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

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Uncorrected Proof

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

14.10 Propagation of Acoustic Waves 445

where T is a very complicated tensor. The nontrivial solutions are determined 432

from the secular equation 433

det | T(q,ω) |=0. (14.112)

434

The roots of this secular equation give the frequencies of the sound waves (two 435

transverse and one longitudinal modes) as a function of q, B

, τ,etc.Actu- 436

ally the solutions ω(q) have both a real and imaginary parts; the real part 437

determines the velocity of sound and the imaginary part the attenuation of 438

the wave. 439

Here, we do not go through the details of the calculation outlined earlier. 440

We will discuss special cases and attempt to give a qualitative feeling for the 441

kinds of eﬀects one can observe. 442

14.10.1 Propagation Parallel to B

443

For propagation parallel to the dc magnetic ﬁeld, it is convenient to introduce 444

circularly polarized transverse waves with 445

= ξ

± iξ

= σ

∓ iσ

(14.113)

We also introduce the parameter β =

4πωσ

ωτ

. Then the nonvanishing 446

components of Γ are Γ

=Γ

=iβσ

and Γ

= −

iω

4π

. Deﬁne the resistivity 447

tensor R by 448

R = σ

−1

. (14.114)

Then R

= σ

−1

and R

= σ

−1

. The secular equation | T |= 0 reduces to 449

two simple equations: 450

= s

∓

zeωB

zmiω

Mτ

(1 − iβ)(σ

− 1)

1 − iβσ

(14.115)

for the circularly polarized transverse waves, and

451

= s

zmiω

Mτ



− 1 −

1+ω



. (14.116)

for the longitudinal waves. In (14.115) and (14.116), s

and s

are the speeds 452

of transverse and longitudinal acoustic waves given, respectively, by 453

and s

1+ω



. (14.117)

From these results, we observe that

454

1. ω has both real and imaginary parts. The real part gives the frequency and 455

hence velocity as a function of B

. The imaginary part gives the acoustic 456

attenuation as a function of ql, ω

τ, B

,etc. 457

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

446 14 Electrodynamics of Metals

2. For longitudinal waves, if we use the semiclassical result for σ

, ω is 458

completely independent of B

. 459

3. In the case where the quantum mechanical result for σ

is used, both the 460

velocity and attenuation display quantum oscillations of the de Haas van 461

Alphen type. 462

4. For shear waves, the right and left circular polarizations have slightly dif- 463

ferent velocity and attenuation. This leads to a rotation of the plane of 464

polarization of a linearly polarized wave. This is the acoustic analogue of 465

the Faraday eﬀect. 466

5. R

does depend on the magnetic ﬁeld, and the acoustic wave shows a 467

fairly abrupt increase in attenuation as the magnetic ﬁeld is lowered below 468

= qv

. This eﬀect is called Doppler shifted cyclotron resonances; DSCR. 469

6. The helicon wave solution actually appears in (14.115), so that the equation 470

for ω

actually describes helicon–phonon coupling. 471

14.10.2 Helicon–Phonon Interaction 472

Look at (14.115), the dispersion relation of the circularly polarized shear waves 473

propagating parallel to B

: 474

= s

∓

zeωB

zmiω

Mτ

(1 − iβ)(σ

− 1)

1 − iβσ

. (14.118)

In the local limit, where σ

is shown in (13.116) and (13.117), we have 475

 1+iωτ ∓ iω

τ. (14.119)

Remember that β 

ωτω

. Therefore, 1 − iβσ

can be written 476

1 − iβσ

 1 − i

ωτω

[1 + iωτ ∓ iω

τ]. (14.120)

Let us assume ω

τ  1, ω

 ω,andβ  1. Then, we can write that 477

1 − iβσ

≈ 1 ∓

, (14.121)

where ω



1 −



is the helicon frequency. Substituting this in

478

(14.118) gives 479

− s

∓ωΩ

ω ∓ω

, (14.122)

where Ω

zeB

is the ionic cyclotron frequency. Equation (14.122) can be 480

rewritten as 481

(ω −s

q)(ω + s

q)(ω ∓ ω

)  ωω

. (14.123)

The dispersion curves are illustrated in Fig. 14.10. The helicon and transverse

482

sound wave of the same polarization are strongly coupled by the term on the 483

right-hand side of (14.123), when their phase velocities are almost equal. The 484

solid lines depict the coupled helicon–phonon modes. 485

Uncorrected Proof

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

14.10 Propagation of Acoustic Waves 447

∝

Fig. 14.10. Schematic of the roots of Eq.(14.123). The region of strongly coupled

helicon–phonon modes for circularly polarized acoustic shear waves

14.10.3 Propagation Perpendicular to B

486

For propagation to the dc magnetic ﬁeld, the resistivity tensor has the 487

following nonvanishing elements: 488

+ σ

= −R

+ σ

= σ

−1

. (14.124)

The secular equation |T

| = 0 again reduces to a 2 × 2matrixanda1× 1 489

matrix, which can be written 490



− A

−A

− A





= 0 (14.125)

and

491



− A



=0, (14.126)

where

492

zmiω

Mτ

(1 − iβ)(σ

− 1)

1 − iβσ



zmq

3M(1 + ω

)

zmiω

Mτ

(

− 1 −

iβσ

1 − iβσ

−

3(1 + ω

)

= −A

zmiω

Mτ

(1 − iβ)σ

1 − iβσ

− ω

zmiω

Mτ

(1 − iβ)(σ

− 1)

1 − iβσ

. (14.127)

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

448 14 Electrodynamics of Metals

The velocity and attenuation of sound can display several diﬀerent types of 493

oscillatory behavior as a function of applied magnetic ﬁeld. Here we mention 494

very brieﬂy each of them. 495

1. Cyclotron resonances When ω = nω

, for propagation perpendicular to B

, 496

the components of the conductivity tensor become very large. This gives 497

rise to absorption peaks. 498

2. de Haas–van Alphen type oscillations Because the conductivity involves 499

sums



n,k

over quantum mechanical energy levels, as is shown in 500

(13.81), the components of the conductivity tensor display de Haas–van 501

Alphen type oscillations exactly as the magnetization, free energy, etc. One 502

small diﬀerence is that instead of being associated with extremal orbits 503

= 0, these oscillations in acoustic attenuation are associated with orbits 504

for which ¯v

= s. 505

3. Geometric resonances Due to the matrix elements ν



iq·r

|ν which behave 506

like Bessel function in the semiclassical limit, we ﬁnd oscillations associated 507

with J



−n

⊥

/ω

) for propagation perpendicular to B

. The physical 508

origin is associated with matching the cyclotron orbit diameter to multiples 509

of the acoustic wavelength. Figure 14.11 shows the schematic of geometric 510

resonances. 511

4. Giant quantum oscillations These result from the quantum nature of the 512

energy levels together with “resonance” due to vanishing of the energy 513

denominator in σ. The physical picture and feeling for the “giant” nature 514

of the oscillations can easily be obtained from consideration of 515

(a) Energy conservation and momentum conservation in the transition 516

)+¯hω

−→ E



+ q

). 517

(b) The Pauli exclusion principle. 518

Suppose that we had a uniform ﬁeld B

parallel to the z-axis. Then, with 519

usual choice of gauge, our states are |nk

, with energies given by 520

Fig. 14.11. Schematic of the origin of the geometric resonances in ultrasonic

attenuation

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

14.10 Propagation of Acoustic Waves 449

Fig. 14.12. Schematic of the transitions giving rise to giant quantum oscillations

)=¯hω



n +



¯h

. (14.128)

Now, we can do spectroscopy with these electrons, and have them absorb

521

radiation. Thus, suppose that an electron absorbs a phonon of energy ¯hω and 522

momentum ¯hq

. Then, energy conservation gives 523



+ q

) − E

)=¯hω

. (14.129)

Thus, only electrons with k

given, with α = n



− n,by 524

¯hq

(ω ∓ αω

(14.130)

will undergo transitions between diﬀerent Landau levels. Let us call this value

525

of k

the parameter K

. Then only electrons with k

= K

can make the 526

transition (n, k

) −→ (n + α, k

+ q

) and absorb energy ¯hω. Figure 14.12 527

shows a schematic picture of the transitions giving rise to giant quantum 528

oscillations. 529

To satisfy the exclusion principle E

) <ζand E

n+α

+ q

)=530

)+¯hω > ζ.Forω

 ω this occurs only when the initial and ﬁnal states 531

are right at the Fermi surface. Then the absorption is “gigantic”; otherwise it 532

is zero. The velocity as well as the attenuation displays these quantum oscil- 533

lations. The oscillations, in principle, are inﬁnitely sharp, but actually they 534

are broadened out due to the fact that the Landau levels themselves are not 535

perfectly sharp, and various other things. However, the oscillations are actu- 536

ally quite sharp, and so the amplitudes are much larger than the widths of 537

absorption peaks. 538

Uncorrected Proof

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

450 14 Electrodynamics of Metals

Problems 539

14.1. Evaluate σ given, for zero temperature, by 540

σ(q, ω)=

4πiω

+ I(q, ω)},

where

541

I(q, ω)=





(ε



) − f

(ε

)



− ε

− ¯hω

k



|kk



|k

∗

and use it to determine E(y)andZ. This is the problem of the anomalous

542

skin eﬀect, with the specular reﬂection boundary condition, in the absence of 543

a dc magnetic ﬁeld. 544

14.2. Derive the result for a helicon wave in a metal propagating at an angle 545

θ to the direction of the applied magnetic ﬁeld along the z-axis. 546

ω 

cos θ

+ c



τ cos θ



One may assume that ω

 ω

. 547

14.3. Investigate the case of helicon–plasmon coupling in a degenerate semi- 548

conductor in which ω

and ω

are of the same order-of-magnitude. Take 549

τ  1, but let the angle θ and ωτ be arbitrary. Study ω as a function 550

of B

, the applied magnetic ﬁeld. 551

14.4. Evaluate σ

, σ

,andσ

from the Cohen–Harrison–Harrison result for 552

propagatin perpendicular to B

in the limit that w = qv

/ω

 1. Calculate 553

to order w

. See if any modes exist (at cyclotron harmonics) for the wave 554

equation ξ

= ε

where ξ = cq/ω. 555

14.5. Consider a semi-inﬁnite metal of dielectric function ε

to ﬁll the space 556

z>0, and an insulator of dielectric constant ε

in the space z<0. 557

1. Show that the dispersion relation of the surface plasmon for the polarization 558

with E

=0andE

=0= E

is written by 559

=0,

where α

and α

are the decay constants in the insulator and metal, 560

respectively. 561

2. Sketch

as a function of

for the surface plasmon excitation. 562

14.6. Assume a vacuum–metal interface at z =0(z<0 is vacuum), and let 563

the electric charge density ρ appearing in Maxwell’s equations vanish both 564

inside and outside the metal. (This does not preclude a surface charge density 565

concentrated in the plane z =0.) 566

Uncorrected Proof

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

14.10 Propagation of Acoustic Waves 451

1. Solve the wave equation in both vacuum and solid by assuming 567

(r,t)=E

iωt−iq

y+α

(r,t)=E

iωt−iq

y−α

where q and ω are given and α

and α

must be determined. 568

2. Apply the usual boundary conditions at z = 0 to determine the dispersion 569

relation (ω vs. q) for surface waves. 570

3. Sketch ω

as a function of q

assuming ωτ  1. 571

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

452 14 Electrodynamics of Metals

Summary 572

In this chapter, we study electromagnetic behavior of waves in metals. The 573

linear response theory and Maxwell’s equations are combined to obtain the 574

condition of self-sustaining oscillations in metals. Both normal skin eﬀect and 575

Azbel–Kaner cyclotron resonance are discussed, and dispersion relations of 576

plasmon modes and magnetoplasma modes are illustrated. Nonlocal eﬀects in 577

the wave dispersions are also pointed out, and behavior of cyclotron waves is 578

considered as an example of the nonlocal behavior of the modes. General dis- 579

persion relation of the surface waves in the metal–insulator interface is derived 580

by imposing standard boundary conditions, and the magnetoplasma surface 581

waves are illustrated. Finally, we brieﬂy discussed propagation of acoustic 582

waves in metals. 583

The wave equation in metals, in the present of the total current j

584

(= j

+ j

ind

), is written as 585

=Γ· E,

where Γ

iω

4π

(ξ

− 1)1 − ξξ

. Here, the spin magnetization is neglected 586

and ξ =

.Thej

and j

ind

denote, respectively, some external current and 587

the induced current j

= σ · E by the self-consistent ﬁeld E. 588

For a system consisting of a semi-inﬁnite metal ﬁlling the space z>0and 589

vacuum in the space z<0 and in the absence of j

, the wave equation reduces 590

to [σ(q,ω) − Γ(q,ω)] · E = 0, and the electromagnetic waves are solutions of 591

the secular equation | Γ − σ |=0. The dispersion relations of the transverse 592

and longitudinal electromagnetic waves propagating in the medium are given, 593

respectively, by 594

= ω

ε(q,ω)andε(q,ω)=0.

In the range ω

 ω and for ωτ  1, the local theory of conduction 595

(ql  1) gives a well-behaved ﬁeld, inside the metal, of the form 596

E(z,t)=E

iωt−z/δ

where q = −i

= −

. The distance δ =

is called the normal skin depth. 597

If l  δ, the local theory is not valid. The theory for this case, in which the 598

q dependence of σ must be included, explains the anomalous skin eﬀect. 599

In the absence of a dc magnetic ﬁeld, the condition of the collective modes 600

reduces to 601

(ω

ε − c

)

ε =0.

Using the local (collisionless) theory of the dielectric function ε ≈ 1 −

, we 602

have two degenerate transverse modes of frequency ω

= ω

+ c

, and a 603

longitudinal mode of frequency ω = ω

. 604

In the presence of a dc magnetic ﬁeld along the z-axis and q in the y- 605

direction, the secular equation for wave propagation is given by 606

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

14.10 Propagation of Acoustic Waves 453



− ξ

−ε

00ε

− ξ



=0.

For the polarization with E parallel to the z-axis we have

607

=1−

4πi

(q,ω),

where σ

(q,ω) is the nonlocal conductivity. For ω

 nω

and in the limit 608

q → 0, we obtain the cyclotron waves given by ω

= n

+O(q

). They prop- 609

agate perpendicular to the dc magnetic ﬁeld, and depend for their existence 610

on the q dependence of σ. 611

For a system consisting of a metal of dielectric function ε

ﬁlling the space 612

z>0 and an insulator of dielectric constant ε

in the space z<0, the waves 613

localized near the interface (z = 0) are written as 614

(1)

(r,t)=E

(1)

iωt−iq

y−α

(0)

(r,t)=E

(0)

iωt−iq

y+α

The superscripts 1 and 0 refer, respectively, to the metal and dielectric. The

615

boundary conditions at the plane z = 0 are the standard ones of continuity 616

of the tangential components of E and H, and of the normal components of 617

D and B. For the polarization with E

= 0, but E

=0= E

, the dispersion 618

relation of the surface plasmon is written as 619

=0,

where α

=(ω

+ q

− ω

)

1/2

and α

=(q

− ε

)

1/2

. 620

The classical equation of motion of the ionic displacement ﬁeld ξ(r,t)in 621

a metal is written as 622

∂

∂t

= C



∇(∇·ξ) − C

∇×(∇×ξ)+zeE +

ξ ×(B

+ B)+F.

Here, C



and C

are elastic constants, and the collision drag force F is F = 623

−

eτ

), where the ionic current density is j

= n

ξ. The self-consistent 624

electric ﬁeld E is determined from the Maxwell equations j

=Γ(q,ω) · E: 625

E(q,ω)=[Γ− σ]

−1

(iωne 1 − σ · Δ) · ξ.

Here, a tensor Δ

is deﬁned by Δ =

eiω



−

iωτ(1+iωτ)



, where

q =

|q|

. 626

The equation of motion is thus of the form T(q,ω) · ξ(q,ω)=0, where T 627

is a very complicated tensor. The normal modes of an inﬁnite medium are 628

determined from the secular equation 629

det | T(q,ω) |=0.

The solutions ω(q) have both a real and imaginary parts; the real part deter-

630

mines the velocity of sound and the imaginary part the attenuation of the 631

wave. 632

Uncorrected Proof

BookID 160928 ChapID 15 Proof# 1 - 29/07/09

15 1

Superconductivity 2

15.1 Some Phenomenological Observations 3

of Superconductors 4

Superconductors are materials that behave as normal metals at high temper- 5

atures (T>T

; however, below T

they have the following properties: 6

1. The dc resistivity vanishes. 7

2. They are perfect diamagnets; by this we mean that any magnetic ﬁeld that 8

is present in the bulk of the sample when T>T

is expelled when T is 9

lowered through the transition temperature. This is called the Meissner 10

eﬀect. 11

3. The electronic properties can be understood by assuming that an energy 12

gap 2Δ exists in the electronic spectrum at the Fermi energy. 13

Some common superconducting elements and their transition temperatures 14

are given in Table 15.1. 15

Resistivity 16

Aplotofρ(T ), the resistivity versus temperature T , looks like the diagram 17

shown in Fig. 15.1. Current ﬂows in superconductor without dissipation. Per- 18

sistent currents in superconducting rings have been observed to circulate 19

without decaying for years. There is a critical current density j

which, if 20

exceeded, will cause the superconductor to go into the normal state. The ac 21

current response is also dissipationless if the frequency ω satisﬁes ω<

¯h

, 22

where Δ is an energy of the order of k

. 23

Thermoelectric Properties 24

Superconducting materials are usually poor thermal conductors. In normal 25

metals an electric current is accompanied by a thermal current that is asso- 26

ciated with the Peltier eﬀect. No Peltier eﬀect occurs in superconductors; the 27

current carrying electrons appear to carry no entropy. 28