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Patterson, Bailey - Solid State Physics Introduction to theory

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10.10 Lattice Absorption, Restrahlen, and Polaritons (B)	569

10.10 Lattice Absorption, Restrahlen, and Polaritons (B)

10.10.1 General Results (A)

Polar solids carry lattice polarization waves and hence can interact with electromagnetic waves (only transverse optical phonons couple to electromagnetic waves by selection rules and conservation laws). The dispersion relations for photons and the phonons of the polarization waves can cross. When these dispersion relations cross, the resulting quanta turn out to be neither photons nor phonons but mixtures called polaritons. One way to view this is shown in Fig. 10.12. We now discuss this process in more detail. We start by considering lattice vibrations in a polar solid. We will later add in a coupling with electromagnetic waves. The displacement of the tth ion in the lth cell for the jth component, satisfies

Mtvtlj = −∑t′h Gttjj′′(h)vtj′,l +h ,					(10.145)
where
G jj′	=	∂ 2U		,	(10.146)
G jj′	=	∂ v j∂ v j′		,	(10.146)
tl,t′,l′=l +h		∂ v j∂ v j′
		tl	t′l′

and U describes the potential of interaction of the ions. If vtl is a constant,

∑t′h Gtt′(h) = 0 .

(10.147)

Fig. 10.12. Polaritons as mixtures of photons and transverse phonons. The mathematics of this model is developed in the text

570 10 Optical Properties of Solids

We will add an electromagnetic wave that couples to the system through the force term.

et E0 exp[i(q l −ωt)] ,

(10.148)

where et is the charge of the tth ion in the cell. We seek solutions of the form

vsl (t) = exp(iq l)vs,q (t) ,

(10.149)

(now s labels ions) with q = K (dropping the vector notation of q, h, and l for simplicity from here on) and t is the time. Defining

Gss′(K ) = ∑h Gss′(h) exp(iKh) ,

(10.150)

we have (for one component in field direction)

= −∑

s′

′(K )v ′

+ e

exp(−iωt) .

(10.151)

s K

Note that

Gss′(K = 0) = ∑h Gss′(h) .

(10.152)

Using the above we find

∑s′Gss′(K = 0) = 0 .

(10.153)

Assuming e1 = |e| and e1 = −|e| (to build in the polarity of the ions), the following equations can be written (where long wavelengths, K 0, and one component of ion location is assumed)

M svs = −∑s′Gss′vs′ + es E0 exp(−iωt) ,

where

Gss′ = ∑h Gss′(h) .

If we assume that

U = ∑l′,h G4 (v1l′ − v2l′+h′)2 ,

where h′ = −1, 0, 1 (does not range beyond nearest neighbors), then

G	(h) =		∂ 2U	= Gδ 0 .
G	(h) =			= Gδ 0 .
11		∂v1l∂v1l +h			h
		∂v1l∂v1l +h
Similarly,
	G		(h′) =Gδ0		,
	22		h′

G11(h) = −G12 (h) ,

(10.154)

(10.155)

(10.156)

(10.157)

(10.158)

(10.159)

10.10 Lattice Absorption, Restrahlen, and Polaritons (B)	571

and
G22 (h) = −G21(h) .	(10.160)
Therefore we can write
M1v1 = G11(v2 − v1) + eE0 exp(−iωt) ,	(10.161)
and
M 2v2 = G22 (v1 − v2 ) − eE0 exp(−iωt) .	(10.162)
We now apply this to a dielectric where
ε = ε0 + P E ,	(10.163)
and
P = ∑i Niαi Eloc,i ,	(10.164)

with Ni = the number of ions/vol of type i and αi is the polarizability. For cubic crystals as derived in the chapter on dielectrics,

Eloc,i = E +

3ε0

Then,

ε = ε0

∑ Niαi E +

3ε0

Let4

B =

∑ Niαi ,

3ε0

ε = ε0 + 3ε0B + B(ε −ε0 ) ,

ε(1− B) = ε0 + 2ε0B ,

and

ε = ε0

1+ 2B

1− B

For the diatomic case, define

N (α

+α

−

) ,

3ε0

(10.165)

(10.166)

(10.167)

(10.168)

(10.169)

(10.170)

(10.171)

4Grosso and Paravicini [55 p342] also introduce B as a parameter and refer to its effects as a “renormalization” due to local field effects.

572 10 Optical Properties of Solids

Nα

ion

(10.172)

ion

3ε0

Then the static dielectric constant is given by

ε(0)

1+ 2[Bel + Bion (0)]

(10.173)

1−[B

+ B (0)]

ion

while for high frequency

ε(∞)

1+ 2Bel

(10.174)

1− B

We return to the equations of motion of the ions in the electric field—which in fact is a local electric field, and it should be so written. After a little manipulation we can write

μv

μG

− v ) +

loc

(10.175)

μv

μG

(v − v

) −

eE .

(10.176)

M 2

loc

M 2

Using

=1 ,

(10.177)

M 2

we can write

μ(v1 − v2 ) + G(v1 − v2 ) = eEloc .

(10.178)

We first discuss this for transverse optical phonons.5 Here, the polarization is perpendicular to the direction of travel, so

Eloc =			P				(10.179)
Eloc =			3ε0				(10.179)
			3ε0
in the absence of an external field. Now the polarization can be written as
P = Pel + Pion = N (α− +α+ )Eloc + Nev , v = v1 − v2 ,							(10.180)
P = N (α+ +α−)				P		+ Nev ,	(10.181)
P = N (α+ +α−)						+ Nev ,	(10.181)
				3ε0
and
P =		Nev			,		(10.182)
P =	1	− B			,		(10.182)
	1	− B
				el

5 A nice picture of transverse and longitudinal waves is given by Cochran [10.7].

10.10 Lattice Absorption, Restrahlen, and Polaritons (B)	573

so the local field becomes
Eloc =		1			Nev		.		(10.183)
Eloc =							.		(10.183)
	3ε0 1− Bel
The equation of motion can be written
μv + Gv =			1			Ne2v		.	(10.184)
μv + Gv =			3ε	0	1− B			.	(10.184)
			3ε		1− B
							el

Seeking sinusoidal solutions of the form v = v0exp(−iωTt) of the same frequency dependence as the local field, then

(1/ 3ε0 )(Ne2 / G)

ωT

−

(10.185)

1− B

We suppose αion is the static polarizability so that

=αion

(10.186)

Eloc

form the equations of motion. So,

Bion (0) =

3ε0

Nαion =

(10.187)

3ε0

(0)

−

ion

(10.188)

− B

For the longitudinal case with q || P we have

= −

+ 1

= −

(10.189)

1oc

ε0

3 ε0

So,

(10.190)

P = Pel + Pion = N (α+ +α−) −

+ Nev = −2BelP + Nev .

3 ε0

Then, we obtain the equation of motion,

μv + Gv = −

Ne2v

(10.191)

3ε

1+ 2B

574 10 Optical Properties of Solids

ω2

(2 / 3ε

)(Ne2

/ G)

1 + 2Bel

By the same reasoning as before, we obtain

(0)

ion

1+ 2B

Thus, we have shown that, in general

ω2

2[B

+ B

(0)]

ion

1+ 2B

and

2 =

−

+ B

(0)

ion

1− B

Therefore, using (10.173), (10.174), (10.194), and (10.195) we find

ε(∞)	=	ωT2 .
ε(0)		ωL2

(10.192)

(10.193)

(10.194)

(10.195)

(10.196)

This is the Lyddane–Sachs–Teller Relation, which was mentioned in Sect. 9.3.2, and also derived in Section 4.3.3 (see 4.79) as an aside in the development of polarons. Compare also Kittel [59, 3rd edn, 1966, p393ff] who gives a table showing experimental confirmation of the LST relation. The original paper is Lyddane et al [10.20]. An equivalent derivation is given by Born and Huang [10.2 p80ff].

For intermediate frequencies ωT < ω < ωL,

ε(ω) −ε(∞)

1+ 2[Bel + Bi (ω)]

−

1+ 2Bel

(10.197)

1-[B

+ B (ω)]

1− B

ε(ω)

ε(∞)

B (ω) .

(10.198)

[1- B − B (ω)](1− B ) i

We need an expression for Bi(ω). With an external field since only transverse phonons are strongly interacting

μv + Gv =	1			Ne2	v + eE ,	(10.199)
	3ε	0	1	− B

				el

10.10 Lattice Absorption, Restrahlen, and Polaritons (B)	575

B (0) =

Nα

(0) =

N e2 .

3ε0

Eloc

3ε0

Seeking a solution of the form v = v0exp(−iωt) we get

2vμ + Gv −

B (0) = eE .

1− Bel

So,

ω2

B (0)

−

1− B

μ(ωT2 −ω2 )v = eE .

So,

αi (ω) =

Eloc μ(ωT2 −ω2 )

Eloc

Using the local field relations, we have

= E +

Nev

loc

3ε0

1− Bel

= E +

3ε

− B

μ (ω2

−ω2 )

so,

Eloc

1+ F

μ ω2 −ω2

where,

F =

Bi (0)

(1 − B )(ω2

−ω2 )

Or,

B (ω) =

Nα

(ω) = (1− B )

1+ F

3ε0

(10.200)

(10.201)

(10.202)

(10.203)

(10.204)

(10.205)

(10.206)

(10.207)

(10.208)

576 10 Optical Properties of Solids

ε(ω)

ε(∞)

3(1− Bel )F /(1+ F )

(10.209)

(1− B

)[(1− B

) − (1− B

)F /(1+ F )]

ε(ω)

ε(∞)

Bi (0)

(10.210)

ε0

1− Bel

(1 − Bel )(ωT2 −ω2 )

Defining

c = 3

Bi (0)

(10.211)

(1− B

after some algebra we also find

cε0

ωT +

= ωL .

(10.212)

ε(∞)

10.10.2 Summary of the Properties of ε(q, ω) (B)

Since n = ε1/2 with σ = 0 (see (10.8)), if ε < 0, one gets high reflectivity (by (10.15) with nc pure imaginary). Note if

ω2	< ω2 < ω2	+	cε0	,	(10.213)
			ε(∞)
T	T

then ε(ω) < 0, since by (10.210), (10.211), and (10.212) we can also write

ε(ω) = ε(∞)	ωL2 −ω2	,
	ωT2 −ω2

and one has high reflectivity (R → 1). Thus, one expects a whole band of forbidden nonpropagating electromagnetic waves. ωT is called the Restrahl frequency and the forbidden gap extends from ωT to ωL. We only get Restrahl absorption in semiconductors that show ionic character; it will not happen in Ge and Si. We give some typical values in Table 10.2. See also Born and Huang [2, p118].

	10.10 Lattice Absorption, Restrahlen, and Polaritons (B)				577

Table 10.2. Selected lattice frequencies and dielectric constants

Crystal	ωT (cm–1)	ωL (cm–1)	ε(0) (cgs)	ε(∞) (cgs)
InSb	185	197	17.88	15.68
GaAs	269	292	12.9	10.9
NaCl	164	264	5.9	2.25
KBr	113	165	4.9	2.33
LiF	306	659	8.8	1.92
AgBr	79	138	13.1	4.6

From Anderson HL (ed), A Physicists Desk Reference 2nd edn, American Institute of Physics, Article 20: Frederikse HPR, Table 20.02.B.1 p.312, 1989, with permission of Springer-Verlag. Original data from Mitra SS, Handbook on Semiconductors, Vol 1, Paul W (ed), North-Holland, Amsterdam, 1982, and from Handbook of Optical Constants of Solids, Palik ED (ed), Academic Press, Orlando, FL, 1985.

10.10.3 Summary of Absorption Processes: General Equations (B)

Much of what we have discussed can be summed up in Fig. 10.13. Summary expressions for the dielectric constants are given in (10.67) and (10.68). See also Yu and Cardona [10.27, p. 251], and Cohen [10.8] as well as Cohen and Chelikowsky [10.9, p31].

Fig. 10.13. Sketch of absorption coefficient of a typical semiconductor such as GaAs. Adapted from Elliott and Gibson [10.11, p. 208]

578 10 Optical Properties of Solids

10.11Optical Emission, Optical Scattering and Photoemission (B)

10.11.1 Emission (B)

We will only tread lightly on these topics, but they are important to mention. For example, photoemission (the ejection of electrons from the solid due to photons) can often give information that is not readily available otherwise, and it may be easier to measure than absorption. Photoemission can be used to study electron structure. Two important kinds are XPS – X-ray photoemission from solids, and UPS ultraviolet photoemission. Both can be compared directly with the valenceband density of states. See Table 10.3. A related discussion is given in Sect. 12.2.

	Table 10.3. Some optical experiments on solids

High-energy re-	The low-energy range below about 10 eV is good for investigating
flectivity	transitions between valence and conduction bands. The use of syn-
	chrotron radiation allows one to consider much higher energies
	that can be used to probe transitions between the conduction-band
	and core states. Since core levels tend to be well defined, such
	measurements provide direct data about conduction band states in-
	cluding critical point structure. The penetration depth is large com-
	pared to the depth of surface irregularities and thus this measure-
	ment is not particularly sensitive to surface properties. Only
	relative energy values are measured.
Modulation	This involves measuring derivatives of the dielectric function to
spectroscopy	eliminate background and enhance critical point structure. The
	modulation can be of the wavelength, temperature, stress, etc. See
	Cohen and Chelikowsky p. 52.
Photoemission	Can provide absolute energies, not just relative ones. Can use to
	study both surface and bulk states. Use of synchrotron radiation is
	extremely helpful here as it provides a continuous (from infrared to
	X-ray) and intense bombarding spectrum.
XPS and UPS	X-ray photoemission spectroscopy and ultraviolet photoemission
	spectroscopy. Both can now use synchrotron radiation as a source.
	In both cases, one measures the intensity of emitted electrons ver-
	sus their energy. At low energy this can provide good checks on
	band-structure calculations.
ARPES	Angle-resolved photoemission spectroscopy. This uses the wave-
	vector conservation rule for wave vectors parallel to the surface.
	Provided certain other bits of information are available (see Cohen
	and Chelikowsky, p.68), information about the band structure can
	be obtained (see also Sect. 3.2.2).

Reference: Cohen and Chelikowsky [10.8]. See also Brown [10.3].

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