Dressel.Gruner.Electrodynamics of Solids.2003

Добавил:

fench Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Национальный исследовательский ядерный университет (МИФИ)

Предмет:

Физика конденсированного тела

Файл:

(q2 − 2qkF) .

2¯m (2qkF − q2) =

(q2 − 2qkF) .

.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 45 46 4748 / 4948 49 > Следующая >>>

458	Appendix F Dielectric response in reduced dimensions

Fig. F.5. Frequency and wavevector dependence of the (a) real and (b) imaginary parts of the dielectric constant ˆ(q, ω) of a free-electron gas at T = 0 in one dimension after Eqs (F.21).

F.2

Dielectric response function in one dimension

459

)

2qk

Electron–hole

hω

excitations

σ1 = 0

transfer

)

2qk

Energy

σ1

≠ 0

−

hω

hω =

(2qk

− q2)

σ1 = 0

2kF

Wavevector q

Fig. F.6. Energy spectrum of the excitations shown as a function of momentum for an electron gas in one dimension. The shaded area indicates the pair excitations possible. Note, at low energies only excitations with momentum transfer q = 0 and q = 2kF are possible.

zero. The only zero energy transitions occur at q = 0 and 2kF. Between these two values, we have

Fig. F.6 shows this one-dimensional excitation spectrum.

Because of the divergency of the absorption at the boundary (shown in Fig. F.3b, for example), in one-dimensional metals collective excitations are only possible outside the continuum of single-particle excitations indicated by the hatched area in Fig. F.6. The plasma frequency has a dispersion

	2N e2
ωp(q) =	sma2 \| ln{qa}\|1/2qa + O(q2) ,	(F.26)

which is linear in ﬁrst approximation.

460	Appendix F Dielectric response in reduced dimensions

References

[And82] T. Ando, A.B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982)

[Cza82] A. Czachar, A. Holas, S.R. Sharma, and K.S. Singwi, Phys. Rev. B 25, 2144 (1982)

[Hau94] H. Haug and S.W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 3rd edition (World Scientiﬁc, Singapore, 1994)

[Lee83] J. Lee and H.N. Spector, J. Appl. Phys. 54, 6989 (1983)

[Mar95] N.H. March and M.P. Tosi, Adv. Phys. 44, 299 (1995)

Further reading

[Hau96] H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors, Springer Series in Solid State Sciences 123 (Springer-Verlag, Berlin, 1996)

[Hee79] A.J. Heeger, Charge-Density Wave Phenomena in One-Dimensional Metals, in: Highly Conducting One-Dimensional Solids, edited by J.T. Devreese, R.P. Evrard, and V.E. von Doren (Plenum Press, New York, 1979)

[Lie97] A. Liebsch, Electronic Excitations at Metal Surfaces (Plenum Press, New York, 1997)

[Ste67] F. Stern, Phys. Rev. Lett. 18, 546 (1967)

[Voi94] J. Voit, Rep. Prog. Phys. 57, 977 (1994)

Appendix G

Important constants and units

Throughout the book we have used cgs, or Gaussian, units. Without doubt, SI units are more convenient when equating numbers and analyzing experimental results. Therefore we provide conversion tables for commonly used quantities. For a discussion of problems of units and the conversion between Gaussian (cgs) and rational SI units (mks), see for example [Bec64, Jac75].

461

462	Appendix G Important constants and units

Table G.1. Conversion table.

In order to convert the equations in cgs units into those in the SI system, the relevant symbols have to be replaced by the corresponding one on the right hand side of the table.

Quantity	Gaussian	SI
	(cgs) systems	(mks) system

Speed of light	c
Electric ﬁeld	E
Electric displacement	D
Scalar potential	φ
Charge density	ρ
Electric polarization	P
Current density	J
Dielectric constant	ˆ
Conductivity	σˆ
Magnetic ﬁeld	H
Magnetic induction	B
Vector potential	A
Magnetization	M
Permeability	µˆ
Impedance	Zˆ
Poynting vector	S
Energy density	u

c = ( µ1 )1/2 0 0

(4π 0)1/2 E

4π 1/2 D

(4π 0)1/2 φ

(4π 1 )1/2 ρ 0

(4π 1 )1/2 P

(4π 1 )1/2 J

σˆ

4π 0

(4π µ0)1/2 H

4π 1/2 B

µ0

4π 1/2 A

µ0

4π 1/2 M

µ0

µˆ µ0

4π 0 Zˆ

4cπ S

4π u

Note: In the case of the electric ﬁeld E we must remember to replace 1/4π by 0.

For convenience, in Table G.2 we also list some fundamental physical constants used throughout the book. These values are only given in SI units since quantitative values are no longer given in cgs units such as erg, dyne, etc. However, some

Appendix G Important constants and units

463

Table G.2. The fundamental physical constants.

Quantity

Value in SI units

Speed of light in a vacuum

c = 2.997 924 58 × 108 m s−1

Permittivity of free space

0 = 8.8542 × 10−12 A s V−1 m−1

Permeability of free space

µ0 = 1.2566 × 10−6 V s A−1 m−1

Elementary charge

e = 1.602 189 × 10−19 A s

Mass of electron

m = 9.010 953 × 10−31 kg

Mass of proton

mp = 1.672 61 × 10−27 kg

Mass of neutron

mn = 1.674 82 × 10−27 kg

Planck constant

h = 6.626 176 × 10−34 J s

h/2π

1.054 589

10−34 J s

Avogadro constant

NA = 6.022 05 × 1023 mol−1

Boltzmann constant

kB = 1.380 66 × 10−23 J K−1

= 5.291 77 × 10−11 m

Bohr radius

rB =

me¯ 2

Rydberg constant

me4

2.179 91 × 10−18 J

Ry = 2h2

Table G.3. Table of units.

Quantity

cgs

Conductivity σ

s−1

1.1 × 10−12 −1 cm−1 −1 cm−1

9 × 1011 s−1

Magnetic ﬁeld H

103

A m−1

4π × 10−3 Oe

4π

Magnetic

10−4 T

104 G

induction B

Magnetic

emu cm−3 Oe−1

4π

cm3 Oe emu−1

4π

susceptibility χm

Pressure p

bar

105 Pa

10−5 bar

7.53 × 10−3 torr

Note: Oe = oersted; G = gauss; T = tesla; Pa = pascal. 1 emu = 1 G cm3; we also found the susceptibility per mass and per mole.

quantities still occur in different units, like the magnetic ﬁeld. In Table G.3 we give only those which are important in our context. Most important are certainly the units of energy and its equivalent such as temperature and frequency; the corresponding conversion is listed in Table G.4, which can also be found in Chapter 8.