Добавил:

fench Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Национальный исследовательский ядерный университет (МИФИ)

Предмет:

Физика конденсированного тела

Файл:

Dressel.Gruner.Electrodynamics of Solids.2003

.pdf

Скачиваний:

Добавлен:

27.08.2013

Размер:

3.91 Mб

Скачать

☆

<<< < Предыдущая 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 4546 / 4946 47 48 49 > Следующая >>>

438	Appendix E Non-local response

where I0 and K0 are the modiﬁed zero order Bessel functions of ﬁrst and second kind, respectively. In the case of higher temperatures but low frequencies (h¯ ωkBT ) we can write

Zˆ S(ω)

hω/π

1/3

hω/ kBT

ω)

1/2

Rn(ω)

tanh{ /2kBT }

3π sinh{ / kBT }

(2 /h

hω

−

hω

3π

2kBT

− 3π 3

(kBT )2

2kBT

coth

ζ (3) coth

−

+ 48 2

h2ω2

(E.19)

if 2 hωkBT ; otherwise

Zˆ S(ω)

π 2

−1/3

= −2i

−i

(E.20)

Rn(ω)

2kBT hω

Finally we want to consider the cases where the frequency exceeds the temperature. In the range kBT h¯ ω ≈ and ω < 2 (0)/h¯ ,

Zˆ S(ω)

hω/2

1/3

exp{− / kBT

π k

1/2

Rn(ω)

E ¯hω/2

hω

}

hω/2

}

{ ¯

}

−

exp{− / kBT }

π kBT

−

π kBT

1/2

(E.21)

hω

hω/2

}

{ ¯

and for large frequencies ω (0)/h but low temperatures

Zˆ S(ω)

= 1

−

√

Rn(ω)

hω

(0)

1 +

−i√3

hω

3 ln

(0)

−

3√3

(E.22)

The simple ω2 dependence of the surface resistance derived in Eq. (7.4.23) is the limit of Eq. (E.21) for small frequencies.

E.5 Non-local response in superconductors

The effects of a ﬁnite mean free path can also be discussed in terms of the kernel. Here we follow the outline of M. Tinkham [Tin96] and G. Rickayzen [Ric65]. In the limit δ, λ the local relations between current and ﬁeld are no longer valid, and we have to use the expressions derived for the anomalous regime (Section E.1),

summarized in Zˆ S/Rn	=	2	σ /σn	−1/3. We are interested in the frequency and
	=		− ˆ
temperature dependent	response and how it varies for different wavevectors.
temperature dependent

E.5 Non-local response in superconductors

439

By comparing Eq. (E.13) with Eq. (4.3.30) we see that

K (0) =	1	=	4π Nse2			(E.23)
					.
	λL2			mc2

This immediately implies that at low temperatures the kernel K as well as the penetration depth λ are frequency independent. Let us ﬁrst discuss the temperature dependence of K (q, T ) in the long wavelength (q → 0) limit. In the expression

J(q, ω) = Jp(q, ω) + Jd(q, ω)

τ →∞ c 2m

2k[A(q, ω)

2k]

lim

1 − f (Ek) − f (Ek+q)

1 − f (Ek) − f (

k+q)

k+q

hω

+ E

k+q

hω

−

ih/τ

− ¯

+ E

× (ukvk+q − vkuk+q)2

f (

k) − f (Ek+q)

f (

k) − f (Ek+q)

−E

hω

−

ih/τ

−E

hω

ih/τ

k+q

+ E

− ¯

+ E

× (ukuk+q + vkvk+q)2

N e2

−

A(q, ω)

(E.24)

the paramagnetic current density simpliﬁes to

−

2e2h2

−

∂ f

= m

m2c

∂ k

Jp(0, T )

k( fk0

fk1)

[A(0)

k]k

in the small ﬁeld approximation. Thus we obtain for the kernel

K p(0, T )

4π N e2

4EF

∂ f

λ−2(0)

∞

∂ f

dζ ,

= −

− ∂

k = −

−∞ −

∂

mc2

where ζ = Ek − EF. The total kernel is given by

E 2)1/2 dE .

K (0, T ) = λL−2(T ) = λL−2(0)

1 − 2

∞ −

∂

(

−

∂

(E.25)

For temperatures T

≥ Tc the material is in the normal state and = 0; since

the integral reduces to

f (0)

= 1/2, we obtain K (0, T

≥ Tc) = 0 as expected.

This corresponds to the exact cancellation of the paramagnetic and diamagnetic currents and no Meissner effect is observed. At low temperatures (T < 0.5Tc), the derivative of the Fermi distribution can be approximated by a linear function, and

440

Penetration depth λ (T) / λ ( 0)

Appendix E Non-local response

1.10

1.05

1.00
0.0	0.1	0.2	0.3	0.4	0.5

10

	0.5		1
		Temperature T / Tc

Fig. E.4. Temperature dependence of the normalized penetration depth λ(T )/λ(0). The low temperature regime (inset) shows an exponential behavior according to Eq. (E.26). In the high temperature range Eq. (E.27) is plotted.

thus the previous expression reduces to

λ(0)	=	2kBT		1/2	− kBT	;

λ(T ) − λ(0)			π	exp			(E.26)

while in the range 0.8Tc < T < Tc, the temperature dependence can be approximated by

λ(0)	=	1 −	Tc		4	−1/2	(E.27)
						.
λ(T )				T

In Fig. E.4 the temperature dependence of the penetration depth is shown according to Eqs (E.26) and (E.27). The temperature dependence of K (0, T ) is plotted in

Fig. E.5a, where K (0, T )/K (0, 0) = [λL(0)/λL(T )]2. As T decreases below Tc, the superconducting gap opens, i.e. (T )/ kBT increases, causing the kernel in

Eq. (E.25) to diminish, and eventually become exponentially small, and K (0, T →

0) = λ−L 2(0).

So far we have restricted our analysis to the q = 0 limit; let us consider next

E.5 Non-local response in superconductors

441

2 K (0, T) λ L(0) = 2 K (0, 0) λ L(T)

K (q) / K (0)

1.0





0.5					K1
					K1


	(a)						K2
							K2



0.0

	0.2	0.4	0.6	0.8		1.0
0.0	0.2	0.4	0.6	0.8		1.0

T / Tc 1.0

0.8

0.6

0.4 1 − (qξ0)2

0.2			(qξ0)−1
0.2			(qξ0)−1
		(b)
0.00		(b)


	1			2	3
				qξ0

Fig. E.5. (a) Temperature dependence of the normalized kernel K (0, T )/K (0, 0) (after [Tin96]). (b) Normalized kernel K (q)/K (0) as a function of the wavevector q.

the q dependence of K (q, T ) for T = 0. If we calculate the experimental value of Jp(q) following Eq. (E.24) we obtain

(q)

(v u

u v

k k+q − k k+q

[k A(q)]k

m2c

+ E

similar to the relation (4.3.31) we arrived at in the case of normal metals. For the

442 Appendix E Non-local response

q dependent kernel

1 − 2

∞ (vku

u vk q)2

dζ

K (q, 0) = λL− (0)

−

;

(E.28)

+ E

k+q

and for small q, we ﬁnd K (q, T )λ2

(

)

≈ 1 − q

2ξ 2

length ξ

0 , where the coherence

3π

is deﬁned in Eq. (7.4.2). If q 1/ξ0 (Pippard limit), K (q, 0) = K (0, 0)

, as

4qξ0

displayed in Fig. E.5b. In both the local and the London limit ξ0 → 0, and the q dependence of the kernel is negligible: K (q, 0) → K (0, 0).

The effect of impurity scattering can be taken into account in a phenomenologi-

cal way similar to Chambers’ approach discussed in Section 5.2.4,

λ2 (T )

K (0, T, )

∞

{−

}

λl2( , T )

= K (0, T, → ∞)

ξ0

J (s , T ) exp

, (E.29)

where we have used the

J (s , T ) function as

deﬁned

0∞ J (s , T ) ds

∞ exp

{−

/ξ

makes

small

and

increased

scattering

rate

( , T )

the

response

local.

(dirty

limit) λ

the

case

λL(T ) (ξ0/ )1/2 [J (0, T )]−1/2.

If we replace J (s , T ) by exp{−s /ξ0}, which is

valid for large , we obtain

λl( , T ) = λL(T ) 1

ξ0

1/2

(E.30)

where Pippard’s coherence length, ξ0, and ξ

are given by

J (0, T )

(E.31)

ξ0

with J (0, T = 0) = 1 and J (0, Tc) = 1.33. A full discussion of non-local effects

is given in [Tin96]. In the q → ∞ London limit (λL ξ0), i.e. the extreme

anomalous regime, K (q) = 3π/(4λ2Lξ0q) leads to λq→∞ = 0.58 λ2Lξ0 1/3 for specular scattering.

The temperature dependence of the penetration depth in the local limit is given

λl(T )

(T )

tanh

(T )

−1/2

(E.32)

λL(0)

ξ0 (0)

2kBT

which resembles Eq. (7.4.21). In the anomalous regime (assuming diffusive scattering), we obtain

λP(T )	=	3π 2 λL(0) (T )	tanh	(T )	−1/3	(E.33)
					,
λL(0)		4 ξ0 (0)		2kBT

where λL(T )/λL(0) is given by Eq. (E.25).

E.5 Non-local response in superconductors

443

Fig. E.6. Schematic representation of the local, London, and Pippard limits in the parameter space given by the three length scales, the coherence length ξ0, the London penetration depth λL, and the mean free path (after [Kle93]).

In Fig. E.6 we compare the three length scales important to the superconducting state; depending on the relative magnitude of the length scales, several limits are of importance. The ﬁrst is the local regime in which is smaller than the distance over which the electric ﬁeld changes, < ξ(0) < λ. When /ξ(0) → 0 the superconductor is in the so-called dirty limit. The opposite situation in which/ξ(0) → ∞ is the clean limit, in which non-local effects are important and we have to consider Pippard’s treatment. This regime can be subdivided if we also consider the third parameter λ. The case ξ(0) > λ is the Pippard or anomalous regime, which is the regime of type I superconductors; and ξ(0) < λ is the London regime, the regime of the type II superconductors.

444	Appendix E Non-local response

References

[Abr63] A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliffs, NJ, 1963)

[Abr88] A.A. Abrikosov, Fundamental Theory of Metals (North-Holland, Amsterdam, 1988)

[Cas67] H.B.G. Casimir and J. Ubbink, Philips Tech. Rev. 28, 271, 300, and 366 (1967)

[Cha52] R.G. Chambers, Proc. Phys. Soc. (London) A 65, 458 (1952); Proc. Roy. Soc. A

215, 481 (1952)

[Cha90] R.G. Chambers, Electrons in Metals and Semiconductors (Chapman and Hall, London, 1990)

[Gei74] B.T. Geilikman and V.Z. Kresin, eds, Kinetic and Nonsteady State Effects in Superconductors (John Wiley & Sons, New York, 1974)

[Kan68] E.A. Kaner and V.G. Skobov, Adv. Phys. 17, 605 (1968)

[Kar86] E. Kartheuser, L.R. Ram Mohan, and S. Rodrigues, Adv. Phys. 35, 423 (1986) [Kit63] C. Kittel, Quantum Theory of Solids (John Wiley & Sons, New York, 1963) [Kle93] O. Klein, PhD Thesis, University of California, Los Angeles, 1993

[Mat58] D.C. Mattis and J. Bardeen, Phys. Rev. 111, 561 (1958)

[Pip54b] A.B. Pippard, Metallic Conduction at High Frequencies and Low Temperatures, in: Advances in Electronics and Electron Physics 6, edited by L. Marton (Academic Press, New Yok, 1954), p. 1

[Pip62] A.B. Pippard, The Dynamics of Conduction Electrons, in: Low Temperature Physics, edited by C. DeWitt, B. Dreyfus, and P.G. deGennes (Gordon and Breach, New York, 1962),

[Reu48] G.E.H. Reuter and E.H. Sondheimer, Proc. Roy. Soc. A 195, 336 (1948)

[Ric65] G. Rickayzen, Theory of Superconductivity (John Wiley & Sons, New York, 1965)

[Sch83] J.R. Schrieffer, Theory of Superconductivity, 3rd edition (W.A. Benjamin, New York, 1983)

[Sok67] A.V. Sokolov, Optical Properties of Metals (American Elsevier, New York, 1967)

[Son54] E.H. Sondheimer, Proc. Roy. Soc. A 224, 260 (1954)

[Tin96] M. Tinkham, Introduction to Superconductivity, 2nd edition (McGraw-Hill, New York, 1996)

Further reading

[Hal74] J. Halbritter, Z. Phys. 266, 209 (1974)

[Lyn69] E.A. Lynton, Superconductivity, 3rd edition (Methuen & Co, London, 1969) [Par69] R.D. Parks, Superconductivity (Marcel Dekker, New York, 1969)

[Pip47] A.B. Pippard, Proc. Roy. Soc. A 191, 385 (1947)

[Pip50] A.B. Pippard, Proc. Roy. Soc. A 203, 98 (1950)

[Pip54a] A.B. Pippard, Proc. Roy. Soc. A 224, 273 (1954)

[Pip57] A.B. Pippard, Phil. Trans. Roy. Soc. (London) A 250, 325 (1957) [Pip60] A.B. Pippard, Rep. Prog. Phys. 33, 176 (1960)

Appendix F

Dielectric response in reduced dimensions

With a few exceptions we have considered mainly bulk properties in the book. The physics of reduced dimensions is not only of theoretical interest, for many models can be solved analytically in one dimension only. A variety of interesting phenomena are bounded to restricted dimensions. On the other hand, fundamental models such as the theory of Fermi liquids developed for three dimensions break down in one or two dimensions. In recent years a number of possibilities have surfaced to explain how reduced dimensions can be achieved in real systems. One avenue is the study of real crystals with an extremely large anisotropy. The second approach considers artiﬁcial structures such as interfaces which might be conﬁned further to reach the one-dimensional limit.

F.1 Dielectric response function in two dimensions

Reducing the dimension from three to two signiﬁcantly changes many properties of the electron gas. If the thickness of the layer is smaller than the extension of the electronic wavefunction, the energy of the system is quantized (size quantization). We consider only the ground state to be occupied. For any practical case, just the electrons are conﬁned to a thin sheet, while the ﬁeld lines pass through the surrounding material which usually is a dielectric. A good approximation of a two-dimensional electron gas can be obtained in surfaces, semiconductor interfaces, and inversion layers; a detailed discussion which also takes the dielectric properties of the surrounding media into account can be found in [And82, Hau94]. In this section we discuss the idealized situation of a two-dimensional electron gas; however, the Coulomb interaction has a three-dimensional spatial dependence. Following the discussion of the three-dimensional case, we ﬁrst consider the static limit. The full wavevector and frequency dependence of the longitudinal response in two dimensions is derived by the formalism used for the three-dimensional case.

445

446	Appendix F Dielectric response in reduced dimensions

F.1.1 Static limit

Let us assume an electric ﬁeld E(q, ω) = E0 exp{i(q · r − ωt)} which is purely longitudinal (q × E0=0) acting on a two-dimensional electron gas conﬁned in the z direction which is surrounded by a vacuum. An external source produces an additional electrostatic potential , which is related to the charge density ρ by Eq. (2.1.7): · ( ) = −4πρ. Here is the dielectric constant of the system

and ρ = ρext + ρind as usual; the particle density is indicated by N . We can express the induced charge density as a function of the local potential, and linearizing it

yields

ρind(r) = −e[N ( ) − N (0)]δ{z} = −e2 (r)						dN
							δ{z} .
						dEF
This allows us to rewrite Poisson’s equation
· ( ) − 2λ (r)δ{z} = −4πρext					,
where we deﬁne the screening parameter in two dimensions as
λ = 2π e2	∂ N		4π N me2				(F.1)
		=		.
	∂EF		vF2
The classical form λDH = 2π N e2	/(kBT ) appears as the two-dimensional form

of the Debye screening length. Not surprisingly, in two dimensions the screening effects are less efﬁcient than in three dimensions. Here we utilized the fact that the Fermi surface for the two-dimensional electron system is a curve and in the simplest case becomes a circle with the radius being the Fermi wavevector kF = (2π N )1/2. The density of states is given by

D(E) =	m	=	N	(F.2)

	π h2		EF
	¯

if only the lowest band is occupied; N is the number of electrons per unit area. Note that the density of states is independent of E. In contrast to the threedimensional case, the screening length λ−1 = h¯ 2/(2me2) for two dimensions is independent of the charge density. In three dimensions we obtained Eq. (5.4.5) as an equivalent to Eq. (F.1), leading to the screened potential given by Eq. (5.4.9). In the two-dimensional case the spatial dependence of the potential is found by a

F.1 Dielectric response function in two dimensions

447

Fourier–Bessel expansion

∞

(r) = q Aq J0(qr) dq ,

where J0 is the Bessel function of the order zero and Aq = e(q + λ)−1. The statically screened potential in two dimensions is therefore

2π e	(F.3)
(q) = q + λ .

Even in the absence of screening, the 1/q dependence of the quasi-two- dimensional Coulomb potential is different from the 1/q2 dependence found in three dimensions. In this approximation, the static dielectric function in two dimensions becomes

1(q) = 1	λ	(F.4)
	+ q

which is analogous to Eq. (5.4.10) derived in the three-dimensional case.

There are a large number of studies dealing with coupled layers since these questions became relevant in connection with the layered high temperature superconductors; for a review see [Mar95].

F.1.2 Lindhard dielectric function

From Eq. (5.4.15) we obtain for the complex dielectric response function

χ (q, ω)

lim

f 0(Ek+q) − f 0(Ek)

(F.5)

1/τ

→

− E

(k)

− ¯

−

0 k

hω

ih/τ

where the summation is taken over all one-electron states. At T = 0 the sum can be evaluated, and the result of the real and imaginary parts is obtained by considering

<<< < Предыдущая 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 4546 / 4946 47 48 49 > Следующая >>>

Соседние файлы в предмете Физика конденсированного тела

#
27.08.2013586.56 Кб22Baranov.Introduction to Physics Of Ultra Cold Gases.pdf
#
27.08.2013204.32 Кб15Chalker.Les.Houches. Integer quantum Hall effect & localization.pdf
#
27.08.2013463.83 Кб12Desbois.Les.Houches. Impurities and quantum Hall effect.pdf
#
27.08.2013522.31 Кб13Dotsenko.Intro to Stat Mech of Disordered Spin Systems.pdf
#
27.08.20131.68 Mб26Dresner, Stability of superconductors.2002.pdf
#
27.08.20133.91 Mб73Dressel.Gruner.Electrodynamics of Solids.2003.pdf
#
27.08.20131.17 Mб15Eschrig - Theory Of Superconductivity, A Primer.pdf
#
27.08.20134.01 Mб24Halilov (ed), Physics of spin in solids.2004.pdf
#
27.08.20135.21 Mб24Manske.Theory of unconventional superconductors.2004.pdf