- Email: [email protected]
Theory of optical generation of longitudinal phonon modes in ionic crystals
Volume 60A, number 5
PHYSICS LETTERS
21 March 1977
THEORY OF OPTICAL GENERATION OF LONGITUDINAL PHONON MODES IN IONIC CRYSTALS A.D. BOARDMAN, MR. PARKER and T.I.Y. ALLOS Department of Pure and Applied Physics, University of Saljbrd, Salford M5 4WT, Lanes., England Received 31 January 1977 The optical generation of bulk phonon polaritons in ionic crystals is treated by the inclusion of the effects of spatial dispersion in classical dispersion theory. The theory is used to explain structural features of the resonance infra-red reflection spectrum of Cr 203.
It is well-established [1] that the propagation of bulk phonon-polaritons in polar ionic crystals may be treated, reasonably satisfactorily, in terms of a local dielectric function. However, certain prominent features of the infra-red reflectivity curves cannot be predicted [11 by such a simple model. This difficulty, as will be shown here, can be overcome with the introduction of nonlocality (i.e. spatial dispersion) into the dielectric function. The existence of non-locality is manifested bythe the presence of at least two independent modes of normal, transverse, electromagnetic (TEM-wave) and of the longitudinal (LU wave) types [2]. Furthermore, since Maxwell’s equations for an unchanged medium require that V D = 0, such waves correspond to solutions in which V E = 0 (TEM) and e(w) = 0 (LU), the latter being strongly frequency rangepropagated in which e(w) = 0. only in the limited The elastic equation for the relative ionic displacement in the polar crystal, extended to include the effects of spatial dispersion, is simply 2 . + w~U+ yu
*
*
*
E /m +DV(Vu), where u ( u÷ u_) is the relative displacement of adjacent ions of opposite charge, WT is the transverse mode resonance frequency and y is a damping parameter. The effective electric field E* is related to the applied field E through consideration of the Lorentz— Lorenz polarisation effects by the relation U
=
e
—
E* = E+ (n/3 CO)((CL + 2)/3)e*u +
((CL —
l)/3)E, (2)
Assuming that the space-time dependence of all variables is of the form exp i(k r wt), equations (1) and (2) can be combined to give (~j~ + 2) D ~ñ ~ E+ ~(n u)n (3) i*
=—~---—
,
C
where n = ck/w (c is the velocity of light in vacuo) 2 2 2 + iY/w + wp(e~+ 2)19w2, (4) = 1 wTIw and w~= Ne*2/e 0m * is the ionic plasma frequency. After some involved algebraic reduction it is possible to derive the non-local form of the dielectric tensor. This tensor, referred to the direction of the polariton propagation, is diagonal and has the form 2/~w2, (5) ~t = ~ = CL—w ~yy = ~L ((CL+2)/3)2~/w2~(~—Dk/w2), ~‘p((~L + 2)/3) ~1= = (6) .
—
—
1’ =
~L
—
The independent propagating modes, with wave numbers kL and kT have dispersion relations given by [2] i.e. longitudinal: transverse : CL
=
— —
0
,
(7)
c2k.~/w2—A(~_Dk~,/w2)=o,(8)
so that in this case c2k~/w2 = c2(l/D)
[~
—
W~((CL+ 2)/3)2/w2CL],
(9)
is ionic charge density, CL is the high frequency dielec- c2k2 1w2 = CL w2((CL + 2)/3)2/~w2, (10) tric constant where e* is the Szigetti charge and m* is T the effective ionic mass [1]. In eq. (l)D (with dimensions These formulae have been used to calculate the re(velocity) [2] is introduced as a further phenomenoloflectivity of p-polarised electromagnetic waves from a gical parameter. 437
N
—
Volume 60A, number 5
L7p~935L.lO
~
80~
PHYSICS LETTERS
dielectric constants are CL = 6.2 and e~ = 12.7, respectively. The results are shown in fig. 1 where it can be
-~z I
6~ 6~, Es 127,
~60L
~.
theory presented here and the published measurements
62,
~
(c I
t 40~
•
__________________________I
.•
of Renneke and Lynch [31.Their double reststrahlen seen that excellent reached the reflectivity peak has,agreement hitherto, isonly beenbetween accounted for by the arbitrary fitting of resonance lunctions to the experimental data. By contrast agreement between theory and experiment is obtained here quite naturally
arid thus demonstrates clearly the influence of spatial dispersion in crystals of this type.
heQret!COI
20.
~
20 24 V~vetength)J m
~b
28
Fig. 1. % Reflectivity of p-polarised electromagnetic radiation from single Cr 2 03 of thickness d. U is the angle of incidence.
single crystal of Cr203 in the wavelength range 8 pm to 28 pm, with the electric field vector perpendicular to the c-axis. In this crystal the high and low frequency
438
21 March 1977
References [1] TA. Bak (ed.), Phonons and phonon interactions, Aarhus Summer School Lectures (W.A. Benjamin, 1964) p. 276. 21 A.D. Boardman and MR. Parker, Phys. Stat. Sol. (h) 71 (1975) 329. 131 DR. Renneke and D.W. Lynch, Pliys. Rev. 138 (1965) 530.
PHYSICS LETTERS
21 March 1977
THEORY OF OPTICAL GENERATION OF LONGITUDINAL PHONON MODES IN IONIC CRYSTALS A.D. BOARDMAN, MR. PARKER and T.I.Y. ALLOS Department of Pure and Applied Physics, University of Saljbrd, Salford M5 4WT, Lanes., England Received 31 January 1977 The optical generation of bulk phonon polaritons in ionic crystals is treated by the inclusion of the effects of spatial dispersion in classical dispersion theory. The theory is used to explain structural features of the resonance infra-red reflection spectrum of Cr 203.
It is well-established [1] that the propagation of bulk phonon-polaritons in polar ionic crystals may be treated, reasonably satisfactorily, in terms of a local dielectric function. However, certain prominent features of the infra-red reflectivity curves cannot be predicted [11 by such a simple model. This difficulty, as will be shown here, can be overcome with the introduction of nonlocality (i.e. spatial dispersion) into the dielectric function. The existence of non-locality is manifested bythe the presence of at least two independent modes of normal, transverse, electromagnetic (TEM-wave) and of the longitudinal (LU wave) types [2]. Furthermore, since Maxwell’s equations for an unchanged medium require that V D = 0, such waves correspond to solutions in which V E = 0 (TEM) and e(w) = 0 (LU), the latter being strongly frequency rangepropagated in which e(w) = 0. only in the limited The elastic equation for the relative ionic displacement in the polar crystal, extended to include the effects of spatial dispersion, is simply 2 . + w~U+ yu
*
*
*
E /m +DV(Vu), where u ( u÷ u_) is the relative displacement of adjacent ions of opposite charge, WT is the transverse mode resonance frequency and y is a damping parameter. The effective electric field E* is related to the applied field E through consideration of the Lorentz— Lorenz polarisation effects by the relation U
=
e
—
E* = E+ (n/3 CO)((CL + 2)/3)e*u +
((CL —
l)/3)E, (2)
Assuming that the space-time dependence of all variables is of the form exp i(k r wt), equations (1) and (2) can be combined to give (~j~ + 2) D ~ñ ~ E+ ~(n u)n (3) i*
=—~---—
,
C
where n = ck/w (c is the velocity of light in vacuo) 2 2 2 + iY/w + wp(e~+ 2)19w2, (4) = 1 wTIw and w~= Ne*2/e 0m * is the ionic plasma frequency. After some involved algebraic reduction it is possible to derive the non-local form of the dielectric tensor. This tensor, referred to the direction of the polariton propagation, is diagonal and has the form 2/~w2, (5) ~t = ~ = CL—w ~yy = ~L ((CL+2)/3)2~/w2~(~—Dk/w2), ~‘p((~L + 2)/3) ~1= = (6) .
—
—
1’ =
~L
—
The independent propagating modes, with wave numbers kL and kT have dispersion relations given by [2] i.e. longitudinal: transverse : CL
=
— —
0
,
(7)
c2k.~/w2—A(~_Dk~,/w2)=o,(8)
so that in this case c2k~/w2 = c2(l/D)
[~
—
W~((CL+ 2)/3)2/w2CL],
(9)
is ionic charge density, CL is the high frequency dielec- c2k2 1w2 = CL w2((CL + 2)/3)2/~w2, (10) tric constant where e* is the Szigetti charge and m* is T the effective ionic mass [1]. In eq. (l)D (with dimensions These formulae have been used to calculate the re(velocity) [2] is introduced as a further phenomenoloflectivity of p-polarised electromagnetic waves from a gical parameter. 437
N
—
Volume 60A, number 5
L7p~935L.lO
~
80~
PHYSICS LETTERS
dielectric constants are CL = 6.2 and e~ = 12.7, respectively. The results are shown in fig. 1 where it can be
-~z I
6~ 6~, Es 127,
~60L
~.
theory presented here and the published measurements
62,
~
(c I
t 40~
•
__________________________I
.•
of Renneke and Lynch [31.Their double reststrahlen seen that excellent reached the reflectivity peak has,agreement hitherto, isonly beenbetween accounted for by the arbitrary fitting of resonance lunctions to the experimental data. By contrast agreement between theory and experiment is obtained here quite naturally
arid thus demonstrates clearly the influence of spatial dispersion in crystals of this type.
heQret!COI
20.
~
20 24 V~vetength)J m
~b
28
Fig. 1. % Reflectivity of p-polarised electromagnetic radiation from single Cr 2 03 of thickness d. U is the angle of incidence.
single crystal of Cr203 in the wavelength range 8 pm to 28 pm, with the electric field vector perpendicular to the c-axis. In this crystal the high and low frequency
438
21 March 1977
References [1] TA. Bak (ed.), Phonons and phonon interactions, Aarhus Summer School Lectures (W.A. Benjamin, 1964) p. 276. 21 A.D. Boardman and MR. Parker, Phys. Stat. Sol. (h) 71 (1975) 329. 131 DR. Renneke and D.W. Lynch, Pliys. Rev. 138 (1965) 530.