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Физика конденсированного тела

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Dressel.Gruner.Electrodynamics of Solids.2003

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168	6 Semiconductors

σ1 (103 Ω −1 cm−1)

ε1 (103)

30
25					(a)
					(a)

20


15


10


5


0



1.2					(b)
1.2
0.8


0.4


0.0


1.0




					(c)
					(c)
0.8
0.8
0.6


0.4


0.2


0.0


	10−1	101	103	105
10−3	10−1	101	103	105

Frequency ν (cm−1)

Fig. 6.14. (a) Optical conductivity σ1(ω), (b) dielectric constant 1(ω), and (c) reﬂectivity R(ω) for a doped semiconductor with an impurity band having ﬁnite dc conductivity σdc = 2 × 104 −1 cm−1 and scattering rate 1/(2π cτ ) = γ = 0.2 cm−1. The excitations across

the energy gap Eg/ hc = ωg/(2π c) = 1000 cm−1 are modeled according to Eq. (6.3.11) by a square root onset σ1(ω) (ω − ωg)1/2(ω2 + ωp2)−1 with ωp/(2π c) = 50 000 cm−1.

6.6 The response for large ω and large q

169

6.6 The response for large ω and large q

Much of what has been said before refers to electronic transitions at energies near to the single-particle gap. Also, with one notable exception, namely indirect transitions, we have considered only transitions in the q = 0 limit; a reasonable assumption for optical processes. The emergence of the gap, and in general band structure effects, lead to an optical response fundamentally different from optical properties of metals where, at least for simple metals, the band structure can be incorporated into parameters such as the bandmass mb – leaving the overall qualitative picture unchanged.

Obviously, opening up a gap at the Fermi level has fundamental consequences as far as the low energy excitations are concerned; there is no dc conduction, but a ﬁnite, positive dielectric constant at zero frequency – and also at zero temperature

– for example. It is expected, however, that such a (small) gap has little inﬂuence on excitations at large frequencies, and also with large momenta; and we should in this limit recover much of what has been said about the high ω and large q response of the metallic state. This is more than academic interest, as these excitations are readily accessible by optical and electron energy loss experiments – among other spectroscopic tools.

The evaluation of the full ω and q dependent response is complicated, and is discussed in several publications [Bas75, Cal59, Coh88]. Here we recall some results, ﬁrst only for ω = 0 and only for three-dimensional, isotropic semiconductors. Starting from Eq. (4.3.20) derived for the longitudinal response in the static limit (ω → 0), the complex dielectric constant reads

4π e2

f 0( k

q,l )

−

f 0

(

kl )

(q) 1

l,l

E +

q, l

exp iq r kl

− E

q,l

| +

{ · }| |

ˆ = − q2 k

(6.6.1)

Let us consider large q values ﬁrst. Here the wavevector dependence of the dielectric constant can, in a ﬁrst approximation, be described by a free-electron gas, with the bandgap and band structure effects, in general, of no importance. This approach leads to Eq. (5.4.18), and we ﬁnd for the real part of the dielectric constant

hωpkF

q 2kF

1(q) = 1 +

¯q F

1 +

−

(6.6.2)

4kF

−

2kF

For large q

hω

1(q) = 1 +

¯ gp

(6.6.3)

approaches the free-electron behavior that we recovered for simple metals in Chapter 5.

170	6 Semiconductors

)

2qk

Electron–hole

hω

excitations

σ1

transfer

= 0

)

2qk

−

Energy

σ1 ≠

ωp(q)

hωp

σ1 = 0

2kF

Wavevector q

Fig. 6.15. Excitation spectrum of a three-dimensional semiconductor. The pair excitations fall within the shaded area. In the region h¯ ω > Eg + (h¯ 2/2m)(2kF + q)q, the absorption vanishes since h¯ ω is larger than the energy difference possible. Also for h¯ ω < Eg + (h¯ 2/2m)(q − 2kF)q we ﬁnd σ1 = 0.

Next, let us examine what happens at high frequencies, in the q = 0 limit; i.e. at frequencies larger than the spectral range where band structure effects are of importance. In this limit, the effects related to the density of states can be neglected, and the inertial response of the electron gas is responsible for the optical response. The Lorentz model in the limit ω ω0 is an appropriate guide to what happens. First, we recover a high frequency roll-off for the conductivity σ1(ω), and

ωp2 τ	(6.6.4)
σ1(ω) ≈ 4π ω2τ 2 .

Second, there is a zero-crossing of the dielectric constant at frequency ω = ωp; this has two important consequences that we have already encountered for metals, namely that ωp is the measure of the onset of transparency (here also just as in

Further reading

[Car68] M. Cardona, Electronic Optical Properties of Solids, in: Solid State Physics, Nuclear Physics, and Particle Physics, edited by I. Saavedra (Benjamin, New York, 1968), p. 737

172	6 Semiconductors

[Cha83] G.W. Chantry, Properties of Dielectric Materials, in: Infrared and Millimeter Waves, Vol. 8, edited by K.J. Button (Academic Press, New York, 1983)

[Coh88] M.L. Cohen and J.R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors (Springer-Verlag, Berlin, 1988)

[Gre68] D.L. Greenaway and G. Harbeke, Optical Properties and Band Structure of Semiconductors (Pergamon Press, Oxford, 1968)

[Har72] G. Harbeke, in: Optical Properties of Solids, edited by F. Abeles` (North-Holland, Amsterdam, 1972)

[Kli95] C.F. Klingshirn, Semiconductor Optics (Springer-Verlag, Berlin, 1995)

[Pan71] J.I. Pankove, Optical Processes in Semiconductors (Prentice-Hall, Englewood Cliffs, NJ, 1971)

[Wal86] R.F. Wallis and M. Balkanski, Many-Body Aspects of Solid State Spectroscopy

(North-Holland, Amsterdam, 1986)

Broken symmetry states of metals

The role of electron–electron and electron–phonon interactions in renormalizing the Fermi-liquid state has been mentioned earlier. These interactions may also lead to a variety of so-called broken symmetry ground states, of which the superconducting ground state is the best known and most studied. The ground states are superpositions of electron–electron or electron–hole pairs all in the same quantum state with total momenta of zero or 2kF; these are the Cooper pairs for the superconducting case. There is an energy gap , the well known BCS gap, introduced by Bardeen, Cooper, and Schrieffer [Bar57], which separates the ground state from the single-particle excitations. The states develop with decreasing temperature as the consequence of a second order phase transition.

After a short review of the various ground states, the collective modes and their response will be discussed. The order parameter is complex and can be written asexp{iφ}; the phase plays an important role in the electrodynamics of the ground state. Many aspects of the various broken symmetry states are common, but the distinct symmetries also lead to important differences in the optical properties. The absorption induced by an external probe will then be considered; it is usually discussed in terms of the so-called coherence effects, which played an important role in the early conﬁrmation of the BCS theory. Although these effects are in general discussed in relation to the nuclear magnetic relaxation rate and ultrasonic attenuation, the electromagnetic absorption also reﬂects these coherence features, which are different for the various broken symmetry ground states. As usual, second quantized formalism, as introduced in Section 4.2, is used to describe these effects, and we review what is called the weak coupling theory of these ground states.

7.1 Superconducting and density wave states

The various ground states of the electron gas are, as a rule, discussed using second quantized formalism, and this route is followed here. The kinetic energy of the

173

174	7 Broken symmetry states of metals

Table 7.1. Various broken symmetry ground states of one-dimensional metals.

	Pairing	Total spin Total momentum Broken symmetry

Singlet superconductor	electron–electron	S = 0	q = 0	gauge
	e+, σ ; e−, −σ
Triplet superconductor	electron–electron	S = 1	q = 0	gauge
	e+, σ ; e−, σ
Spin density wave	electron–hole	S = 1	q = 2kF	translational
	e+, σ ; h−, −σ
Charge density wave	electron–hole	S = 0	q = 2kF	translational
	e+, σ ; h−, σ

electron gas is

Hkin =		p2	(r) = k,σ	h2k2	ak+,σ ak,σ = k,σ	Ekak+,σ ak,σ (7.1.1)
	dr (r) 2m			¯2m

in terms of the creation and elimination operators deﬁned in Section 4.2. Next we describe the interaction between the electrons; this, in its general form, is given by


Hee =		dr dr N (r)V (r, r )N (r ) =		dr dr (r) (r)V (r, r ) (r ) (r )
=	k,k	,l,l ,σ,σ	Vk,k ,l,l ak+,σ ak+,σ al,σ al ,σ ;

where N is the particle density, and V denotes the potential energy due to the interaction. With these – the kinetic and the interaction – terms, the pairing Hamiltonian is cast into the form

H =		k,k	,l,l ,σ,σ
H =	Ek ck+,σ ck,σ +		Vk,k ,l,l ak+,σ ak+,σ al,σ al ,σ .	(7.1.2)

k,σ

In order to see its consequences, the interaction term has to be speciﬁed to include only terms which lead to – in the spirit of the BCS theory – formation of electron (or hole) pairs. For the term

k,l
Vk,l ak+,σ a−+k,σ a−l,−σ al,σ	,	(7.1.3)

,σ

for example, electron pairs are formed with total momentum q = 0 and total spin S = 0, the well known Cooper pairs. This, however, is not the only possibility. The situation is simple in the case of a one-dimensional metal where the Fermi

7.1 Superconducting and density wave states

175

surface consists of two points at kF and −kF. Then the following pair formations can occur:

k = −k	l = −l	σ = −σ	singlet superconductor
k = −k	l = −l	σ = σ	triplet superconductor
k = k − 2kF	l = l − 2kF	σ = −σ	spin density wave
k = k − 2kF	l = l − 2kF	σ = σ	charge density wave

The ﬁrst two of these states develop in response to the interaction Vk,l = Vq for which q = 0; this is called the particle–particle or Cooper channel. The resulting ground states are the well known (singlet or triplet) superconducting states of metals and will be discussed shortly. The last two states, with a ﬁnite total momentum for the pairs, develop as a consequence of the divergence of the ﬂuctuations at q = 2kF (Table 7.1); this is the particle–hole channel, usually called the Peierls channel. For these states one ﬁnds a periodic variation of the charge density or spin density, and consequently they are called the charge density wave (CDW) and spin density wave (SDW) ground states. In the charge density wave ground state

ρ = ρ1 cos{2kF · r + φ}

(7.1.4)

where ρ1 is the amplitude of the charge density. In the spin density wave case, the spin density has a periodic spatial variation

S = S1 cos{2kF · r + φ} .

(7.1.5)

Throughout this chapter we are concerned with density waves where the period λDW = π/|Q| is not a simple multiple of the lattice translation vector R, and therefore the density wave is incommensurate with the underlying lattice. Commensurate density waves do not display many of the interesting phenomena discussed here, as the condensate is tied to the lattice and the phase of the ground state wavefunction does not play a role.

A few words about dimensionality effects are in order. The pairing, which leads to the density wave states with electron and hole states differing by 2kF, is the consequence of the Fermi surface being two parallel sheets in one dimension. The result of this nesting is the divergence of the response function χˆ (q, T ) at Q = 2kF at zero temperature as displayed in Fig. 5.14. Parallel sheets of the Fermi surface may occur also in higher dimensions, and this could lead to density wave formation, with a wavevector Q related to the Fermi-surface topology. In this case the charge or spin density has a spatial variation cos{Q · r + φ}.

176	7 Broken symmetry states of metals

We will outline the solution which is obtained for Cooper pairs, i.e. pairs with total momentum zero and with spin zero. The Hamiltonian in this case is

H =
H =	Ekak+,σ ak,σ +	Vk,l ak+,σ a−+k,−σ a−l,−σ al,σ .	(7.1.6)
	k,σ	k,l,σ

It is modiﬁed by seeking a mean ﬁeld solution; this is done by assuming that a−k,−σ ak,σ have non-zero expectation values, but that ﬂuctuations away from average are small. We write formally

a−k,−σ ak,σ = bk + (a−k,−σ ak,σ − bk)

and neglect bilinear terms. Inserting these into the original Hamiltonian results in


H = Ekak+,σ ak,σ + Vk,l(ak+,σ a−+k,−σ bl + bk+a−l,−σ , al,σ − bk+bl)
k,σ	k,l

with

bk = a−k,−σ , ak,σ th

where th denotes the thermal average. We also introduce the notation

	k = − l	Vk,l a−l,−σ , al,σ th ,
a complex gap, as we will see later. With this we have

H = Ekak+,σ ak,σ − kak+,σ a−+k,−σ + ka−k,−σ ak,σ − kbk+
k,σ	k

, (7.1.7)

(7.1.8)

. (7.1.9)

We intend to ﬁnd a solution by diagonalization: we introduce new states – which will be the quasi-particle states – which are related to our original states by a linear transformation, and require that the Hamiltonian is diagonal with respect to these new states. We write

ak,σ	=		γ	k,0	+	v γ		+		(7.1.10a)
		uk				k		k,1
+	=	v γ		k,0	+ uk		γ	+	,	(7.1.10b)
a−k,−σ		k						k,1

where |uk|2 + |vk|2 = 1. We insert the above into the Hamiltonian and require that terms which would not lead to a diagonalization, the terms of γkγl or γl γk in abbreviated notation, are zero. The end results are as follows:

	1		1 −	ζk
1	− \|uk\|2 = \|vk\|2 =				;	(7.1.11)
		2		Ek
the energies of the quasi-particle excitations are
	Ek = \| k\|2 + ζk2 1/2			.		(7.1.12)

7.1 Superconducting and density wave states

177

There is an energy minimum (for ζk = 0) for the excitation of the quasi-particles; this is the well known superconducting gap. We will assume that k is real. This is not a solution yet, but we require selfconsistency by using the above coefﬁcients to write k in Eq. (7.1.8), and using the Fermi distribution function

f (Ek) = exp Ekk−TEF + 1 −1

to describe their population probability at a temperature T . This leads to the socalled gap equation:

k = − l

Vk,l l

−

2 f (

)

= − l

Vk,l l tanh

(7.1.13)

2kBT

We assume that Vk,l is a constant up to a cutoff energy; this yields, by converting the summation

energy Ec and is zero above this over l to an integral,

1 = D(0)V 0 E	E{E} dE ,	(7.1.14)
	c tanh

where D(0) is the density of states in the normal state, i.e. the metallic state above the transition.

The end results are as follows: there is a ﬁnite transition temperature Tc which is related to the gap (which does not depend on the momentum) by

2 (T = 0) = 3.528kBTc ;

(7.1.15)

both parameters are non-analytical functions of V , the interaction potential which leads to the superconducting state. The gap (T ) is temperature dependent and approaches zero in a fashion familiar for order parameters of a second order (mean ﬁeld) phase transitions; this temperature dependence is displayed in Fig. 7.1. The density of the quasi-particle state Ds(E) is

D(0)	=	0		if	<
D(0)	=	( 2 2)1/2			E
Ds(E)			E	if	E >	(7.1.16)
			E	if	E >
			E −\| \|

and diverges at the gap. The source of this divergence is similar to that encountered in one-dimensional semiconductors, a situation similar to perfect nesting, as every k, σ state will have its −k, −σ counterpart. The above density of states is sampled by tunneling experiments.

The procedure outlined above can also be used to discuss the other broken symmetry ground states. The gap equation, the form of the density of states, remains the same; what is different are the energy scales involved and the character of the ground states. The BCS superconducting state arises as the consequence of

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