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Optical conductivity in alkali metals
J. Phys. Chem. Solids
Pergamon Press 1969. Vol. 30, pp. 959-961.
OPTICAL
CONDUCTIVITY
Printed in Great Britain.
IN ALKALI
METALS
H. BiiTTNER and E. GERLACH Battelle-Institut e.V., Frankfurt/Main, Germany (Received
14 August
29 May 1968; in revisedform
1968)
Ah&act-It is shown that the Mayer-Naby peak in the optical conductivity of alkali metals possibly arises from transitions from the Fermi sea to the Brillouin zone boundaries.
scale and the correct magnitude and fulfills most requirements to explain the experiments. Nettel[2] also considers transitions to the zone boundaries, but does not give a detailed discussion of this point.
1. INTRODUCTION
large number of attempts to explain the Mayer-Naby peak[l, 21 in the optical conductivity of alkali metals, no satisfactory solution to the problem has been obtained as yet. An extensive critical analysis of the subject was given in [l]. In a more recent paper, Nettel[2] followed a suggestion by Mayer [l] and Ferrell[ 11 and studied phonon-aided optical absorption (for Na). We considered the general problem from a similar point of view using simple perturbation theory and taking electron phonon interaction explicitly into account. It is shown that the band gap parameter EB = 2V is the crucial quantity determining the various processes of optical absorption. We find that in the framework of the nearly free electron approximation phonon-assisted transitions to the zone boundary yield a transition rate which is of lower order in the band gap parameter than those to other regions and is also of lower order than the direct interband [3] transition. (The ordinary Drude transition is negligible in the Mayer-Naby frequency range[ 11). We derive a formula which shows that the resulting absorption peak has the correct position on the frequency DESPITE
I
A
(n, klH,,ln’, k’+K’)
2. THEORY
We consider a system of nearly free electrons in a weak periodic potential whose wave functions are modified by the phonon scattering operator H,. In first order we have Jn,k+K) +
x n’,K,K’
bW+K’)(n’,k’+K’IH,ln,k+K) E(n,k+K)-E(n’,k’+K’) (1)
where ln,k+ k> is the wave function of the non-interacting electron phonon system of energy E = E,, + Ephon; k lies in the reduced zone; K is a reciprocal lattice vector of b.c.c. structures. Now, transitions due to an electromagnetic interaction H,, = fielimA - V
(2)
from the Fermi sea to unoccupied states in the lower and upper band are given by the following matrix element:
= O(&-k)O(lk’+K’I
-/tF) _k
)
F
+C B(lk’+K, K,
= ln,k+K)
(ktHolk+K)(n,k+K,IH,ln',k'+K')
Edd-EE,dk’+&)
_-k ) (k’+K,1H,lk’+K’)(n’, F
Ed4
k’+K#Z,ln, --E,dk’+W
k)* .
(3)
960
H. Bi_?tTNER and E. GERLACH
Here we have neglected the small contributions from the phonon energies in the denominators. First we consider the electron phonon matrix element using the deformation potential approximation (n, k+K,]H,]n’, k’+K’) = il( &/2MNc,)“* { (n + 1)*‘*8& -
,+
rc*s . ..~+J
(4) where 1 is an energy constant of about 27 eV [4] (atomic unit), q is the phonon wave vector, M the ionic mass, N the ionic density, c, the longitudinal velocity of sound and n the Bose
Hence with equations (2) and (6), the virtual optical transition at the zone boundary gives the following matrix element: -(K/2)H,IK/2)+=
(ef~/2m) K-A
(8)
which does not depend on I/. Now we neglect the ordinary Drude term [ll and all contributions proportional to V (indirect transitions not too close to the zone boundaries and direct transitions). Inserting (5) and (8) into (3) we obtain two contributions, @ = 2hW’/o(A]*, to the optic conductivity arising from transitions of rate W’ to the upper and lower band, respectively
I
distribution for a given T (whereas the pure electron system is assumed to be at T = 0, if the Fermi level is well below the band edge). For T s 0, (Debye-temperature) equation (4) can be approximated by if(k,T/2MNc,*)
“*{6,,+1+ S,,,t+,}.
(5)
The electron photon matrix element was studied by a number of authors [3] in the context of direct interband transitions to states not too close to the boundaries and was found to be proportional to I/ (first Fourier coefficient of the lattice potential). In addition, in our problem virtual optical transitions close to the zone boundary have to be considered, whose matrix elements turn out to be independent of I/. The nearly free electron wave functions in the vicinity of the band edge can easily be obtained from a degenerate perturbation treatment which yields. ]K/2)’ = (2L3)“* [exp (iK*r/2 & exp (--iK*r/2)] corresponding
(6)
to the eigenvalues
E;,’ (K/2) = (fiK/2) 2/2m + V .
(7)
The factor 2 arises from phonon emission and absorption, the factor 4 from time averaging the electromagnetic field. The k’ summation is carried out by putting k’ equal to K’/2 and multiplying by the volume of a narrow wave vector shell at the zone boundary. From the breakdown of the non-degenerate perturbation theory, the width of the shell is estimated to be mV/‘lhK. In the lowestorder approximation a constant gap width is assumed to exist all over the zone boundary, which we take to be spherical. The constant gap approximation corresponds to a reasonable simplification of Ham’s [5] results (except for Na). This is reconfirmed later from the good agreement of our curves with the experimental ones. For a more accurate treatment detailed knowledge of the band structure of the zone boundary would be required, but this is expected to lead only to minor corrections. We consider only transitions between the conduction and the first excited band. This implies that K’ and k’ are not independent so that only & of the shell has to be considered. Summation over K’ gives a factor 32 (/AIn-/a)* (a = lattice constant).
OPTICAL
Carrying obtain
CONDUCTIVITY
out the k-summation
we finally
(TV= 1 e2PkBTm1j2Eg( Eel* - hw ) u2 6
fi2NMa3c 2 (fiw)3 ; fl(ho-E.$+E,).
(10)
The optical conductivity cr= u++ ucalculated for Na and K gives the order of 1014see-’ in agreement with the experiments. However, it should be emphasized that there is some uncertainty in the sound velocity, the band edges, the interaction parameter 1, and also in the shell width. In Fig. 1 an example is
b
hw (ev) Fig. 1. Calculated optical conductivity in the MayerNaby frequency range (broken line: averaged by 0.4 eV intervals).
given for the o-dependence of u (normalized to 1014see-l), which still has to be superimposed and smeared out by the ordinary Drude and the direct interband term. 3. CONCLUSIONS
We have shown that the Mayer-Naby peak can be explained by simple indirect transitions from the Fermi sea to the zone boundaries. We find the calculated peak for all alkali metals to have the correct position on the
IN ALKALI
METALS
961
frequency scale and to be of the order of 1014set-’ as required to explain the MayerNaby experiments. The curves in Fig. 1 show an asymmetry in the same sense as the experimental ones, the steepest descent being on the low energy side. The fact that our peak for Na is more pronounced than the experimental one may be due to the constant gap approximation, which in this case is not as well justified as for the other alkali metals. Further, the temperature dependence of (10) is in agreement with experimental results obtained for Na; it does not explain the behavior of K and Cs. The interference dip [l] at the threshold of the Mayer-Naby peak is expected to appear if the 1V12-contributions and the Drude term are taken into account. It can be shown that there is no interference between the Drude term and our approximation. Since we used only small portions of the detailed band structure which are uncritical in transitions to the liquid state, no major deviations are to be expected in this case, in agreement with the experiments (see [23). After completion of the present work new experimental data on Na and K were published by Neville V. Smith[6], which do not exhibit the Mayer-Naby peak. As suggested by the author this discrepancy may be explained by the fact, that the experiments were performed on evaporated films rather than on bulk material. REFERENCES 1. ABELES F., (Editor) Optical properties and electronic structure of metals and alloys. North Holland, Amsterdam (1966). 2. NETTEL S. J., Phys. Rev. 150, 421 (1966); see also PHILLIPSJ. C.,SolidSt. Phys. l&55 (1966). 3. BUTCHER P. N., Proc. phys. Sot. A64,765 (1951); HOPFlELD J. J., Phys. Rev. 139, A 419 (1965); OVERHAUSER A. W., Phys. Rev. 156, 844 (1967); APPELBAUM J. A., Phys. Reo. 144, 453 (1966); ANIMALU A. 0. E., Phys. Rev. 163, 557; 562 (1967). 4. KITTEL C., Quantum Theory of Solids. Wiley, New York (1963). 5. HAM F. S., Phys: Rev. 128,82 (1962). 6. SMITH N. V., Phys. Rev. Lett. 21,96 (1968).
Pergamon Press 1969. Vol. 30, pp. 959-961.
OPTICAL
CONDUCTIVITY
Printed in Great Britain.
IN ALKALI
METALS
H. BiiTTNER and E. GERLACH Battelle-Institut e.V., Frankfurt/Main, Germany (Received
14 August
29 May 1968; in revisedform
1968)
Ah&act-It is shown that the Mayer-Naby peak in the optical conductivity of alkali metals possibly arises from transitions from the Fermi sea to the Brillouin zone boundaries.
scale and the correct magnitude and fulfills most requirements to explain the experiments. Nettel[2] also considers transitions to the zone boundaries, but does not give a detailed discussion of this point.
1. INTRODUCTION
large number of attempts to explain the Mayer-Naby peak[l, 21 in the optical conductivity of alkali metals, no satisfactory solution to the problem has been obtained as yet. An extensive critical analysis of the subject was given in [l]. In a more recent paper, Nettel[2] followed a suggestion by Mayer [l] and Ferrell[ 11 and studied phonon-aided optical absorption (for Na). We considered the general problem from a similar point of view using simple perturbation theory and taking electron phonon interaction explicitly into account. It is shown that the band gap parameter EB = 2V is the crucial quantity determining the various processes of optical absorption. We find that in the framework of the nearly free electron approximation phonon-assisted transitions to the zone boundary yield a transition rate which is of lower order in the band gap parameter than those to other regions and is also of lower order than the direct interband [3] transition. (The ordinary Drude transition is negligible in the Mayer-Naby frequency range[ 11). We derive a formula which shows that the resulting absorption peak has the correct position on the frequency DESPITE
I
A
(n, klH,,ln’, k’+K’)
2. THEORY
We consider a system of nearly free electrons in a weak periodic potential whose wave functions are modified by the phonon scattering operator H,. In first order we have Jn,k+K) +
x n’,K,K’
bW+K’)(n’,k’+K’IH,ln,k+K) E(n,k+K)-E(n’,k’+K’) (1)
where ln,k+ k> is the wave function of the non-interacting electron phonon system of energy E = E,, + Ephon; k lies in the reduced zone; K is a reciprocal lattice vector of b.c.c. structures. Now, transitions due to an electromagnetic interaction H,, = fielimA - V
(2)
from the Fermi sea to unoccupied states in the lower and upper band are given by the following matrix element:
= O(&-k)O(lk’+K’I
-/tF) _k
)
F
+C B(lk’+K, K,
= ln,k+K)
(ktHolk+K)(n,k+K,IH,ln',k'+K')
Edd-EE,dk’+&)
_-k ) (k’+K,1H,lk’+K’)(n’, F
Ed4
k’+K#Z,ln, --E,dk’+W
k)* .
(3)
960
H. Bi_?tTNER and E. GERLACH
Here we have neglected the small contributions from the phonon energies in the denominators. First we consider the electron phonon matrix element using the deformation potential approximation (n, k+K,]H,]n’, k’+K’) = il( &/2MNc,)“* { (n + 1)*‘*8& -
,+
rc*s . ..~+J
(4) where 1 is an energy constant of about 27 eV [4] (atomic unit), q is the phonon wave vector, M the ionic mass, N the ionic density, c, the longitudinal velocity of sound and n the Bose
Hence with equations (2) and (6), the virtual optical transition at the zone boundary gives the following matrix element: -(K/2)H,IK/2)+=
(ef~/2m) K-A
(8)
which does not depend on I/. Now we neglect the ordinary Drude term [ll and all contributions proportional to V (indirect transitions not too close to the zone boundaries and direct transitions). Inserting (5) and (8) into (3) we obtain two contributions, @ = 2hW’/o(A]*, to the optic conductivity arising from transitions of rate W’ to the upper and lower band, respectively
I
distribution for a given T (whereas the pure electron system is assumed to be at T = 0, if the Fermi level is well below the band edge). For T s 0, (Debye-temperature) equation (4) can be approximated by if(k,T/2MNc,*)
“*{6,,+1+ S,,,t+,}.
(5)
The electron photon matrix element was studied by a number of authors [3] in the context of direct interband transitions to states not too close to the boundaries and was found to be proportional to I/ (first Fourier coefficient of the lattice potential). In addition, in our problem virtual optical transitions close to the zone boundary have to be considered, whose matrix elements turn out to be independent of I/. The nearly free electron wave functions in the vicinity of the band edge can easily be obtained from a degenerate perturbation treatment which yields. ]K/2)’ = (2L3)“* [exp (iK*r/2 & exp (--iK*r/2)] corresponding
(6)
to the eigenvalues
E;,’ (K/2) = (fiK/2) 2/2m + V .
(7)
The factor 2 arises from phonon emission and absorption, the factor 4 from time averaging the electromagnetic field. The k’ summation is carried out by putting k’ equal to K’/2 and multiplying by the volume of a narrow wave vector shell at the zone boundary. From the breakdown of the non-degenerate perturbation theory, the width of the shell is estimated to be mV/‘lhK. In the lowestorder approximation a constant gap width is assumed to exist all over the zone boundary, which we take to be spherical. The constant gap approximation corresponds to a reasonable simplification of Ham’s [5] results (except for Na). This is reconfirmed later from the good agreement of our curves with the experimental ones. For a more accurate treatment detailed knowledge of the band structure of the zone boundary would be required, but this is expected to lead only to minor corrections. We consider only transitions between the conduction and the first excited band. This implies that K’ and k’ are not independent so that only & of the shell has to be considered. Summation over K’ gives a factor 32 (/AIn-/a)* (a = lattice constant).
OPTICAL
Carrying obtain
CONDUCTIVITY
out the k-summation
we finally
(TV= 1 e2PkBTm1j2Eg( Eel* - hw ) u2 6
fi2NMa3c 2 (fiw)3 ; fl(ho-E.$+E,).
(10)
The optical conductivity cr= u++ ucalculated for Na and K gives the order of 1014see-’ in agreement with the experiments. However, it should be emphasized that there is some uncertainty in the sound velocity, the band edges, the interaction parameter 1, and also in the shell width. In Fig. 1 an example is
b
hw (ev) Fig. 1. Calculated optical conductivity in the MayerNaby frequency range (broken line: averaged by 0.4 eV intervals).
given for the o-dependence of u (normalized to 1014see-l), which still has to be superimposed and smeared out by the ordinary Drude and the direct interband term. 3. CONCLUSIONS
We have shown that the Mayer-Naby peak can be explained by simple indirect transitions from the Fermi sea to the zone boundaries. We find the calculated peak for all alkali metals to have the correct position on the
IN ALKALI
METALS
961
frequency scale and to be of the order of 1014set-’ as required to explain the MayerNaby experiments. The curves in Fig. 1 show an asymmetry in the same sense as the experimental ones, the steepest descent being on the low energy side. The fact that our peak for Na is more pronounced than the experimental one may be due to the constant gap approximation, which in this case is not as well justified as for the other alkali metals. Further, the temperature dependence of (10) is in agreement with experimental results obtained for Na; it does not explain the behavior of K and Cs. The interference dip [l] at the threshold of the Mayer-Naby peak is expected to appear if the 1V12-contributions and the Drude term are taken into account. It can be shown that there is no interference between the Drude term and our approximation. Since we used only small portions of the detailed band structure which are uncritical in transitions to the liquid state, no major deviations are to be expected in this case, in agreement with the experiments (see [23). After completion of the present work new experimental data on Na and K were published by Neville V. Smith[6], which do not exhibit the Mayer-Naby peak. As suggested by the author this discrepancy may be explained by the fact, that the experiments were performed on evaporated films rather than on bulk material. REFERENCES 1. ABELES F., (Editor) Optical properties and electronic structure of metals and alloys. North Holland, Amsterdam (1966). 2. NETTEL S. J., Phys. Rev. 150, 421 (1966); see also PHILLIPSJ. C.,SolidSt. Phys. l&55 (1966). 3. BUTCHER P. N., Proc. phys. Sot. A64,765 (1951); HOPFlELD J. J., Phys. Rev. 139, A 419 (1965); OVERHAUSER A. W., Phys. Rev. 156, 844 (1967); APPELBAUM J. A., Phys. Reo. 144, 453 (1966); ANIMALU A. 0. E., Phys. Rev. 163, 557; 562 (1967). 4. KITTEL C., Quantum Theory of Solids. Wiley, New York (1963). 5. HAM F. S., Phys: Rev. 128,82 (1962). 6. SMITH N. V., Phys. Rev. Lett. 21,96 (1968).