the term in the response function one order higher than the dipolar.
NONLINEAR RESPONSE FUNCTION The quantity of interest in this paper is that part of the current induced in a solid which is proportional to the square of the external field. The second order nonlinear response function, Q, relates the space and time Fourier transform of the current J”(q’w) to the product of the transformed vector potentials Afi (Q’w’) and Av (Q”, w”) in the following way: 2. THE
Je(q,
W) = lssj
dw’dw” dq’dq”
x aW (q, q’, Q”, w’f
con, w’)
x Ar(q’, w’) Av(q”, co”) S(w’+ w”The
response
function,
obtained
w).
(1)
in an earlier
1114
P. L. KELLEY
paper,(‘) is given by Q”PY(Q,q’, q”, w’+d,
The problem to be considered in the following calculations is that of frequency doubling. In this WI and w’+ w”= wz=2wl. case, w’ = w”= Here, wl is the first harmonic frequency and ~2 is the second harmonic frequency.
cd)
=& 1 c
exp[i(q+q’+q”)
x
rw
. %]
(q, q’, q”, cd+ co”, co’),
I
3. THE DIPOLE APPROXIMATION
(2)
where WCis the volume of a unit cell, R, is a lattice vector, and r =fi”(q, q’, q”, w’+ con,w’)
~x,_q-*‘~ii(Q'+Q")I$oJ~
+247vo+ w”) @z-w))>L-4, P (@)Is,, +;
[(7(9o+w’)
&,-,*$a
[(7(9o+w’+J>P
(q)),
(3)
(@)I), %, n*,jY (@?I>*
In the above equation, p(ts) is the equilibrium one electron distribution operator, j(q) is the cellular transform of the momentum density operator $+I)
= N
j
dr exp(GI - r> p(r),
(4) rapv
cell
and (Fq) is the transform of the mass density operator. 7(w) is given by l/w+ A where l --f O+. The Liouville operator, 2’0, has been introduced and is defined by 2z,A
E l/ti [#I), A] ;
in addition, its analog operator, is defined by
K, for the
wave
vector
(2~1,
wl)
Here, k is the wave vector operator terms of Bloch states {n’k’) by =
k’ln’k’
>.
defined
in
(7)
The response function may now be evaluated by writing the trace as a sum over the Bloch states of the solid.
-sv
2
f( w&))
1
1 X-
1.wnn,(k)+2wl+ie
wnne(k)+wl+ie
X &n*&,n*&n(k)
1
-_ wn,n-(k)
(6)
=
k=k’zk” n,n’,n”
(5)
KA = [&A].
kln’k’ )
In the preceding Section, the second order nonlinear response function has been given as a trace over the one-electron states of the solid. In this Section, the dipole approximation will be made and the response function will be evaluated for a simple model band structure. The aim here is to obtain a picture of the dipole transition schemes and give an order of magnitude estimate of the response function for the various processes. The dipole approximation consists of making the approximation exp(iq - t) N 1 within the cellular integrals, under the assumption that Iq - (11-g 1 (where a is a lattice spacing). Furthermore, the q’s in the Kronecker deltas in equation (3) are set equal to zero. The last approximation constrains the interband transitions to be vertical. Although the transitions are not quite vertical, the error made in the vertical approximation is also of the order of ]q * al. With these approximations in mind, equation (3) can be written explicitly in terms of matrix elements between Bloch states {nk}, {n’K’} and (0”) as
+2w1+
1
ic
( wnn-(k)
X PL*&,n*&ta(k)
+ ~1 + ie
SECOND
HARMONIC
GENERATION
where tiw,,&) = (nkj.%‘slnk)(n’Kf&‘eln’~>. Note that the solid considered for the dipole process cannot have inversion symmetry, since the states of such a system can be chosen to have definite parity (these are not the Bloch states) and the product of three dipole operators cannot connect states of the same parity. (See Appendix.) The results can now be discussed in terms of transitions between energy bands. The three states {nR}, (n’k’), and (n”K”) may correspond to states in one, two, or three bands. If only one band were involved, expression (8) would vanish, since the four terms in (8) would cancel. Thus, the states must involve either two or three bands. The two-band scheme consists of two interband transitions and one intraband transition; the threeband scheme consists of three interband transitions. The magnitude of I’ will be calculated for the two and three band schemes. The energy bands will be taken to be parabolic about k = 0 in wave vector space. Figure I represents the band structure of the idealized solid. The valence band (band 1) has energy versus wave vector dependence (lkfS’ollk>
lizkz = El(k) = --=
(9)
1
while the conduction bands (bands 2 and 3) have the dependence <2klA=‘o\%) = Ez(k) = EgeZptt
(10) 2
SOLIDS
Pk2 2m
(11)
3
Band 1 will be considered completely filled, while bands 2 and 3 are empty. The, transitions will be taken to be virtual (Eg, Eg > 284; in other words, none of the denominators in (8) are allowed to vanish. This implies that the energy of the radiation field is conserved but that there is mixing of the modes of the electromagnetic field by the solid.
1115
for a Bloch state @, k) is given by
(12)
and is, therefore, an odd function of k. On converting the sum over k in equation (8) into an integral, it is evident that only those terms in the remainder of the integrand (i.e. apart from the
3
----- i/
mm--
-
i
_f
I
E!J
Et3
I
k--l 1
e-m---
FIG. 1.
4
Three-band model.
intraband matrix element) which are odd in K will contribute. Therefore, if the product of the two interband momentum matrix elements is expanded as a series in k, only the odd powers in k give a nonvanishing contribution. This expansion will be accomplished by means of a k rp perturbation calculation, where, for the sake of simplicity, only the term linear in K will be kept. This approximation is valid in the limit where the bands are far apart in energy. k lp perturbation theory contains l
The two-band calculation First, the two-band scheme is considered. The interband transitions are taken between bands 1 and 2. The intraband momentum matrix element
mm-
_u_ -I--_
and <3k~&‘o~3k> = l&(k) = E;+
IN
1116
P. L. KELLEY
energy denominators and interband momentum matrix elements. The assumption made here is that the bands are far enough apart so that the denominators are large enough to make the series convergent. The linear part of the product of the interband matrix elements is found to be (&(+&k))l’““a’
= - (&(k)$&(k>)““ear
The behavior of (15), as a function of WI, is not too transparent. However, it can be pointed out that (15) does not exhibit any strong singularities. If one goes to the limit of narrow bands, with 2Awr near enough to the band gap so that Eg-2Awl is small compared to the band width (in other words, a- Q & < a+, b*), the inverse tangent functions can be expanded to obtain
(13) where
3
___-
x (A;;Y+A;;P)
Pz’l
(14) In the last expression, the matrix elements of p refer to the center of the zone. The sum over j is taken so that the wu’s do not vanish. Note that if we were to include only bands 1 and 2 in the sum, then A would vanish. Other bands must be involved in the calculation of A. Substituting (9), (lo), (12) and (13) into (8), the following is obtained in the spherical zone approximation:
+
(a+)3j2
tan-l
tan-l --@&-I,
E-(b+)3/2
@+)1/s
(15) wherepm = hkM, and KM is the wave vector at the spherical Brillouin zone boundary. The a’s and b’s are defined by a* 3 2/.q2(E,+2fiwr),
(16)
b* = 2p12(Eg f
(17)
and The interband fined by
fiw1).
mass CL, introduced
pip =
mtmj -* mz+mj
above, is de-
(18)
PM
3
(19)
It should be mentioned that the two-band model places a restriction on the response tensor. PRICE(*) has shown that the third rank response tensor for a two-band process must be antisymmetric under the interchange of the two indices corresponding to the interband dipole matrix elements. This result may be obtained by applying Kramer’s theorem to the interband elements. However, those crystals which have a nonvanishing third rank tensor are not necessarily constrained to be symmetric under the interchange of two arbitrary indices. Therefore, the two-band scheme will not be forbidden by crystal symmetry. The three-band calculation Next, the three-band calculation where all the momentum matrix elements are interband is treated. Since these matrix elements are complicated functions of k, it is advisable to make a simplifying assumption and replace the matrix by the band edge matrix element; in other words, (nkljwjnk)
1: (nOJp+‘O)
= p&r.
(20)
Here, it is assumed that symmetry will allow transitions between the band edges. For a particular symmetry type, this assumption must be tested by studying the transformation properties of the wave functions at the band edge. Substituting (9), (lo), (11) and (16) into (8), the leading term in the three-band process gives: l?“P (2w1, WI) = 2
(2~12) (2~1s)
SECOND x
HARMONIC
-.L_{(a-ytan-l -&(c)i’s
GENERATION
tan-l
(21) where c- = 24E;-Awl).
(22)
Again assuming narrow bands, however, with 2tiwl near Eg (i.e. a- 4 p& Q c-), the inverse tangent functions can be expanded to obtain rep” (2Wl, Wl) = --&
(24
c&3)
x P”l2&&
PM
X
1113(E;,-Awl)--~~12(E~-2Aw~)
’
r2 Band _ r3 Band
N
2p12 -* 4J
(24)
Thus, it is seen that, under the assumption of bands narrow compared to the gap, the three band process will be dominant. It should be remembered, in considering the above results, that the amplitude of the second harmonic is proportional to l? and local field corrections. These local field corrections are dependent on the dielectric tensor which usually shows a strong singularity near the band gap. Thus, it may well be that any dependence of the second harmonic on the difference between the band edge and the second harmonic frequency is due, in large measure, to the linear polarizability of the medium at the
1117
SOLIDS
second harmonic frequency. The increased difficulty in phase matching will play an important role also, as the second harmonic approaches the band edge. 4. HIGHER ORDER EFFECTS In solids with inversion symmetry the dipolar process vanishes and higher order processes must be considered. An order of magnitude estimate will be made of the term in I’ which is one order higher than the dipolar; this is that part of I’ which is linear in Q. The linear term will then be compared with the dipolar term. Because of the complicated nature of the higher order processes only the first commutator in (3) will be evaluated here. The aim is to explicitly show for the simplified model used here that a typical term linear in q is of order pa or q/KM smaller than the dipolar. The other terms in (3) should also be of order q/kM smaller.
(23)
Whether I’ has a singularity, and the position of this singularity, is determined by the gaps Eg and Eg and by the interband masses ~12 and ~1s. The limit taken above constrains the second harmonic to be below both band gaps. This is in contrast t,o the previou0 three-band model where Eg > 2fiwl > Eg, in which case the medium will probably be absorbing to second harmonic radiation. The ratio of the two-band to the three-band process can now be estimated as
IN
{r 6pv (2wl, wI))(linear i* 4) =
c 8
- e3(27r)3
2m3c2Vh
k-k’
n,n’
XL1
wnn,(k) Iarm(k)
l +
2wl
1 -t wn,n(k) + 2USJ%a*
Pi&~$44 Ptw4 ]s
(25)
where q1 is the wave vector of the first harmonic. The matrix elements of the momentum operators will be taken between band 1 and band 2. As in (20), these matrix elements will be replaced by those at the band edge. Converting the sum in (25) to an integral, and carrying out the integration, the following is obtained 239 {r”pY (2wl, wl)}(ltnear in e) = m%%i2 8
c
x8 q1 $v P:2 p21 x
E
((a+)112 tan &+(+I2
x-
PM
(a-)1/2
- 2(a)l’a tan-1 -
tan-l
1118
P.
L.
Here, a* is defined by (16), and a = 2plzEg. Again going to the limit of narrow bands, but with 2Awl close to Eg (i.e. a-< p& < a, a+), equation (26) becomes {r EFV(2wl, wl)}(linear fn q) =
x
4!L
1112 T--
Pf2 p;2 PM.
It will now be shown that (3) vanishes for systems with inversion symmetry in the dipole approximation. This is most readily shown by introducing a complete orthonormal set of states defined in the following way:
(27)
Comparing this result with the dipolar term, it is found that the ratio is approximately Jhlear
in g
rdipolar
m ql/k&f w 10-d.
6. ARMSTRONGJ. A., BLOEMBERGBN N., DUCXIINGJ. and PERSHAN P. S., Pkys. Rew. 127, 1918 (1962). 7. KELLEYP. L., J. Phys. Chem. Solids 26,607 (1963). 8. PRICEP. J., Private communication.
APPENDIX
27X+3 _ mscatia
c 8
KELLEY
(28)
Thus, the second harmonic intensity, which is proportional to the square of r, is M 10ms smaller in crystals with inversion symmetry than in crystals without inversion symmetry. 5. CONCLUSIONS
The preceding has given an evaluation of the second harmonic response function for two- and three-band transition schemes in the dipole approximation, as well as an estimate of the response function for a two-band scheme for electromagnetic processes which are of higher order than the dipolar. The calculation has been carried out for a solid with highly idealized bands. This idealized model may be recognized as a first step toward the methods to be used in the calculation of the nonlinear response of an actual solid. The details of the calculation for a real solid would necessarily involve its symmetry properties and actual energy surfaces.
Ink +
> =42-&,
InO +
> =
+k)
*
1% -k)),
k,
< 0
k # OX
InK + )
k=O
jn,O>
= In,K>
(29)
k = K
{n,O) and {n,K}, respectively, indicate states at k = 0 and symmetry points on the zone face. As is well known, these are energy eigenstates 3Eaoltl, k +
> = &(k)ln,k
+
>.
(30)
It is easily shown for k # 0 that these are states of definite parity. If #z (k, r) = exp(ik
- r) unk(r) = (r[nk + ),
and the choice of phase then it can be shown that #‘,k: (4
U”L(-r)
=
= t&r)
is
(31) made,
(32)
+ $q&>*
The states at k = O,K, namely {n,O} and {n,K}, can also be chosen to have definite parity in systems with inversion symmetry; however, the + at k = 0,K does not necessarily indicate parity + . In taking the trace in (3) if the sum is carried out over the states defmed by (29), the following is obtained in the dipole approximation
Acknowledgements-The author would like to thank Dr. W. H. KLEINER and Dr. A. L. MCWHORTER for their critical reading of this manuscript. REFERENCES 1. FRANKENP. A., HILL A. E., PETERSC. W. and WEINFUXICH G., Phys. Rev. Letters 7, 118 (1961). 2. BASS M., FRANKEN P. A., HILL A. E., PETERS C. W. and WEINREICHG., Phys. Rev. Letters 8, 18 (1962). 3. GIORDMAINE J. A., Phys. Reo. Letters 8, 19 (1962). 4. MAKER P. D., TEFCHUNER. W., NISENOFF M. and SAVAGEC. M., Pkys. Rev. Letters 8, 21 (1962). 5. LAX B., MAVROIDES J. G. and EDWARDSD. F., Pkys. Rw. Letters 8, 166 (1962).
(nk f
Ip+z’k
+ ) (n’k If: Jp+“k
x
?
+ >
>
1 w,*,*(k) + 2~l+
in
1 w,tnn(k) + WI + k x (n’k f
jj+z”k
{nk + Jj+z’k f
) (d’k
+ )
k If+k
+ >
SECOND
+
1 w,e,( k) + WI+ ic
1
GENERATION
(nk f jjh’jn’k & )
x (n’k + Jj+“k
+
HARMONIC
f
\ ) (d’k + Jj+zk + )) 1
awn(k) +2w + ic own(k) + WI+ ic
IN
x (nk + jj+t’k
1119
SOLIDS
f
) (n’k + jj+“k
x (d’k + l$+zk + >I. I
+ ) (33)
The operator 5 is nonvanishinn only for states of opposite parity; hen& an odd product of p’s is nonvanishing only for states of opposite parity. Since the products of the p matrix elements have the same final as initial state, the product must vanish.