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Физика конденсированного тела

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Dressel.Gruner.Electrodynamics of Solids.2003

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2kF

448	Appendix F Dielectric response in reduced dimensions

the limit for τ → ∞. For the real part of the Lindhard function we obtain:

−e2 D(EF)

1 − C− qF 2kF − qvF

− 1

1/2

2kF

qvF

− +

−

for

> 1 <

−

1/2

)

−

χ1(q, ω)

−

(F.6a)

for

> 1 >

2kF

− qvF

2kF

+ qvF

1/2

2kF

qvF

−

for

< 1 <

2kF

qvF

2kF

qvF

−

e D( F)

for

< 1 >

−

where we deﬁne C± = sgn ± qωvF , and the density of states at the Fermi energy is D(EF) = N/EF in two dimensions; the imaginary part of the Lindhard function is

e2 D(EF) qF 1 −

2kF

− qvF

1/2

−

1 − 2kF

+ qvF

1/2

for

> 1 <

−

1/2

2kF

qvF

−

for

> 1 >

2kF

qvF

2kF

qvF

−

χ (q, ω)

(F.6b)

1/2

e2 D(

F) kF

2kF

qvF

−

for

< 1 <

2kF

qvF

2kF

qvF

−

for

< 1 >

2kF

qvF

2kF

qvF

−

The real

and imaginary parts of

χ (q, ω)

in Fig.

F.1 as a function of

are plotted

reduced frequency and wavevector. In two dimensions the Fermi surface is a circle

F.1 Dielectric response function in two dimensions

449

Fig. F.1. Frequency and wavevector dependence of (a) the real and (b) the imaginary parts of the Lindhard response function χˆ (q, ω) of a free-electron gas at T = 0 in two dimensions evaluated using Eqs (F.6).

450	Appendix F Dielectric response in reduced dimensions

and χ1 is constant up to q = 2kF over the entire semicircle. In this range there are no excitations possible and thus χ2 vanishes. In the static limit (ω = 0) the imaginary part of the response function χ2(q, 0) = 0 everywhere, and the real part becomes

−e2 D(EF)

χ1(q, 0) = −e2 D(EF)

1 − 2qF	2kF		− 1	1/2
k		q	2
k		q

for q < 2kF

for q > 2kF ,

(F.7)

where the long wavelength limit is given by −e2 D(EF) = −2N e2/mvF2 = −e2 N /EF.

F.1.3 Dielectric constant

The complex dielectric constant ˆ(q, ω) in two dimensions is immediately derived from Eq. (F.5):

(q, ω)

2e2

f 0(Ek+q) − f 0(Ek)

(F.8)

− π q2

E(k + q) − E(k) − h(ω + i/τ )

The real and imaginary parts of the dielectric constant ˆ(q, ω) have the form

1 − h¯ vFq2 1 − C− q

2kF − qvF

− 1

1/2

4e2kF

1/2

2kF

qvF

−

for

1 <

2kF

qvF

2kF

qvF

−

1/2

−

1(q, ω)

−

(F.9a)

for

1 >

2kF

− qvF

2kF

+ qvF

1/2

4e2k

hvFq

2kF

qvF

− ¯

−

for

< 1 <

2kF

qvF

2kF

qvF

−

for

< 1 >

−

F.1 Dielectric response function in two dimensions

451

Fig. F.2. Frequency and wavevector dependence of (a) the real and (b) the imaginary parts of the dielectric constant ˆ(q, ω) of a free-electron gas at T = 0 in two dimensions after Eqs (F.9). The zero-crossing of 1(q, ω) indicates the plasmon frequency.

452	Appendix F Dielectric response in reduced dimensions

with C± = sgn

, and

2kF

qvF

− qvF

− 1 − 2kF + qvF

hvFq

1 − 2kF

1/2

4e2kF2

for

> 1 <

2kF

qvF

2kF

qvF

−

1/2

2kF

qvF

hvFq

−

for

> 1 >

2kF

qvF

2kF

qvF

−

(q, ω)

(F.9b)

1/2

2kF

qvF

hvFq

−

for

< 1 <

2kF

qvF

2kF

qvF

−

for

< 1 >

−

Both functions are displayed in Fig. F.2.

At q = 2kF we see a change of slope in the dielectric function which reﬂects that the screening is cut off for large wavelengths. This leads to Friedel oscillations in the response of the system to a localized perturbation, which also occur in the three-dimensional case (Eq. (5.4.20)). For large distances we obtain

(r)	eλ4kF2	cos{2kFr}	(F.10)
	(2kF + λ)2 (2kFr)2

for the spatial dependence of the potential in a two-dimensional electron gas. For static ﬁelds (ω → 0), Eq. (F.9a) gives the approximation:

for

2kF

1(q, 0)

2kF

2 1/2

≤

(F.11)

q > 2kF

−

for

which for small q leads to the result derived in Eq. (F.4) for static screening. For q > 2kF the screening effects fall off much more rapidly. It is interesting to note that via kF the screening now depends on the charge concentration.

F.1.4 Single-particle and collective excitations

In the two-dimensional electron gas the conditions for single-particle excitations are essentially the same as discussed in Section 5.3 for three dimensions. There is a particular kind of singularity at E(q) = | 2h¯m2 (q2 − 2qkF)| which becomes more

(r) =

F.2 Dielectric response function in one dimension

453

important for one dimension [Cza82], as we will see in the following section. Up to q = 2kF, but in particular near q = 0, excitations are possible, and Eqs (5.3.1)– (5.3.3) derived in three dimensions apply.

Plasmons are the longitudinal collective excitations of a two-dimensional electron gas which are sustained if 1(q, ω) = 0. For long wavelengths (mω hqk¯ F), we obtain

		ω2	=	mω2	2
q2	−				,	(F.12)
		c2		2π N e2

where the right hand side can be neglected for very long wavelengths (q < 2π N e2/mc2). For short wavelengths Eq. (F.9a) leads to the plasma frequency in

two dimensions
ωp2 ≈	2π N e2q	+	3	q2vF2 .	(F.13)
	m		4
The well known square root plasma dispersion of the leading term ωp					√		is
						q

not affected by the ﬁnite thickness of the electron gas or by correlation effects; it is obtained for degenerate as well as non-degenerate electron gases and has been observed in various systems, such as electrons on liquid helium or inversion layers in semiconductors [And82].

F.2 Dielectric response function in one dimension

The one-dimensional response functions are of great theoretical importance although the realization can only be achieved as the limiting case of a very anisotropic material, such as quasi-one-dimensional conductors. In recent years the problem gained relevance due to the progress in conﬁning the two-dimensional electron gas at semiconductor interfaces or arranging metallic atoms at surfaces along lines. As already pointed out in the discussion of the two-dimensional case, we consider an idealized situation of a one-dimensional electron gas, but the electric ﬁeld lines extend in three-dimensional space. Concerning the size quantization effect, we again consider only the ground state of the system to be occupied.

F.2.1 Static limit

The charge density of the screening cloud around a point charge decreases very slowly at long distances. In one dimension the q dependence of the potential cannot be deﬁned in the same way as for two and three dimensions (cf. Eqs (F.3) and (5.4.9)); the Coulomb potential can be approximated by [Hau94]

454 Appendix F Dielectric response in reduced dimensions

The case of a quantum wire with ﬁnite thickness d is discussed in [Lee83]. The one-dimensional density of charge carriers is N = 2kF/π ; the density of states diverges at the band edge with

D(E) =	2m	=	N		(F.14)
				.
	π h2k		2E
	¯

F.2.2 Lindhard dielectric function

As in three dimensions, the complex dielectric function in one dimension is obtained by solving

χ (q, ω)	=	e2	lim	k	f 0(Ek+q ) − f 0(Ek )	.	(F.15)
					E(k + q) − E(k) − h¯ (ω + i/τ )
ˆ		1/τ →0

Since in one dimension the Fermi surface consists of two lines, the sum over

k space is reduced to 2

f 0(Ek )

analytical form

# E(k+q)−E(k)−h¯ ω .

After some algebra, we obtain the

ω+i/τ

qvF

+ Ln

2kF

qvF

χˆ (q, ω) = −e2 D(EF)

F −

ω+i/τ

2kF

−

qvF

−

2kF

qvF

−

(F.16)

The real part and the imaginary part of the Lindhard dielectric function have the following form:

−

+ 1

2kF

qvF

2kF

qvF

,(F.17a)

χ1

(q, ω)

= −

−

< 1

qvF

2kF

e2 D(

< 1

χ (q, ω)

−

(F.17b)

F − 2kF

F +

2kF

−

> 1 .

qvF

2kF

In one dimension the density of states is

2 F

π h2kF

for both spin

directions. Both parts of the complex Lindhard function are plotted in Fig. F.3.

The Fermi surface in one dimension contains two points and we have perfect nesting for q = 2kF indicated by the peak at zero energy (ω = 0). The semicircle is given by h¯ ω = 2h¯m (q2 −2qkF). The region around the plasma frequency ωp shows a zero-crossing similar to the three-dimensional case (Fig. 5.15). The real part of the Lindhard function is plotted in Fig. F.3a for different frequencies and wavevectors. In Fig. 5.14 the static limit of the dielectric response function χ1(q) is also shown

F.2 Dielectric response function in one dimension

455

Fig. F.3. Frequency and wavevector dependence of (a) the real and (b) the imaginary parts of the Lindhard response function χˆ (q, ω) of a free-electron gas at T = 0 in one dimension as calculated using Eqs (F.17). The peaks are artifacts of the numerical procedure.

456

Lindhard function χ1(q) / χ1(q = 0)

Appendix F Dielectric response in reduced dimensions

Ε F = 2 kBT

2kF

Wavevector q

Fig. F.4. One-dimensional response function χ1(q) at ﬁnite temperatures as a function of wavevector normalized to the q = 0 value. The temperatures indicated are normalized to the Fermi energy EF.

in the one-dimensional case. It has a pronounced singularity at q = 2kF, which reﬂects that all states with the momentum difference 2kF have the same energy:

	e2k		q	+	2kF
χ1(q, 0) = −	F	D(EF) ln		+		(F.18)
χ1(q, 0) = −	q	D(EF) ln	q	−	2kF	(F.18)
				−

assuming a linear dispersion relation around EF; this approximation is only valid near the Fermi wavevector kF. For small q values χ11D(q) is given by the Thomas– Fermi approximation Eq. (5.4.19). In contrast to a three-dimensional electron gas the response function diverges at q = 2kF in one dimension. For ﬁnite temperatures the divergency of χ11D(q) at q = 2kF decreases and the peak broadens as shown in Fig. F.4. According to the mean ﬁeld theory we ﬁnd

χ	1D	(2k	F	, T )	= −	e2 D(	EF	) ln	1.14EF	.	(F.19)
	1
									kBT

− 1

+ 1

F.2 Dielectric response function in one dimension

457

F.2.3

Dielectric constant

In one dimension the equation for the dielectric constant is given by

ω+i/τ

+ Ln

ω+i/τ

−

qvF

ˆ(q, ω) = 1

F +

hq3vF

ω+i/τ

2kF

−

2kF +

−

qvF

−

qvF

(F.20)

Using Eq. (3.2.7) the real and imaginary parts of the dielectric constant are

4e2kF

−

(q, ω)

2kF

qvF

+ hq3vF

−

, (F.21a)

< 1

(q, ω)

< 1 <

. (F.21b)

qvF

2kF

hq¯

−

2kF

q F

2kF

4π e2k

qvF

−

2kF

> 1

The real part of the dielectric constant is plotted in Fig. F.5, together with the imaginary part. In the static case (ω → 0), the real part of the dielectric constant simpliﬁes to

		8e2m		q	+	2kF	.
1(q, 0) = 1	+		ln	q	+	2kF		(F.22)
1(q, 0) = 1	+	h2q3	ln	q	−	2kF		(F.22)
	¯				−

If the wavevector q is comparable to the Fermi wavevector, Eq. (F.20) tends to

				8e2v	F		/h
(q, ω)	=	1	−		F		¯		,	(F.23)
(q, ω)		1					¯	2	,	(F.23)
ˆ				(ω + i/τ )		2	−	q4vF
				(ω + i/τ )			−	4kF2

which is similar to the value from the semiclassical treatment for large kF.

In Eq. (5.4.10) we arrived at the three-dimensional dielectric constant in the ω → 0 limit, i.e. the well known Thomas–Fermi dielectric response function. In one dimension the dielectric constant in the limit ω qvF, or ωτ 1, is given by

1(q, ω) ≈ 1	4π N e2	(F.24)
	− mq2vF2 .

Interestingly, the difference between one and three dimensions is just a factor 2/3. We also arrive at this factor during the discussion of one-dimensional semiconductors in Section 6.3.1.

F.2.4 Single-particle and collective excitations

In one dimension, the single-particle excitation spectrum is signiﬁcantly different from the threeand two-dimensional cases. For 0 < q < 2kF, Emin is no longer

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