Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

14.2 Skin Eﬀect in the Absence of a dc Magnetic Field 425

which can be written 44



−

+ ε(q,ω)0 0

0 −

+ ε(q,ω)0

00ε(q,ω)



=0. (14.13)

Here, we have introduced the dielectric function

ε(q,ω)=1 −

4πi

(q,ω), (14.14)

andwehaveassumedthatε is diagonal (this is true for an electron gas in the 46

absence of a dc magnetic ﬁeld). The transverse electromagnetic waves which 47

can propagate in the medium are solutions of the equation 48

= ω

ε(q,ω). (14.15)

In addition, there is a longitudinal wave which is the solution of the equation

ε(q,ω)=0. (14.16)

Normal Skin Eﬀect

In the absence of a dc magnetic ﬁeld, the local theory of conduction gives 51

σ =

1+iωτ

τ/4π

1+iωτ

. (14.17)

Therefore, we have

ε(q,ω)=1−

iω

τ/ω

1+iωτ

(14.18)

ε(q,ω)=

1+(ω

− ω

)τ

1+ω

− i

ωτ(1 + ω

)

(14.19)

Usually in a good metal ω

 10

/s, a frequency in the ultraviolet. Therefore, 54

in the optical or infrared range ω

 ω. The parameter τ can be as small as 55

−14

sec or as large as 10

−9

s in very pure metals at very low temperatures. 56

Letusﬁrstconsiderthecaseωτ  1. Then, since ω

 ω,wehave 57

ε(q,ω) −

. (14.20)

Substituting this result into the wave equation c

= ω

ε(q, ω)gives 58

q = ±i

= ±

. (14.21)

We choose the well-behaved solution q = −

so that the ﬁeld in the metal is 59

of the form 60

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

426 14 Electrodynamics of Metals

E(z,t)=E

iωt−z/δ

. (14.22)

What we ﬁnd is that electromagnetic waves do not propagate in the metal 62

(for frequencies lower than ω

), and that the electric ﬁeld in the solid drops 63

oﬀ exponentially with distance from the surface. The distance δ =

is called 64

the normal skin depth. 65

If ωτ  1, (this is usually true at rf frequencies, even at low temperatures 66

with pure materials) we have 67

ε(q,ω)  1 − i

∼−i

ωτ

when ω

 ω. (14.23)

The solution of the wave equation is given by 68

q = ±



ωτ



1/2

(1 − i), (14.24)

so that the ﬁeld E is of the form 69

E(z,t)=E

iωt

−(i+1)

(

ωτ

)

1/2

. (14.25)

Thus, the skin depth is given by

δ =



ωτ



1/2

. (14.26)

Ifthemeanfreepathl is much greater than the skin depth, l  δ, then the

local theory is not valid. In good metals at low temperatures, it turns out that 72

l  v

τ  10

nm and δ  10nm, so that l  δ, and we must use the nonlocal 73

theory. 74

Anomalous Skin Eﬀect 75

The normal skin eﬀect was derived under the assumption that the q depen- 76

dence of σ was unimportant. Remember that this assumption is valid if 77

ql = qv

τ  1. We have found that the electric ﬁeld varies like e

−z/δ

.If 78

δ turns out to be smaller than l = v

τ, our initial assumption was certainly 79

incorrect. The skin depth δ is of the order of 80

δ 

. (14.27)

Therefore, if

ωc

<ωτ, (14.28)

the theory is inconsistent because the ﬁeld E(z) changes appreciably over a

mean free path l contradicting the assumption that the q dependence of σ can 83

be neglected. The theory for this case in which the q dependence of σ must 84

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14.3 Azbel–Kaner Cyclotron Resonance 427

be included is called the theory of the anomalous skin eﬀect. In the nonlocal 85

theory, we can write, (we take the y-axis to be perpendicular to the metal’s 86

surface), 87

(y)=





σ(y, y



) ·E(y



), (14.29)

which is true for an inﬁnite medium. However, we have to take into account 89

the surface of the metal here. We shall do this by using the formalism for the 90

inﬁnite medium, and imposing appropriate boundary conditions, namely, the 91

method of images. 92

14.3 Azbel–Kaner Cyclotron Resonance 93

The theory of the anomalous skin eﬀect in the presence of a dc magnetic 94

ﬁeld aligned parallel to the surface is the theory of Azbel–Kaner cyclotron 95

resonance in metals. We shall present a brief treatment of this eﬀect, and 96

leave the problem of the anomalous skin eﬀect in the absence of a dc magnetic 97

ﬁeld as an exercise. 98

Let us choose a Cartesian coordinate system with the y-axis normal to the 99

surface, and the z-axis parallel to the dc magnetic ﬁeld (see Fig. 14.1). For a 100

polarization in which E(r,t) is parallel to the z-axis, the wave equation can 101

be written, since q is along the y-direction, 102



−q

)E(q, ω



4πiω

(q, ω). (14.30)

This comes from the Fourier transform of the wave equation. For the case

103

of self-sustaining oscillations of an inﬁnite medium, we would set j

equal to 104

Fig. 14.1. The coordinate system for an electromagnetic wave propagating parallel

to the y-axis with the surface at y = 0. A dc magnetic ﬁeld B

is parallel to the

z-axis

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

428 14 Electrodynamics of Metals

Fig. 14.2. A semi-inﬁnite medium in terms of inﬁnite medium picture. An electric

ﬁeld E

(y) in the metal is shown near the surface y =0

the induced electron current given by σ

(q, ω)E(q, ω). For the semi-inﬁnite 105

medium, however, one must exercise some care to account for the boundary 106

conditions. The electric ﬁeld in the metal will decay in amplitude with distance 107

from the surface y = 0. There is a discontinuity in the ﬁrst derivative of 108

E(r,t)aty = 0. Actually the term −q

E in the wave equation came from 109

making the assumption that E(y) was of the form e

−iqy

.Onecanusethis 110

“inﬁnite medium” picture by replacing the vacuum by the mirror image of 111

the metal as shown in Fig. 14.2. The ﬁctitious surface current j

∝ δ(y)must 112

be introduced to properly take account of the boundary conditions. By putting 113

(q, ω)=σ

(q, ω)E(q, ω), we can solve the wave equation for E(q, ω). The 114

ﬁctitious surface current sheet of density j

is very simply related to the 115

magnetic ﬁeld at the surface 116

(y)=

2π

H(0)δ(y)orj

(q)=

cH(0)

2π

. (14.31)

Solving (14.30) for E(q, ω)gives

117

E(q, ω)=

2iωH(0)/c

−q

−

4πiω

(q, ω)

. (14.32)

By substituting this result into the Fourier transform

118

E(y)=

2π



∞

−∞

dq e

−iqy

E(q, ω), (14.33) 119

and the deﬁnition of the surface impedance Z 120

Z =

4π

E(0)

H(0)

, (14.34)

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

14.3 Azbel–Kaner Cyclotron Resonance 429

Fig. 14.3. The coordinate system for an electromagnetic wave propagating parallel

to the y-axis to the metal surface (y = 0) for the case of polarization in the x-

direction. A dc magnetic ﬁeld B

is parallel to the z-axis

we can easily obtain the electric ﬁeld as a function of position and the surface 121

impedance of the metal. In (14.34), E(0) and H(0) are the electric ﬁeld and 122

the magnetic ﬁeld at the surface, respectively. 123

For the case of a wave polarized in the x-direction instead of the 124

z-direction, σ

is replaced by 125

= σ

. (14.35)

This is equivalent to assuming that electrons are specularly reﬂected at the

126

boundary y = 0. Figure 14.3 shows the coordinate system for a wave prop- 127

agating perpendicular to the boundary of the metal with polarization in the 128

x-direction normal to the dc magnetic ﬁeld. One can see that although E(y) 129

is continuous, its ﬁrst derivative is not. Therefore, in deﬁning the Fourier 130

transform of

∂

E(y)

∂y

we must add terms to take account of these continuities. 131



∞

−∞

dy e

iqy

∂

E(y)

∂y

=ΔE



− iqΔE

− q

E(q), (14.36)

where

132

ΔE





∂E(y)

∂y



y=0

−



∂E(y)

∂y



y=0

−

, (14.37)

and

133

ΔE

= E(0

) − E(0

−

). (14.38)

For the case of specular reﬂection we take ΔE

=0andΔE



=2E



). This 134

adds a constant term to the wave equation 135

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

430 14 Electrodynamics of Metals



−q



E(q, ω)=−ΔE



4πiω

(q, ω). (14.39)

This added term can equally well be thought of as a ﬁctitious surface current

136

(y)givenby 137

(y)=−

ΔE



4πiω

δ(y). (14.40)

So that the inﬁnite medium result (or inﬁnite medium wave equation) can be

138

used if we take for j

(q, ω) 139

(q, ω)=j

(q, ω)+j

(q). (14.41)

Actually, the results just derived are valid in the absence of a magnetic ﬁeld

140

as well as in the presence of a dc magnetic ﬁeld. In the absence of a magnetic 141

ﬁeld the conductivity tensor is given by 142

σ(q, ω)=

4πiω

+ I(q, ω)}, (14.42)

where

143

I(q, ω)=





(ε



) − f

(ε

)



− ε

− ¯hω

k



|kk



|k

∗

. (14.43)

14.4 Azbel–Kaner Eﬀect 144

If we use the Cohen–Harrison–Harrison expression for σ

(q, ω), (13.125), we 145

have 146

=3σ

∞



n=−∞

(w)

1 − iτ (nω

− ω)

, (14.44)

147

where 148

(w)=



−1

d(cos θ)cos

θJ

(w sin θ). (14.45)

Since w  1 (i.e., we assume w ≡

 1 for the values of q of interest 149

in this problem) we can replace J

(w sin θ) by its asymptotic value for large 150

argument. 151

lim

z→∞

(z) ≈

πz

cos[z −



n +



]. (14.46)

Substituting (14.46) into the expression for c

(w) (14.45) gives 152

(w) ≈

. (14.47)

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

14.5 Magnetoplasma Waves 431

Therefore, we have 153

≈

3ω

4πi˜ω

∞



n=−∞

1+nω

/˜ω

. (14.48)

Making use of the fact that

154

∞



n=−∞

1+nω

/ω

πω

cot

πω

(14.49)

and −icotx =cothix, one can easily see that

155

≈

3ω

16qv

coth



π(1 + iωτ)



. (14.50)

This can be rewritten

156

≈−

3iω

16qv

cot



π(ω − i/τ)



. (14.51)

157

In the limit ωτ  1, this function has sharp peaks at ω  nω

.Inthislast 158

expression, we have substituted ω −i/τ for ω. In the limit ωτ  1, σ

shows 159

periodic oscillations as a function of ω

. These oscillations also show up in the 160

surface impedance. 161

In using the Cohen–Harrison–Harrison expression for σ

,wehaveobvi- 162

ously omitted quantum oscillations. By using the quantum mechanical expres- 163

sion for σ, for example σ

(q, ω) given by (13.154), one can easily obtain the 164

quantum oscillations of the surface impedance. 165

14.5 Magnetoplasma Waves 166

We have seen that if we omit spin magnetization, the Maxwell equations for 167

awaveoftheforme

iωt−iq·r

can be written 168

=Γ(q,ω) · E. (14.52)

The total current is usually the sum of some external current and the induced

169

electron current. If one is interested in the self-sustaining oscillations of the 170

system, one wants the external driving current to be equal to zero. Then the 171

electron current is given by j

= σ · E, and this is the only current. Thus, 172

=Γ· E = σ · E,sothatwehave 173

| Γ − σ |= 0 (14.53)

gives the dispersion relation for the normal modes of the system.

174

In the absence of a dc magnetic ﬁeld (14.53) reduces to 175

(ω

ε − c

)

ε =0. (14.54)

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

432 14 Electrodynamics of Metals

Fig. 14.4. The dispersion curves of transverse and longitudinal plasmon modes in

the absence of a dc magnetic ﬁeld

In the local (collisionless) theory, the dielectric function is given by 176

ε ≈ 1 −

, (14.55)

so the normal modes are two degenerate transverse modes of frequency

177

= ω

+ c

, (14.56)

and a longitudinal mode of frequency

178

ω = ω

. (14.57)

Figure 14.4 shows the dispersion curves of transverse and longitudinal plasmon

179

modes in the absence of a dc magnetic ﬁeld. There are no propagating modes 180

for frequencies ω smaller than the plasma frequency ω

. 181

Now, consider the normal modes of the system in the presence of a dc 182

magnetic ﬁeld. We choose the z-axis parallel to the magnetic ﬁeld, and let the 183

wave vector q lie in the y − z plane. The secular equation can be written 184



− ξ

−ε

− ξ

0 ξ

− ξ



=0. (14.58)

In writing down (14.58) we have introduced ξ =

, and now assume a local 185

theory of conductivity in which the nonvanishing elements of ε are 186

(ω)=ε

(ω)=1−

− ω

(ω)=−ε

(ω)=−i

/ω

− ω

, (14.59)

(ω)=1−

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

14.5 Magnetoplasma Waves 433

Fig. 14.5. The dispersion curves of magnetoplasma modes in a metal when the

wave propagates in the z-direction parallel to the dc magnetic ﬁeld B

Because the dielectric constant is independent of q, the secular equation turns 187

out to be rather simple. It is a quadratic equation in q

188

αq

+ βq

+ γ =0, (14.60)

where

189

α = ε

(ω)sin

θ + ε

(ω)cos

θ,

β = −ω

(ω)ε

(ω)(1 + cos

θ)+[ε

(ω)+ε

(ω)] sin

, (14.61)

γ = ω



(ω)+ε

(ω)



Here, θ is the angle between the direction of propagation and the direction

190

of the dc magnetic ﬁeld. For θ = 0 (14.60) reduces to 191

(ω)

− ω

(ω)]

+ ω

(ω)

=0. (14.62)

192

The roots can easily be plotted; there are four roots as are shown in Fig. 14.5. 193

The longitudinal plasmon ω = ω

is the solution of ε

(ω)=0.Thetwo194

transverse plasmons start out at q =0asω =



+(ω

/2)



1/2

.Atvery 195

large q they are just light waves, but there is a diﬀerence in the phase velocity 196

for the two diﬀerent (circular) polarizations. Their diﬀerence in phase velocity 197

is responsible for the Faraday eﬀect–the rotation of the plane of polarization 198

in a plane polarized wave. The low frequency mode is the well-known helicon. 199

For small values of q it begins as 200

ω =

, (14.63)

and it asymptotically approaches ω = ω

for large q. 201

For θ =

(14.60) reduces to 202

− ω

(ω)]

(ω) − ω

[ε

(ω)+ε

(ω)]

=0. (14.64)

The mode corresponding to q

= ω

(ω) is a transverse plasmon of fre- 203

quency ω =



+ c



1/2

. The helicon mode appears no longer. The other

204

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BookID 160928 ChapID 14 Proof# 1 - 29/07/09

434 14 Electrodynamics of Metals

Fig. 14.6. The dispersion curves of magnetoplasma modes in a metal when the

wave propagates in the z-direction normal to the dc magnetic ﬁeld B

two modes have mixed longitudinal and transverse character. They start at 205

ω =









1/2

(14.65)

for q = 0. For very large q one mode is nearly transverse and varies as

206

ω ≈ cq while the other approaches the ﬁnite asymptotic limit ω =

+ ω

. 207

The roots of (14.64) are sketched in Fig. 14.6. For an arbitrary angle of 208

propagation

the helicon mode has a frequency (we assume ω

 ω

) 209

ω 

cos θ

+ c



τ cos θ



. (14.66)

Here, we have included the damping due to collisions. For very large values

210

, the two ﬁnite frequency modes are sometimes referred to as the hybrid- 211

magnetoplasma modes. Their frequencies are 212



+ ω







+ ω



− ω

cos



1/2

. (14.67)

213

For propagation at an arbitrary angle we can think of the four modes as 214

coupled magnetoplasma modes. The ω

modes described above for very large 215

values of q are obviously the coupled helicon and longitudinal plasmon. 216

14.6 Discussion of the Nonlocal Theory 217

By considering the q dependence of the conductivity one can ﬁnd a number 218

of interesting eﬀects that have been omitted from the local theory (as well as 219

the quantitative changes in the dispersion relation which are to be expected). 220

Among them are: 221

M.A. Lampert, J.J. Quinn, S. Tosima, Phys. Rev. 152, 661 (1966).