PART VI. Physics & Applications of Optical …...Application of Quantum Mechanics to Part VI: Physics & Applications of Optical Measurements Condensed Matter Physics Nai-Chang Yeh

Application of Quantum Mechanics to Part VI: Physics & Applications of Optical Measurements Condensed Matter Physics

Nai-Chang Yeh Caltech Ph135a (2017 – 2018)

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PART VI. Physics & Applications of Optical Measurements References:

1) “Principles of the Theory of Solids”, J. M. Ziman, Chapter 8. 2) “Quantum Theory of Solids”, C. Kittel, Chapter 15.

In this chapter we consider the physical principles of optical measurements of condensed matter by

considering the interaction of electromagnetic waves with solids and the resulting response in optical measurements that are characteristic of the physical properties of the material and are generally frequency dependent. We begin in PartVI.1 with general descriptions of the macroscopic theory of light propagation in solids, followed by discussion of the dispersion and absorption of lights in Part VI.2 based on microscopic consideration. In Part VI.3 we consider the photon-phonon interaction in solids, which yields absorption lines in optical spectra that are associated with the lattice vibrations of solids and are determined by the selection rules. Useful information about the electronic properties of semiconductors and ionic crystals may be derived from optical measurements by considering the direct interband transitions and the phonon-assisted indirect interband transitions, as well as transitions among exciton states, as discussed in Part VI.4. Finally, the interaction of photons with conduction electrons typical of metallic systems is discussed in Part VI.5.

VI.1. Macroscopic Theory

To understand light-matter interaction, we begin with considering macroscopically how electromagnetic waves propagate in a neutral medium. The Maxwell equations for this problem are given as follows:

4, 0,

, 0.

c t c

c t

EH E E

HE H

(VI.1)

For simplicity, we assume that the medium is non-magnetic so that the permeability in CGS units is taken as = 1. We remark that the presence of the conductivity is the primary difference between propagation in free space and in a medium. From EQ. (VI.1), we obtain the differential equation for the electric field E:

2

22 2 2

4

c t c t

E E

E . (VI.2)

Therefore, if the electric field of given in a plane wave form:

0 exp i t E E K r , (VI.3)

we find that the wavevector K |K| becomes a complex quantity from EQs. (VI.2) and (VI.3):

1 22

2

2 2

4 4, Ni K i

c c c c

K , (VI.4)



2

where N (n + i) is defined as the complex refraction index. Therefore, if the electromagnetic wave is propapating along the z-direction into the medium, EQ. (VI.3) becomes

0 exp expn

i z t zc c

E E , (VI.5)

so that the velocity of the wave in the medium is reduced by a factor n and is also damped as it propagates. The damping is associated with absorption of the medium through the induced current density J and the resulting joule heating.

To evaluate the absorption of the medium, we consider the induced current density J from EQ. (VI.1):

24 4Ni i

c t c c c c

E

J H E E E . (VI.6)

Therefore, the joule heating is given by

2 2 24 4 2Re Re

ni E E E

c c c c

J E . (VI.7)

Hence, the absorption coefficient , which refers to the fraction of energy absorbed in passing a unit thickness of the material, is given by

2

Re 2

nE c

J E

. (VI.8)

Typically when we study optical properties of materials, we would shine light on the surface of the

material under consideration. Therefore, it is useful to consider the situation of radiating light on a planar surface of a material. For simplicity, let’s consider the normal incident condition, and assume that the material occupies the space z > 0 and the light incidents from free space at z < 0 towards the material with the direction of propagation along positive z-axis. If the electric field amplitudes for the transmitted, incident and reflected waves are given by E0, E1 and E2, respectively, we find that the electric field in z > 0 is given by

0

N0 expxE z E i z t

c

, (VI.9)

and the electric field in z < 0 takes the form

1 20 exp expx

z zE z E i t E i t

c c

. (VI.10)

At the interface z = 0, both the electric field E and magnetic field H must be continuous for a neutral material. Therefore, we have the following two boundary conditions:

0 1 2E E E , (VI.11)

0 1 2NE E E . (VI.12)



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Therefore, we find that the ratio of the complex amplitude of the reflected and incident waves is

2

1

1- N

1+N

E

E . (VI.13)

The corresponding reflection coefficient R is thus given by

22 2

2 2

11- NR

1+N 1

n

n

. (VI.14)

The above macroscopic analysis suggests that by measuring both R and , we will be able to determine

the values of n and . However, the quantities n and are not completely independent of each other. They are actually related by their frequency dependent dispersion relations. In fact, this type of relations are common if we consider linear response functions and causality, which we elaborate below.

The frequency-dependent linear response of a system D() to an external perturbation F() may be

expressed in terms of a general susceptibility (), which is a complex quantity ’+ i”, as follows:

D F . (VI.15)

The corresponding Fourier transform of D() is simply

.

i t i t

i t i t i t

D t d e D d e F

d e dt t e dt F t e

dt t t F t dt t t F t

. (VI.16)

Applying the condition of causality, we have

0 for .t t t t (VI.17)

Hence, the general susceptibility may be expressed as follows:

0

i t i tdt t e dt t e

. (VI.18)

Next, we consider as a complex variable so that ' + i and 0+. If we further assume that

there is no singularity above the real-axis (') in the complex plane and that the magnitude of general susceptibility vanishes as || ∞, we may apply Cauchy’s theorem to EQ. (VI.18), which yields:

1d

i

. (VI.19)

Noting that () = () from EQ. (VI.18), we may separate out the real and imaginary parts of the general susceptibility in EQ. (VI.19) and obtain the following Kramers-Kronig relations:



4

2 20

2const.d

, (VI.20)

and 2 20

2d

. (VI.21)

Returning to our discussion on the complex refractive index N, we have

2 2 2N 2n i n , (VI.22)

so that

2 2

2 20

22const.

nn d

, (VI.23)

and

2 2

2 20

22

nn d

. (VI.24)

Consequently, by measuring either the reflection or the absorption coefficient of a material over a wide range of frequency, we should be able to determine the values of n and .

VI.2. Microscopic Theory for Dispersion and Absorption

Next, we want to consider how the macroscopic optical properties are related to the microscopic physical system of the material. Our starting point is to consider how atoms in a medium respond to the perturbation of an electromagnetic field. In the linear response limit, we may keep our theory to first-order time-dependent perturbation. For simplicity, we further consider that the atoms are effectively non–interacting, each with a ground state wavefunction 0 ( ) r and excited states wavefunctions ( )j r in the absence of

external perturbation. In the presence of a local electric field

0ˆ i t i tt x E e e E , (VI.25)

the perturbation potential is a dipole field potential given by (e E r), and the perturbed wavefunction for the atom is given by

0 0, exp expj j jjt i t c t i t r E E , (VI.26)

where E0 and Ej are the ground state and excited state energies, respectively. From EQ. (VI.26) and the Schrodiner’s equation we find that the coefficients cj satisfy the differential equation:

0*0

ji tjj

dci d e t e

dt r r E r r

E E. (VI.27)

The solution for cj is therefore given by



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0

0

0

0 0

*00

*0 00

0 00

0 0

0 0

1

1

1

1 1.

j

j

j

j j

t i t

j j

t i ti t i tj

t i ti t i tj

i t i t

j

j j

c t dt d e t ei

dt e e E e d exi

dt e e E e exi

e eeE x

r r E r r

r r r

E E

E E

E E

E E E E

E E E E

(VI.28)

From EQ. (VI.28) it is clear that the coefficients cj would be rapidly oscillating with time except for the special condition

0j j E E . (VI.29)

Additionally, we remark that the matrix element of the dipole moment exj0 must satisfy the selection rules so that it is non-vanishing, and the selection rules may be determined by symmetry consideration according to group theory. Therefore, in the event that the transition from the ground state 0 to the excited state j is not

forbidden, we expect a photon-induced quantum transition between 0 and j when j .

Given EQs. (VI.26) and (VI.28), we may evaluate the dipole moment of the atom under the time-

dependent perturbation of the dipole field to the first order as follows:

*

*0 0

2200

, ,

1 1.

i t i tj j j jj

i t i tjj

j j

ex t d t ex t

ex c t e ex c t e

Ee x e e

r r r

(VI.30)

From EQ. (VI.30), we identify the corresponding “general susceptibility” for the dipole moment in response to an external electromagnetic field, which is called the atomic polarizability and is given by

22

0

2 2

2j j

jj

e x

. (VI.31)

We may further employ the Thomas-Reiche-Kuhn sum rule:

2

02

21j jj

mx

(VI.32)

and define the oscillator strength f j for the transition to the jth excited state as follows:

2

02

2, 1.j j j jj

mf x f

(VI.33)



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Thus, EQ. (VI.31) is rewritten into the following form:

2

2 2

j

jj

fe

m

. (VI.34)

To connect the above microscopic picture to macroscopic theory, we consider a system consisting of a

total number of N0 identical atoms that are subjected to the dipole field perturbation. The total polarization of the system is therefore given by P = N0()E, and the corresponding dielectric constant becomes

02

202 2 2 2

1 4

41 1 ,j j

pj jj j

Nf fN e

m

(VI.35)

where p is plasma frequency. However, EQ. (VI.35) implicitly assumes infinite lifetime for all oscillators, which is unphysical. For a more realistic system, we introduce the relaxation rates j << j so that EQ. (VI.35) is rewritten into

2

2 21 j

p jj j

f

i

. (VI.36)

Let’s consider the frequency dependence of the dielectric constant in different limits. Noting that the

lowest value of j typically corresponds to an optical or infrared frequency, we have p ~ j in general. At low frequencies, the dielectric constant may be approximated by its static value, which is given by

2

20 1 1p

jjj

f

. (VI.37)

In contrast, for high frequencies > all j, we have

2

21 .p

(VI.38)

We also consider the special cases when ~ j, known as regions of anomalous dispersion. If the dielectric constant is negative, from EQ. (VI.4) we find that the refractive index N becomes purely imaginary. Thus, we have

1 20, .n (VI.39)

From (VI.39) and (VI.14), we find that R = 1, implying total reflection from the surface of the medium. For j

, the medium would first appear transparent although with a very high refractive index, suddenly become opaque and totally reflecting, and then become transparent again when approaching the next excitation level.

The aforementioned dispersion relation of the dielectric constant in EQs. (VI.37), (VI.38) and (VI.39) is schematically shown in Fig.VI.1. However, these descriptions did not include the imaginary part of the dielectric constant, which would be inconsistent with the Kramers-Kronig relations. Comparing EQ. (VI.23) with EQ. (VI.35), we find that the following condition must be satisfied:



7

222 p j jj

n f

. (VI.40)

Therefore, from EQs. (VI.35) and (VI.40) the dielectric constant must include an imaginary part and takes the form

2

2 2

2 2 2

2 2

11

2

11

2

p j jjj

p j jjj

if

if

. (VI.41)

The real and imaginary parts of the dielectric constant near the frequency of a specific excitation state j are shown in Fig. VI.2(a).

The effect of finite lifetimes for all oscillators may be included by adding (ij /2) to j where j << j. Thus, the dielectric constant becomes

2 2

22 22 2 2 2 2 2 2 2

2

2 2

1

1 ,

j jp jj

j j j j

jp j

j j

f i

f

i

(VI.42)

which recovers EQ. (VI.36). The expression given in EQ. (VI.42) provides a reasonable description for the occurrence of a series of discrete lines in the absorption spectrum with linewidth broadening, as illustrated in Fig. VI.2(b), and each discrete line corresponds to a specific eigen-state of the system under consideration.

However, we note that the derivation for EQ. (VI.42) assumes that the local field that polarizes each atom is identical to the macroscopic field applied to the system. This assumption is, strictly speaking, not precise because other atoms in the system would result in a net polarization on each local atom. For an isotropic medium, this effect can be corrected by considering an averaged polarization P N0()Elocal to the local field Elocal:

local macro

4

3 E E P . (VI.43)

This Lorentz correction, together with the relations P e()Emacro and () 1 + 4e() [or, equivalently, D ()Emacro = Emacro + 4P, where Emacro represents the macroscopic electric field], leads to e() = N0()/[1(4/3)N0()] and therefore the following Clausius-Mosotti relation:

0

1 4

2 3N

. (VI.44)

The aforementioned discussion for electronic excitations in a collection of atoms can in fact be

generalized to the consideration of photon-induced phonon excitations, although the corresponding frequency range would be quite different due to much heavier ionic masses. For instance, we may define the effective “plasma frequency” p for an ionic crystal as p

2 = 4N0e/Meff, where Meff denotes the effective mass of the ionic crystal. We further define the dielectric constant for ∞ as (∞), and denote the optical



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phonon modes by j. Therefore, the dispersion relation of the dielectric constant for frequencies lower than those of electronic excitations may be expressed by the form similar to EQ. (VI.35) as follows:

22 2

,jp j

j

f

(VI.45)

For simplicity, if we consider a diatomic linear crystal so that there is only one transverse optical mode T, we find that in the low frequency limit we have

220 .p T (VI.46)

Similar to our prior discussion on electronic excitations, if the frequency of the incident light approaches the transverse optical phonon mode, when the frequency slightly exceeds T, total reflection would appear. On the other hand, if the crystal is more than one dimension, then there will be both longitudinal (L) and transverse optical modes and L > T. With continuing increase of frequency, no light could propagate because of the energy gap between the transverse and longitudinal optical phonon modes, which is known as the Reststrahlen effect. However, once the frequency approaches the longitudinal optical phonon mode L from below, the dielectric constant becomes positive again so that light can propagate through the medium. Therefore, we find

2 2 2 22 20 ,L p T L L T p . (VI.47)

From EQs. (VI.46) and (VI.47), we obtain the following Lyddane-Sachs-Teller relation:

20L T . (VI.48)

The relation given in EQ. (VI.48) is in fact quite general for most polar materials. In the case of a three dimensional ionic crystal, T would be the frequency of a transversely polarized optical mode, whose dipole moment interacts strongly with a transverse electromagnetic wave. This mixed mode of light and crystal polarization is a true mode of propagation in the material and is known as a polariton. On the other hand, it can be shown that there is no coupling between photons and the longitudinal phonon mode. In Problem Set #5 we will investigate the explicit dispersion relation of the polariton spectrum and prove the absence of coupling between the longitudinal phonon fields and the photon.

Figure VI.1: Dispersion of light in a medium, showing sharp changes from large positive to large negative values in the real part of the dielectric constant () near the eigen frequencies of discrete excitation states j.

1 2

()

(0)

0 1



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Figure VI.2: (a) The real and imaginary parts of the refraction index according to EQ. (VI.41). (b) The real and imaginary parts of the broadened refraction index according to EQ. (VI.42).

VI.3. Photon-Phonon Interactions

The absorption of photons can assist transitions between the optical phonon branches ( optical) and acoustic phonon branches ( acoustic). The transitions must conserve energy as well as momentum. However, for infrared and optical photons, the corresponding photon momenta are much smaller than the reciprocal lattice vector, and are therefore negligible. Hence, the photon-assisted transitions may be considered as “vertical transitions”, as schematically shown in Fig. VI.3. From conservation of energy we have

optical acoustic . (VI.49)

Whereas vertical transitions imply that

optical acousticq q . (VI.50)

Figure VI.3: (a) Absorption of a photon by an optical phonon mode via a “vertical transition” in a one-dimensional crystal of two atoms per unit cell so that there are two phonon branches, one optical and the other acoustic. (b) Absorption of photons by the optical phonon modes via “vertical transitions” in a three-dimensional crystal of two atoms per unit cell. There are a total of 3 optical phonon branches and 3 acoustic phonon branches. (c) Other vertical transitions between the acoustic and optical phonon branches in a three-dimensional crystal.

(a) n ()

()

j

j

1

(b)

()

1

j

Re ( )

Im ( )

2j

(a) (b) (c)



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In addition to photon-assisted transitions from acoustic to optical phonon branches, creation of two phonons with opposite momenta is also possible with the absorption of a photon. In this case, we have the following energy and momentum conservation relations:

optical acoustic , (VI.51)

optical acoustic q q . (VI.52)

However, the two-phonon process involves the non-linear contribution from the lattice displacement to the dipole moment, which is a second-order effect. Therefore, it is generally much weaker than the one-phonon process. Moreover, the probability of two-phonon creation process is a sensitive function of the temperature because the process involves second-order lattice displacements, which would contain squares of the

following two matrix elements that involve the phonon creation and annihilation operations †aq and aq :

1 2†1 1n a n n q q q q , (VI.53)

1 21n a n n q q q q . (VI.54)

Here nq denotes the number of phonons of momentum q and frequency q, whose average number is of the Bose-Einstein type and so is a strong function of the temperature:

11Bk Tn e

qq

. (VI.55)

In optical measurements, the aforementioned photon-phonon interactions will give rise to inelastic

diffraction of light. Therefore, an incident photon ,k is transformed ito another photon , k with the

emission of a phonon ,q , which gives rise to a shift in the frequency of some of the light scattered by the

matter with the conservation rules k k q and . In this context, the first order Raman

effect is associated with the one-phonon absorption, whereas the second-order Raman spectrum contains signals from the two-phonon processes. While the selection rules for allowed optical transitions in a perfect crystal are determined by symmetry, the presence of impurities could lead to forbidden absorption. Typically Raman scattering techniques have been widely applied to the studies of phonon modes and impurities in solids and of molecular vibration modes in molecules.

VI.4. Interband Transitions and Excitons

Our description in Part VI.2 for the response of matter to optical light is generally valid for both photon-phonon and photon-electron interactions. Therefore, if we consider the Bloch electrons in a crystal, their interactions with photons could result in transitions from broad energy bands of filled to empty Bloch states. Such a broad absorption band of the interband transition may be approximated as vertical transitions (Fig.VI.4) because the momentum of typical optical photons (~107 m1) is generally much smaller than the size of the Brillouin zone (~109 m1) and so may be neglected.

To evaluate the frequency dependent dielectric constant (k,) in a semiconductor under interband transitions, we follow the same first-order time-dependent perturbation theory as in PartVI.2, except that the perturbation interaction is now replaced by the Coulomb interaction. Thus, if we denote the valence and conduction electronic states by k and k q g , respectively, where q denotes the photon momentum



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and g is the reciprocal lattice constant, we find an expression for the dielectric constant similar to the expressions given by EQs. (VI.35) and (VI.31):

22

22 2 2

24, 1

iee

q

q r

k

k k q g k k q gq

k k q g

E E

E E,

2

2 2

41

e f

m

k

kk

, (VI.56)

where (4e2/q2) is the matrix element of Coulomb interaction, fk is the oscillator strength for the transition between k and k q g and k is the energy difference between the states of the valence and

conduction bands. Assuming vertical transitions so that q = 0 and noting that k is continuous (see Fig.VI.4), we may simplify EQ. (VI.56) into the following form:

2

2 2

40, 1 dfe

dm

N

, (VI.57)

where d d N is the number of states with a vertical energy difference between the valence and

conduction bands in the frequency range d , and f denotes the oscillator strength.

As before, the imaginary part of the dielectric constant may be obtained via the Kramers-Kronig relations, which yields:

24

22 d

en d f

m

N ,

2 22

d

ef

m

N . (VI.58)

The function Nd () represents the joint density of states for the valence and conduction bands

separated by an energy . Given that both the valence and conduction bands are continuous periodic functions in reciprocal space, Nd () is also a smooth function except special critical points in the reciprocal space known as the van Hove singularities. Among the critical points, the most important one for the interband transitions is the absorption band edge, which is associated with the minimum vertical energy

Figure VI.4: Schematics of the photon-induced vertical interband transitions in a semiconductor. Here “Z.B.” refers to the “zone boundary” of the Brillouin zone.



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difference 0 of the interband transitions in a semiconductor due to the dispersion of energy bands. Here

we note that the absorption band edge 0 is not necessarily the energy gap Egap defined as the energy

difference between the bottom of the conduction band and the top of the valence band, as exemplified in Fig.VI.5(a). For a three-dimensional crystal (with principle axes along q1, q2 and q3 in the momentum space), we may express the frequency spectrum near 0 by the following relation

2 2 2

0 1 1 1 2 2 2 3 3 3c c cq q q q q q q , (VI.59)

where qc = (q1, q2, q3) is the critical point associated with 0, and the coefficients 1, 2 and 3 are all positive numbers because 0 is a local minimum. Given the dispersion relation of (q) near 0 in EQ. (VI.59), the joint density of states near the absorption edge of interband transitions becomes

1/2

01/221 2 3

1

4d

N , (VI.60)

which is obtained by considering the definition of Nd () = dNcv()/d, where Ncv() is the number of states between conduction and valence bands separated by energy . From EQ. (VI.59), the frequency volume vq enclosed by the ellipsoidal constant surfaces surfaces around qc is

3/2

01/2

1 2 3

4

3qv

. (VI.61)

Therefore, we obtain Ncv() within the volume vq:

3/2 3/2

0 03 1/2 1/23 2

1 2 3 1 2 3

4

3 8 62q

cv

vN

(VI.62)

and the joint density of states for > 0 is given by

1/2

01/221 2 3

1

4cv

d

dN

d

N . (VI.63)

In fact, the joint density of states near different types of critical points in different dimensions would

have different frequency dependences. For instance, if all the coefficients 1, 2 and 3 in EQ. (VI.59) are negative, then the critical point 0 would be a maximum rather than a minimum. In this case we find that Nd() ( 0)1/2 for 0 > . On the other hand, if 1 > 0, 2 > 0 and 3 < 0, the critical point is a saddle point. Similarly, 0 is also a saddle point if 1 > 0, 2 < 0 and 3 < 0. We further remark that the density of states of all types of critical points in three dimensional crystals are continuous. On the other hand, for two dimensional crystals, the saddle point (with 1 > 0 and 2 < 0) is logarithmically divergent at 0 whereas the maximum (with 1 < 0 and 2 < 0) and minimum (with 1 > 0 and 2 > 0) remain continuous. Finally, for one dimensional crystals, the maximum (with 1 < 0) and minimum (with 1 > 0) critical points are both singular at at 0. These results associated with the critical points of the joint electronic density of states are mathematically identical to what we have discussed in Part II.

It is also possible to observe optical transitions that correspond to gap q E through indirect,

phonon-assited transitions by simultaneously allowing emission or absorption of a phonon of a frequency



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q in a second-order process, as exemplified in Fig. VI.5(b). Given that q is a small energy (on the order of

tens of meV) relative to a typical energy gap (on the order of 1 eV), it would seem that the absorption would start from Egap. However, the probability for second-order indirect transitions is much smaller than that of direct transitions, and is also a strong function of the temperature.

In addition to well defined energy bands in perfect crystals, donor (acceptor) impurities may form impurity states near the bottom (top) of the conduction (valence) bands in doped semiconductors, as illustrated in Fig. VI.6 (a). In this case, electrons from the acceptor impurity states may be excited into the donor impurity states and leave behind holes. The electron-hole pairs may become bound and have a finite lifetime before they recombine. Therefore, these exciton bound states can result in sharp absorption lines similar to those associated with hydrogen atomic energy levels below the absorption band edge. There are typically two types of excitons: the Wannier excitons for expanded hydrogenic orbits in semiconductors with high dielectric constant and light carriers, as illustrated in Fig. VI.6 (b), and the Frenkel excitons for tightly bound, highly localized orbits in molecular and ionic crystals, as shown in Fig. VI.6 (c).

Finally, we remark that interband optical transitions can also be observed in metals, provided that electrons may not have completely filled a band or may have fallen into different zones. Such interband transitions follow all aforementioned theory, except that the phenomena would be somewhat masked by the high reflectivity associated with the high electrical conductivity of metals.

(a) (b)

Figure VI.5: (a) Example of a direct interband transition energy 0 that

does not correspond to the energy gap Egap. (b) Example of an indirect, phonon-assisted transition.

Figure VI.6: (a) Exciton states between the conduction band and the valence band. (b) Wannier excitons in semiconductors. (c) Frenkel excitons in molecular and ionic crystals.

(a) (b) (c)



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VI.5. Interaction with conduction electrons in good metals

In this section we examine the optical properties of good conductors. Assuming dominant electrical conductivity, we find that the complex refractive index N can be approximated as follows:

1 2 1 2 1 2

4 4 2N 1n i i i i

. (VI.64)

Therefore, the reflection coefficient R from EQ. (VI.14) becomes

2 2 22 1 22

2 2 2 2 22

1 211- N 4R 1 1 2

1+N 21 21

n nn n

n n nn

. (VI.65)

The relation in EQ. (VI.65) is known as the Hagen-Rubens relation for good conductors.

Next, we consider classical electrical conductivity by defining a relaxation time so that

2

0N e

m

, (VI.66)

where N0 is the density of conducting electrons. Using EQs. (VI.65) and (VI.66), we find that

1 21 21 2

2 20

2 2R 1 2 1 2 1 2

2 4 p

m

N e

. (VI.67)

Generally R given in EQ. (VI.67) is very close to 1 even if approaches infrared frequencies. However, in both EQs. (VI.66) and (VI.67) we have implicitly assumed that is independent of . This assumption is no longer valid if the optical frequency is so high that electrons cannot follow the oscillation of the electric field and that the classical description of relaxation via collisions completely breaks down. In other words, the Hagen-Rubens relation no longer holds if .

To investigate the optical properties of metals over a wide range of frequency, we may approach the problem by either considering the semi-classicial Boltzmann equation, or by taking the general Kubo formula for the linear response function, which is associated with the retarded Green function expression for the correlation function of relevant physical quantities. In the following we use the conductivity formalism derived from the Boltzmann equation (to be elaborated in Part VII) and discuss the frequency dependence of various optical properties in metals.

[Electrical conductivity derived from the Boltzman equation]

The conductivity tensor of a metal may be considered as a linear response of electrons in the metal to an external electric field E so that the resulting electrical current density J is related to E by the expression J = Eσ . As detailed in Part VII, the current density J is related to the velocity kv and the local

concentration of the carrier fk r in state k, which is found to satisfy the following relation:



15

3 23

02

3

22

8

1,

4F

dkd k e f d k d e f

d

dS fd e

v

k k k k

k kk

J v v

v v E

EE

EE

(VI.68)

where the change in carrier momentum k over the relaxation time by an external electric field E is given by k = ( / ) ( / | )|e k kE v v . Therefore, the conductivity tensor becomes

2

3,

4 1Fe dS

v i

k k

k k

v vq

v q. (VI.69)

We further note that the expression 0f kE in EQ. (VI.68) represents the equilibrium fermion distribution

function so that

1

0 0exp 1 ,B

f fk T

k

k k

E E (VI.70)

where denotes the chemical potential.

[Frequency dependent optical properties in a metal of cubic symmetry]

To get better quantitative sense of the results shown in EQ. (VI.69), we consider a cubic metallic crystal so that the conductivity tensor may be replaced by a scalar conductivity. For typical optical light, kv q .

Thus, we have

2

3 2 2 2 2

1 10

12 1 1F

v i iedS

. (VI.71)

If we take = 1, then the real and imaginary parts of N2 become

2 22 2

2 2 2 2

4 01 1

1 1pn

, (VI.72)

2

2 2 2 2

4 02

1 1pn

. (VI.73)

With EQs. (VI.72) and (VI.73), we may consider the frequency dependence of various optical properties

in three different regions: [1] 1 ; [2] 1 p ; and [3] p . The schematic behavior of

various optical properties is summarized in Fig.VI.6.



16

[1] 1

In this low frequency limit, the metal is highly reflecting because the imaginary part of N2 is much larger than the real part and that the Hagen-Rubens relation is valid. The absorption coefficient = 2n/c is nearly independent of the frequency according to EQs. (VI.8) and (VI.73). The real part of N2 is negative from EQ. (VI.72) and is much larger than unity in magnitude.

[2] 1 p

This frequency range is known as the relaxation region, where the term 22 in the denominators of EQs.

(VI.72) and (VI.73) dominate so that

22 2

21 pn

,

2

2

12 pn

. (VI.74)

The absorption coefficient ()2 falls off rapidly with frequency, and the metal is still strongly reflecting, with the reflection coefficient R being

1 2

0

1 2R 1 1

p

. (VI.75)

Figure VI.6: Schematic frequency dependence of the optical properties of metals, showing the real and imaginary parts of the dielectric constant , reflection coefficient R, absorption coefficient , and the refraction coefficients n and .



17

[3] p

In this limit the real part of N2 becomes positive, and the reflection coefficient R falls off to zero,

implying that the metal becomes nearly transparent, with a small absorption coefficient

2

2

2 1pn

c c

. (VI.76)

Finally, we note that in the low frequency limit, electromagnetic waves are attenuated rapidly as the

waves enter the metal. From EQs. (VI.5) and (VI.64), we define the damping distance associated with the skin effect of as the classical skin depth :

1 22

c c

. (VI.77)

Clearly the skin depth decreases with increasing frequency.

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PART VI. Physics & Applications of Optical …...Application of Quantum Mechanics to Part VI: Physics & Applications of Optical Measurements Condensed Matter Physics Nai-Chang Yeh