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Patterson, Bailey - Solid State Physics Introduction to theory

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10.2 Macroscopic Properties (B)	549

There are some other ways to write these relationships,
εr (ω) =	1	P∫∞	εi (ω) dω =	1	P ∫∞ εi (ω) dω +			∫0	εi (ω) dω	. (10.37)

	π	−∞	ω − a	π		0	ω − a	−∞	ω − a

But, the second term can be written

− ∫−∞

εi (ω)

dω = ∫−∞

[−εi (ω)]

d(−ω) = ∫∞

[−εi (−ω)]

dω ,

(10.38)

0 ω − a

−ω + a

0 ω + a

and ε*(r, t) = ε(r, t), so ε(−q,−ω) = ε*(q, ω). Therefore,

ε (−ω) = ε(ω) ;

(10.39)

εr (−ω) = εr (ω) ,

(10.40)

and

εi (−ω) = −εi (ω) .

(10.41)

We get

P∫0

εi (ω)

dω = ∫∞

εi (ω)

dω .

(10.42)

−∞ ω − a

0 ω + a

We can thus write the real component of the dielectric constant as

εr (a) =

∫

∞ εi (ω)dω

+ ∫∞ εi (ω)dω

∫∞ ωεi (ω)dω

π 0 ω − a

0 ω + a

π 0 ω2 − a2

and similarly the imaginary component can be written

εi (a) = −

∫

∞ εrdω

= −

∫∞ εrdω + ∫0 εrdω

π −∞ ω − a

0 ω − a

−∞ ω − a

= −

∫∞ εrdω + ∫0 εr (−ω)d(−ω)

ω − a

−∞

−ω + a

= −

∫∞ εr (ω)dω

+ ∫0 εr (ω)dω

ω − a

∞ ω + a

= −

∫∞ εr (ω)dω

− ∫∞ εr (ω)dω

= −

2aP

∫∞ εr (ω)dω .

ω − a

ω + a

π 0 ω2 − a2

(10.43)

(10.44)

In summary, the Kronig–Kramers relations can be written, where εr(ω) → εr(ω) − 1 should be substituted


εr (a) =	P		∫∞	Im[ε(ω)]dω	=	2P		∫∞ ωεi (ω)dω ,		(10.45)
				ω − a
	π −∞					π 0 ω2 − a2
εi (a) = −	P	∫∞		Re[ε(ω)]dω	= −		2Pa		∫∞ εr (ω)dω .	(10.46)
				ω − a
	π −∞						π 0 ω2 − a2

550 10 Optical Properties of Solids

10.3 Absorption of Electromagnetic Radiation–General (B)

We now give a fairly general discussion of the absorption process by quantum mechanics (see also Yu and Cardona [10.27 Chap. 6] as well as Fox op. cit. Chap. 3). Although much of the discussion is more general, we have in mind the absorption due to transitions between the valence and conduction bands of semiconductors. If −e is the electronic charge, and if we assume the electromagnetic field is described by a vector potential A and a scalar potential φ, the Hamiltonian describing the electron in the field is in SI

H =	1	[p + eA]2 − e(φ +V ) ,	(10.47)
	2m

where V is the potential in the absence of an electromagnetic field; V would be a periodic potential if the electron were in a solid. We will use the Coulomb gauge to describe the electromagnetic field so φ = 0, A = 0 and the fields are given by

E = −	∂A	, B = × A .	(10.48)
	∂t

The Hamiltonian can then be written

H =	1	[ p2 + eA p + ep A + e2 A2 ] − eV .	(10.49)
	2m

The terms quadratic in A will be ignored as they are normally small compared to the terms linear in A. Further in the Coulomb gauge, we can write

p Aψ (Aψ) = ( A)ψ + (A )ψ = A ψ ,						(10.50)
so that the Hamiltonian can be written
H = H 0 + H ′ ,		H 0 =		p2	− eV ;
H = H 0 + H ′ ,		H 0 =	2m		− eV ;
			2m
where the perturbation is
H ′ =	e	A p .				(10.51)
H ′ =	m	A p .				(10.51)
	m

We assume the matrix element responsible for electronic transitions will be in the form f|H|i , where i and f refer to the initial and final electron states and H′ is the perturbing Hamiltonian. We assume the vector potential is given by

A(r,t) = ae{exp[i(k r −ωt)] + exp[−i(k r −ωt)]} ,

(10.52)

where e k = 0 and a2 is given by

a = E2 2ω2 ,

(10.53)

10.4 Direct and Indirect Absorption Coefficients (B)	551

	2
¯¯		is the averaged squared electric field. Then,
where E		is the averaged squared electric field. Then,
		Pabsorption		= 2π a2 e2	f	exp(ik r)e p i	2 ,	(10.54)
		i→f		m2
				m2
and for emission
		Pemission	= 2π a2 e2		f	exp(−ik r)e p i	2 .	(10.55)
		i→f		m2
				m2

10.4 Direct and Indirect Absorption Coefficients (B)

Let us now look at the absorption coefficient. Using Bloch wave functions (ψk = eik·ru(r)), we have

f exp(ik r)e p i = ∫ u f	exp[i(k − k f	+ ki ) r]	e kiuidΩ
+ ∫ u f	exp[i(k − k f	+ ki ) r]	e puidΩ .

The first integral can be written as proportional to

∫ψ f ψi exp( ik r )dΩ = ∑ j exp[ i( k − k f + ki ) Rj ] ×∫Ωc u f exp[ i( k − k f + ki ) r ]uidΩ

N ∫Ωc ψ f ψidΩ 0 ,

(10.56)

(10.57)

by orthogonality and assuming k is approximately zero, where we have also used

∑ j exp[i(k − k f + ki ) R j ] = δkk f −ki (N ) ,

(10.58)

and Ωc is the volume of a unit cell. The neglect of all terms but the k = 0 terms (called the electric dipole approximation) allows a similar description of the emission term. Following a similar procedure for the second term in (10.56), we obtain for absorption,

f exp(ik r) e p i = N ∫	u f e p uidΩ ,	(10.59)
Ωc

with k = 0 and ki = kf.

Notice in the electric dipole approximation since ki = kf, we have what are called direct optical transitions. If something else such as phonons is involved, direct transitions are not required but the whole discussion must be modified to include this new physical ingredient. The electric dipole transition probability for photon absorption per unit time is

Piabs→ f =	2π	∑k		E2	e2 N 2	∫Ωc u*f e p uidΩ	2

					m2		δ[Ec (k) − Ev (k) − ω] . (10.60)
			2ω2

552 10 Optical Properties of Solids

The power (per unit volume) lost by the field due to absorption in the medium is the transition probability per unit volume P multiplied by the energy of each photon (where in carrying out the sum over k in (10.60), we will assume we are summing over k states per unit volume). Carrying out the manipulations below, we finally find an expression for the absorption coefficient and, hence, the imaginary part of the dielectric constant. The power lost equals

P ω = −					dI			,
P ω = −					dt			,
					dt
where I is the energy/volume. But,
dI		dI dx						c
− dt = −				dt	=		αI			,
− dt = −		dx		dt	=		αI		n	,
where α = 2niω/c, and ni = εi/2n. Thus,
− dI	= εiωI				= P ω .
dt			n2
Using
	1		2
I =	1	n	2			2	×2 ,
I =	2	n	ε0 E				×2 ,
	2

(10.61)

(10.62)

(10.63)

(10.64)

where n = (ε/ε0)1/2 if μ = μ0 and the factor of 2 comes from both magnetic and electric fields carrying current, we find

εi (ω) =	P	1	.	(10.65)
	ε0

		E2

Using the Kronig–Kramers relations, we can also derive an expression for the real part of the dielectric constant. Defining

	M vc	=	∫ u f e puidΩ	,

			Ω
we have (using (10.65), (10.66), and (10.60))

	π		e 2	∑k		2

εi =					M vc		δ (Ec − Ev −	ω) ,

	ε0	mω

and by (10.45)

Mvc

εr =1

∑k

mε0

m ω

−

(10.66)

(10.67)

(10.68)

10.4 Direct and Indirect Absorption Coefficients (B)	553

(where Ec − Ev ≡ ωcv and δ(ax) = δ(x)/a has been used). Recall that the Σk has to be per unit volume and the oscillator strength is defined by

fvc =	2	M vc	2	.	(10.69)


	m ωcv

Classically, the oscillator strength is the number of oscillators per unit volume with frequency ωcv. Thus, the real part of the dielectric constant can be written

		e2		fvc
εr =1	+		∑k			.	(10.70)
εr =1	+	mε0	∑k	2	2	.	(10.70)
		mε0	ωcv −ω

We want to work this out in a little more detail for direct absorption edges. For direct transitions between parabolic valence and conduction bands, effective mass concepts enter because one has to deal with both the valence band and conduction band. For parabolic bands we write

Evc = Eg +					2k		2	,
Evc = Eg +					2μ			,
					2μ
where
1		=	1	+	1	.
	μ	=	m	+		.
	μ		m		m
			c		v

The joint density of states per unit volume (see (10.94)) is then given by

μ3/ 2

− E

where

> E

2 3

and

D j

= 0

where

Evc < Eg .

(10.71)

(10.72)

(10.73)

(10.74)

Thus, we obtain that the imaginary part of the dielectric constant is given by

						Ω >1 ,
εi (ω) = K			Ω −1,			Ω >1 ,		(10.75)
		0	,			Ω <1

where
K =	2e2	(2μ)3 2 Mvc				2 Eg	,	(10.76)
K =		m2ω2 3					,	(10.76)
		m2ω2 3
and
		Ω =		ω	.			(10.77)
		Ω =			.			(10.77)
				Eg

554 10 Optical Properties of Solids

Fig. 10.3. (a) Direct transitions and indirect transitions due to band filling; (b) Indirect transitions, where kph is the phonon wave vector; (c) Vertical transitions dominate indirect transitions when energy is sufficient to cause them. Emission and absorption refer to phonons in all sketches

From this, one then has an expression for the absorption coefficient (since α = ωεi/nc). Thus for direct transitions and parabolic bands, a plot of the square of the absorption coefficient as a function of the photon energy should be a straight line, at least over a limited frequency. Figure 10.3 illustrates direct and indirect transitions and absorption. Indirect transitions are discussed below.

The fundamental absorption edge due to the bandgap determines the apparent color of semiconductors as seen by transmission.

We now want to discuss indirect transitions. So far, our analysis has assumed a direct bandgap. This means that the k of the initial and final electronic states defining the absorption edge are almost the same (as has been mentioned, the k of

10.4 Direct and Indirect Absorption Coefficients (B)	555

the photon causing the absorption is negligible, compared to the Brillouin zone width, for visible wavelengths). This is not true for the two most common semiconductors Si and Ge. For these semiconductors, the maximum energy of the valence band and the minimum energy of the conduction band do not occur at the same k vectors, one has what is called an indirect bandgap semiconductor. For a minimum energy transition across the bandgap, something else, typically a phonon, must be involved in order to conserve wave vector. The requirement of having, for example, a phonon being involved reduces the probability of the event; see Fig. 10.3b, c, Fig. 10.4, (10.82), and consider also Fermi’s Golden Rule.

Even in a direct bandgap semiconductor, processes can cause the fundamental absorption edge to shift from direct to indirect, see Fig. 10.3a. For degenerate semiconductors, the optical absorption edge may be a function of the carrier density. In simple models, the location of the Fermi energy in the conduction band can be estimated on the free-electron model. When the Fermi energy is above the bottom of the conduction band, the k vector of the minimum energy that can cause a transition has also shifted from the k of the conduction band minimum. Now direct transitions will originate from deeper states in the valence band, they will be stronger than the threshold energy transitions, but of higher energy.

Fig. 10.4. Indirect transitions: hfa = Eg − Ephonon, hfb = Eg + Ephonon, Eg = (hfa + hfb)/2, sketch

Phonon Electron

Photon Electron

Electron

Fig. 10.5. An indirect process viewed in two steps

For indirect transitions, we can write the energy and momentum conservation conditions as follows:

k′ = k + K ± q ,

(10.78)

556 10 Optical Properties of Solids

where K = photon 0 and, q = phonon (=kph in Fig. 10.3b). Also
E(k′) = E(k) + ω ± ωq ,	(10.79)

where ω = photon, and ωq = phonon. Note: although the photon makes the main contribution to the transition energy, the phonon carries the burden of insuring that momentum is conserved. Now the Hamiltonian for the process would look like

H ′ = H ′

+ H ′

(10.80)

photon

phonon

where

H ′

p A ,

(10.81)

photon

and

H ′

∑σkq

(a+

+ a )c+

+q,σ

k,σ

(10.82)

phonon

−q

q k

One can sketch the indirect process as a two-step process in which the electron absorbs a photon and changes state then absorbs or emits a phonon. See Fig. 10.5.

We mention as an aside another topic of considerable interest. We discuss briefly optical absorption in an electric field. The interesting feature of this phenomenon is that in an electric field, optical absorption can occur for photon energies lower than the normal bandgap energies. The increased optical absorption due to an electric field can be qualitatively understood by thinking about pictures such as in Fig. 10.6. This figure does not present a rigorous concept, but it is helpful.

Very simply, we can think of the triangular area in the figure as a potential barrier that electrons can “tunnel” through. From this point of view, one perhaps believes than an electric field can cause electronic transitions from band 2 to band 1 (This is called the Zener effect). Obviously, the process of tunneling would be greatly enhanced if the electron “picked up some energy from a photon before it began to tunnel.” Further details are given by Kane [10.15].

It is not hard to see why the Zener effect (or “Zener breakdown”) can be considered as a tunneling effect. The horizontal line corresponds to the motion of an electron (if we describe electrons in terms of wave packets, then we can speak of where they are at various times and we can label positions in terms of distances in the bands). Actually, we should realize that this horizontal line corresponds to the electric field causing the electron to make transitions to higher and higher stationary states in the crystal. When the electron reaches the top of the lower band, we normally think of the electron as being Bragg reflected. However, we should remember what we mean by the energy gap.

The energy gap, Eg, does not represent an absolutely forbidden gap. It simply represents energies corresponding to attenuated, nonpropagating wave functions. The attenuation will be of the form e−Kx, where x represents the distance traveled (K is real and greater than zero) and K is actually a function of x, but this will be ignored here. The electron gains energy from the electric field E as |eEx|. When

10.4 Direct and Indirect Absorption Coefficients (B)	557

Eg (energy gap)

Band 1

E (electric field)

Band 2

Fig. 10.6. Qualitative effect of an electric field on the energy bands in a solid

the electron has traveled x = |Eg/eE|, it has gained sufficient energy to get into the bottom of the upper band if it started at the top of the lower band. In order for the process to occur, we must require that the electron’s wave function not be too strongly attenuated, i.e. Zener breakdown will occur if 1/K >> |Eg/eE|. To see the analogy to tunneling, we observe that the electron’s wave function in the triangular region also behaves as e−Kx from a tunneling viewpoint (also with K a function of x), and that the larger we make the electric field, the thinner the area we have to tunnel across, so the greater a band-to-band transition. A more quantitative discussion of this effect is obtained by evaluating K not from the picture, but directly from the Schrödinger equation. The x dependence on K turns out to be fairly easy to handle in the WKB approximation.

Finally, we can summarize the results for many cases in Table 10.1. Absorption coefficients α for various cases (parabolic bands) can be written

	A		+ β − Eg )γ ,
α =		(hf		(10.83)

hf

where γ, β depend on the process as shown in the table. When phonons are involved we need to add both the absorption and emission (±) possibilities to get the total absorption coefficient.2 A very clean example of optical absorption is given in Fig. 10.7. Good optical absorption experiments on InSb were done in the early days by Gobeli and Fan [10.15]. In general, one also needs to take into account the effect of temperature. For example, the indirect allowed term should be written

	A′		(hf	+ β − Eg )2		(hf	− β − Eg )2
α =					+			exp(β / kT )	,	(10.84)
α =					+	exp(β / kT ) −1		exp(β / kT )	,	(10.84)
hf		exp(β / kT ) −1				exp(β / kT ) −1

where A′ is a constant independent of the temperature, see, e.g., Bube [10.4] and Pankove [10.22].

2An additional very useful reference is Greenaway and Harbeke [10.16]. See also Yu and Cardona [10.27].

558 10 Optical Properties of Solids

Table 10.1. Absorption coefficients

	γ	β
Direct, allowed	1/2	0	See (10.75)
Direct, forbidden	3/2	0
Indirect, allowed	2	±hfq (phonons)
Indirect, forbidden	3	±hfq (phonons)

γ and β are defined by (10.83).

	104
–1·eV)	103
ω(cm
α·	102
	10
	0	0.2	0.4	0.6	0.8

ω (eV)

Fig. 10.7. Optical absorption in indium antimonide, InSb at 5 K. The transition is direct because both conduction and valence band edges are at the center of the Brillouin zone, k = 0. Notice the sharp threshold. The dots are measurements and the solid line is ( ω − Eg)1/2. (Reprinted with permission from Sapoval B and Hermann C, Physics of Semiconductors, Fig. 6.3 p. 154, Copyright 1988 Springer Verlag, New York.)

10.5 Oscillator Strengths and Sum Rules (A)

Let us define the oscillator strength by

fij = bωij i e r j 2 .

We will show this is equivalent to the previous definition with the proper choice of b by using commutation relations to cast it in another form. From [x, px] = i we can show

[H , x] = −	i	px .	(10.85)
	m

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