Single Particle Versus Collective Electronic Excitations 1

Home , Electron hole, Plasmon

Single Particle versus Collective Electronic Excitations

Philip B. Allen

Department of Physics, SUNY, Stony Brook, NY 11794, USA

Abstract

As a rst approximation, a metal can b e mo delled as an electron gas. A non-interacting electron

gas has a continuous sp ectrum of electron-hole pair excitations. At eachwavevector Q with

jQj less than the maximum Fermi surface spanning vector 2k there is a continuous set of

electron-hole pair states, with a maximum energy but no gap the minimum energy is zero.

Once the Coulombinteraction is taken into account, a new collective mo de, the plasmon, is built

from the electron-hole pair sp ectrum. The plasmon captures most of the sp ectral weightinthe

scattering cross-section, yet the particle-hole pairs remain practically unchanged, as can b e seen

from the success of the Landau Fermi-liquid picture. This article explores howeven an isolated

electron-hole pair in non-interacting approximation is a form of charge densitywave excitation,

and how the Coulombinteraction totally alters the charge prop erties, without a ecting many

other prop erties of the electron-hole pairs.

Intro duction

The low-lying excitations of a non-interacting electron gas are simple rearrangements of the

o ccupancy of the single electron plane-wave orbitals. Because real metals, according to Landau

theory,have a lot in common with non-interacting quantum electron gases, this sub ject is well

known to all who study solids. If we neglect band structure, the electron orbitals lab eled by

quantum numb ers k =k; have energy =h k =2m: The ground state has all orbitals

o ccupied which lie inside the Fermi surface, with wavevectors ob eying jk j

ob eying < , where k and are the Fermi wavevector and energy. The simplest excitation,

k F F F

known as an electron-hole e-h pair, consists of moving an electron out of a state k b elow the

~ ~

Fermi surface, putting it into a state k + Q ab ove. This is shown in Fig. 1.

The Coulombinteraction is numerically not small, but Landau theory argues that never-

theless, the consequences of the Coulombinteraction are less drastic than one might supp ose.

However, one drastic e ect certainly happ ens, namely, out of the e-h pair excitation sp ectrum,

the Coulombinteraction creates a new, collective excitation, the plasmon. This typically lies

at quite a high energy. Fig. 2 shows the sp ectrum predicted in random phase approximation

RPA for so dium metal. The plasmon has an energyh! Q which starts at 6eVatQ =0

and disp erses upwards in energy. Also shown in this picture is the continuous sp ectrum of e-h

~ ~

pairs. This sp ectrum is easily understo o d from Fig. 1. Atany xed value of Q with jQj < 2k ,

it is p ossible to nd orbitals k just b elow the Fermi surface such that the corresp onding orbital

This article was published in the b o ok From Quantum Mechanics to Technology, edited byZ.Petru, J.

Przystawa, and K. Rap cewicz Springer, Berlin, 1996 pp. 125-141. 1 k+Q k

~ ~

Figure 1: An electron-hole pair excitation, denoted jk; k + Q>, in a non-interacting electron

gas.

k + Q lies just ab ove the Fermi surface. This means that pair excitations exist with arbitrarily

small excitation energies for Q<2k , whereas for larger Q, a gap exists to the lowest single

pair excitation. For any Q there is also a maximum energy e-h pair which can b e created,

~ ~

namely when the hole state k lies just b elow the Fermi surface in the direction of Q, and the

corresp onding electron state then lies as far as p ossible outside the Fermi sea. This excitation

2 2

has energyh k + Q =2m h k =2m. This formula gives the upp er edge of the e-h pair

continuum shown in Fig. 2. Surprisingly, in spite of the totally new plasmon found in RPA,

nevertheless, the sp ectrum of e-h pairs is not altered from the free electron value in RPA.

This sub ject has b een understo o d for more than 30 years. Nevertheless, the standard treat-

ments in solid state and most many-b o dy texts [1] do not discuss certain asp ects which I nd

paradoxical. This article is intended as p edagogical, aiming to state and then to explain these

paradoxes as clearly as p ossible. Of course, there is no actual paradox in the existing theory,

which provides successful approximate metho ds for calculating many prop erties of metals, but

the most p opular approaches use language in which these interesting paradoxical issues are

never apparent.

The paradoxes are forcefully apparentintwovery interesting exp erimental studies of Ra-

man scattering by electronic excitations, by Contreras, So o d, and Cardona [2]. The op ening

paragraph of the rst of these pap ers reads:

\Metals and heavily dop ed degenerate semiconductors can scatter

light either through single-particle or collective excitations of free car-

riers. The single-particle excitations corresp ond to charge-density uc-

tuations and, as such, they are screened at low frequencies in a self-

consistent manner by the electrons themselves. Thus, in simple, free-

electron-like carrier systems, no low-frequency scattering is observed.

Instead, a p eak at the plasma frequency is seen."

The \single-particle" excitations referred to ab ove are just the e-h pair excitations. How close is 2 10

8 Plasmon

4 Electron-Hole Energy (eV) Continuum

0 0.0 0.5 1.0 1.5 2.0

Wavevector (Q)

Figure 2: The plasmon disp ersion curve and the e-h pair sp ectrum as calculated in RPA using

parameters appropriate to metallic so dium in free electron approximation. The wavevector is

in units of A .

this \corresp ondence" of e-h pairs to charge-density uctuations? In myown case, the closeness

was hard to appreciate at rst. After all, the charge density di erence b etween an excited e-h

2 2

pair state and the ground state elab orated later must b e just = j j j j , whichis

k +Q k

zero for plane-wave states. I shall show later that the corresp ondence is actually p erfect. To

nuclear physicists this corresp ondence is quite familiar. Referring to excitations of the nucleus,

Brown [3] says

\... we wish to talk ab out vibrations, which are density uctuations or

{ in quantum-mechanical language { particle-hole excitations ..."

To summarize, the apparent paradoxes are these:

1. How can an e-h pair excitation b e equivalenttoa charge-density uctuation?

2. If screening by the Coulombinteraction eliminates nearly all the low-frequency scat-

tering of lightinfavor of collective plasma oscillations, how can it b e that the low-lying

e-h pair sp ectrum remains unaltered in other exp eriments sp eci c heat, susceptibility,

conductivity and in Landau theory?

History and Standard Interpretation

Electron density oscillations were susp ected in electrical discharges in gases, and this sub ject

was elucidated, b oth exp erimentally and theoretically,byTonks and Langmuir [4]. Apparently

the corresp onding e ect for electrons in metals was seen exp erimentally b efore b eing understo o d 3

theoretically. Exp eriments by Ruthemann [5] and Lang [6] stimulated Kronig, Korringa, and

Kramers to recognize the connection to the classical plasma oscillations seen byTonks and

Langmuir. Slater [7] has summarized the history. Bohm and Pines then wrote a series of pap ers

which recognized the imp ortance of the collective plasma degrees of freedom for understanding

the interacting electron gas problem. In a review article, Pines [8] coined the term \plasmon",

and ever since, the solid state texts have recognized the plasmon as one of the elementary

excitations, or quasiparticles, of solid state physics.

The standard derivation of the frequency of plasma oscillations, app earing in all the texts,

is identical to an argument from Tonks and Langmuir [4]. Consider a slab of thickness D and

in nite transverse size, emb edded in an in nite sample of metal with electron density n. Imagine

that the electrons in this slab are all displaced by the same small amount u in the direction

normal to the slab call this the \upward" direction. Then a thin layer of charge of thickness

u and surface charge density = neu accumulates at the upp er surface of the slab and +neu

at the lower surface. Therefore there is a capacitor-typ e E - eld of magnitude 4 =4 neu in

the upward direction inside the slab. This eld exerts a force eE in the downward direction

on every electron in the slab. This is a restoring force Ku prop ortional to the displacementof

each electron. This causes each electron in the slab to oscillate at frequency K=m, namely,

2 2

! =4ne =m 1

The plasmons in metals are quantized versions of these classical oscillators, with energyh! .

The other standard textb o ok result is that the frequency of plasma oscillations is b est

understo o d or calculated by lo oking for the zero of the real part of the complex dielectric

function Q; ! +i Q; ! : Even b etter, the Fourier transform S Q; ! of the density-density

1 2

correlation function <r;tr ; 0 >, gives the sp ectrum of density oscillations in a material.

Van Hove showed that in Born approximation, the inelastic scattering cross section for particles

like x-rays and electrons which couple to electron densityisgiven by S Q; ! , which is also

called the inelastic structure factor. Finally, the same function, S Q; ! , is directly prop ortional

2 2

to Im1=Q; ! = = + . Thus the zeros of Q; ! corresp ond to p eaks in S Q; !

2 1

1 2

and to p eaks in the inelastic scattering cross section, provided the damping, given by Q; !

is small. When Q; ! is calculated in RPA, the zeros of give the plasmon disp ersion shown

in Fig. 2. The b oundaries of the e-h continuum coincide with the region of Q; ! where

di ers from zero. In that region, plasmons are very heavily damp ed, and merge smo othly into

the e-h pair sp ectrum.

The b est test of the ideas of collective plasmon excitations is exp eriment. In the past,

inelastic electron scattering away from Q = 0 and also inelastic x-ray scattering with energy

resolution b etter than 1eV have b oth b een dicult. Recently, synchrotron x-ray sources have

made the latter exp erimentmuch easier, and we can exp ect many new results on collective

electron b ehavior [9]. A go o d example is the measured disp ersion curve of plasmons in Na, as

determined by inelastic electron scattering byvom Felde et al. [10] and shown in Fig. 3 These

results demonstrate that at least in certain simple metals, sharp plasmons exist, and that the

RPA is at least qualitatively very successful in explaining the sp ectrum.

Charge density of an Electron-Hole Pair 4 12

10 Na 8

4 Plasmon Energy (eV)

0 0.0 0.2 0.4 0.6 0.8 1.0

Wavevector (Q)

Figure 3: Plasmon disp ersion for so dium measured in Ref. [10 ]. The vertical bars are the

measured widths of the plasmon resonances. Q is measured in units A . The solid curve

marks the edge of the e-h pair sp ectrum as given in Fig. 2. The datum at the largest Q had

such a broad lineshap e that no sensible width could b e assigned.

Contreras et al [2] state that an e-h pair is a charge-densitywave excitation. The simple

truth of this statement do es not emerge in any of the textb o ok treatments I know. Let us

establish some notation. In a non-interacting gas, the ground state can b e written as

occ

j0 >= c jvac >: 2

Here I am using the notation k as a shorthand for all the quantum numb ers of the orbital, that

is, the wavevector k as well as the spin quantum numb er and in a real material the band

index. An e-h pair excitation is denoted by

jk; k + Q>= c c j0 > 3

k +Q

This state vanishes unless the orbital k is o ccupied in the ground state, and the orbital k + Q

is empty.

The charge densityis e times the electron density. I will leave out the factor e and refer

to it as charge densityanyway. To nd the charge density of the e-h pair state, one would

calculate the exp ectation value of the charge density op erator,

^ ^

^~r; t= ~r; t ~r; t 4

where the eld op erator can b e represented in terms of the one electron states as

~r; t= ~r c t 5

k k

k 5

In a one-electron approximation, or else in the \interaction representation", c t has the form

expi t=h c . In the ground state, the particle densityis

k k

0 0

tj0 > 6 ~r c tc ~r; t=< 0j^~r; tj0 >=< 0j ~r

p 0 p

p p

p;p

Only the diagonal terms p = p which are o ccupied give a non-zero contribution to this sum,

and we get the familiar result, indep endent of time,

occ

~r = j ~r j 7

0 p

Rep eating the calculation for the e-h pair state jk; k + Q>,we get the same answer except that

the list of o ccupied states is di erent. If we lo ok at the change in the charge densitybetween

the ground state and the e-h pair state, we get

2 2

~r; t= j ~r j j ~r j : 8

k +Q k

For free electrons, the orbitals ~r are plane waves expik ~r , and vanishes. For the Blo ch

states of a real metal, the orbitals are plane waves mo dulated by p erio dic functions u ~r . For

small Q, the p erio dic function u is almost the same as u , so the change in charge density

k +Q k

vanishes as Q go es to zero. There is no evidence of a charge-densitywave.

Nevertheless, Contreras et al. [2] are right. There is a charge-densitywave hidden in the

pair state. The way to see it is to consider what would happ en if a small amount of e-h pair

excitation were mixed with the ground state. In other words, consider the state

j > j0 > + jk; k + Q>=1+ c c j0 > 9

k;k+Q k

k +q

where is a small complex numb er, and it is assumed that the pair excitation do es not vanish,

i. e. that k is inside the Fermi distribution and k + Q outside. The calculation now yields to

rst order in the small parameter

i t=h

k +Q k

= ~r ~r e +c:c: 10

k +Q

where c.c. stands for \complex conjugate." For small Q, the electron-hole pair energy

k +Q

~ ~

=h is Q ~v where ~v is the group velo city, @ =@ k .At the same level in smallness of Q,we

k k k k

can write the pro duct of wavefunctions as

2 iQ~r

~r ~r = j ~r j e 11

k k

k +Q

This is just the zeroth order result of a \k p~ " p erturbation theory. Substituting these changes

into Eq. 10, we get the result

~r; t= 2j jj ~r j cos[Q ~r ~v t+ ] 12

k k

where is the phase of .Nowwe see that the e-h pair excitation, relative to the ground state,

is a charge-densitywave! The propagation direction is of course given by Q, and the phase 6

^ ~

velo city is the comp onent Q ~v of the electron group velo city in the direction of Q. The group

velo city,however, is ~v . The simple cosine oscillation of electron densityisvery reminiscentofa

phonon. However, unlike phonons whichhave 3 branches for each atom in the cell, the number

of e-h pairs at a xed Q is a macroscopic numb er. Sp eci cally, for free electrons with Q> 2k ,

the numb er of pairs with wavevector Q is the same as the numb er of o ccupied electron states,

whereas for Q<2k , the numb er is reduced by a factor 3=2Q=2k [1 Q=2k =3].

F F F

It is evident from the electrostatic argumentofTonks and Langmuir that once the Coulomb

interaction is taken into account, there will b e a large additional restoring force on the charge

oscillation, and the result will b e that the charge density will oscillate at the plasma frequency

rather than the non-interacting frequency Q~v . This will b e shown explicitly in the next section.

One should pause to ask what prop erty of the medium p ermits wave propagation Eqn. 12

of electron charge b efore the Coulombinteraction is turned on. If there is no interaction, a

classical gas do es not supp ort propagating waves. The answer is the Fermi degeneracy, which

forces an energy cost if the lo cal density is altered.

In uence of CoulombInteraction on Charge of the e-h Pair

It is convenient to formulate this as a linear resp onse problem. This waywe exp ect the

Coulombinteraction to enter nicely as a dielectric screening. Therefore, wewant a p erturbing

eld which will create the e-h pair and which can b e represented as a term in the Hamiltonian.

The following op erator do es the trick:

H =h c c t: 13

k +Q

This inserts an electron-hole pair at time zero. The dimensionless strength of the insertion

eld, , will b e taken to b e a small numb er. Linear resp onse theory provides formal answers

for the alterations of system prop erities which can b e measured at a later time, to rst order

in . In this section, we explore the resulting charge-density resp onse, that is, the exp ectation

value of the op erator ^ of Eqn. 4. Later we lo ok at the current resp onse.

Standard techniques of linear resp onse theory [11 ] tell us that the resp onse is

0 0 0 0y 0

~r; t= dt h0j^tH t H t ^tj0i: 14

This is a standard Kub o-typ e formula, except that since the Hamiltonian H is not Hermitean,

so the commutator is replaced by a slighly more complicated form. Inserting Eqn. 13 for H ,

the charge density resp onse is

~r; t = 2Im f Dtg t 15

D t=< 0jT ^~r; tc c j0 > 16

k +q

where tis1ift>0 and zero otherwise. D is a typ e of two-particle Green's function, and

can b e evaluated p erturbatively in p owers of the Coulombinteraction by standard Feynman

metho ds using time-ordered Green's functions. The time-ordering op erator T has b een inserted 7

into Eqn. 16. Since ultimately we only need to know D for p ositive times, this do esn't change

anything but facilitates the p erturbation theory.

First, evaluate the charge resp onse function without any Coulombinteractions, D . By

standard metho ds one gets

i! t iQ~r

e D ! 17 D t= e

0 0

f 1 f f 1 f

k +Q k k k +Q

18 D ! =i

~ ~

! + i Q ~v ! i Q ~v

k k

For p ositive t, the contour is closed in the lower complex ! plane, and only the rst term in

D ! contributes. The answer is

~ ~

~r; t=2j j sinQ ~r Q ~v t + j ~r j f 1 f 19

k k k k +Q

This is just the same as the previous result for the charge densitychange which o ccurs when

an e-h pair is present with amplitude at all times, Eqn. 12, except for a phase change of

=2. This is b ecause our earlier pair oscillator started at time zero with given amplitude

whereas the present oscillator at time zero was given \velo city" by the impulsive insertion. The

presentversion also contains explicitly the Fermi factors f and 1 f which are zero or

k k +Q

one dep ending on whether the orbital is within or without the Fermi distribution.

The Coulombinteraction has the form

0 0

V = ^ v Q ^ 20

Q Q

where the op erator ^Q is the Fourier transform of the charge density op erator Eqn. 4. For

free electrons this is just

^ = c c 21

Q k

k +Q

Toevaluate the resp onse function D exactly in the presence of Coulombinteractions is of course

imp ossible. Diagrammatic p erturbation theory allows a classi cation of the correction terms.

The RPA is a standard approximation whichkeeps an in nite set of leading diagrams which

contain the leading small Q divergence, or the long-range part of the Coulombinteraction.

The result at RPA level is

i! t

D ! e d!

iQ~r

22 D t= e

RP A

1 v Q Q;!

where is the usual non-interacting susceptibility \bubble" diagram, and the subscript T

indicates that it is the \time-ordered" rather than the usual \retarded" or causal resp onse

2 2

function that is needed. The factor v Qis4e =Q , and the denominator 1 v Q is

0R 0

the dielectric function Q; ! . These formulas are written for the uniform electron gas with no

background of actual atoms. Generalizing to real electrons interacting with a crystalline array

of atoms is not hard, but complicates the notation b ecause of the matrix nature of .At this 8

stage it is also easy to include the dominant impurity e ects, simply by including them in .If

the mean free path ` = v is short enough that Q` 1, the standard Drude result applies,

Q; ! =1 23

! ! + i=

It is imp ortant to convert this to the retarded form, which can b e done using the general

relations

Re ! =Re ! 24

R T

Im ! 25 Im ! = tanh

T R

2k T

Atlow T , the hyp erb olic tangentis1 and to convert a retarded function to a time-ordered

function, one just changes the sign of the imaginary part when !<0: To a go o d approximation

the Drude resp onse function can b e written

1 1 ! + i= ! + i=

2! ! ! + i=2 ! + ! + i=2

R P P P

Because ! 1= , the rst term of Eqn. 26 is only imp ortant when !>0 and the second

term is only imp ortant when !<0. Therefore this is converted into a time-ordered resp onse

function by the simple approximate exp edient of taking the complex conjugate of the second

term,

1 ! i= ! ! + i=

27 =

2! ! ! + i=2 ! + ! i=2

T p p p

Putting this into Eqn. 22 and closing the contour in the lower half of the complex ! plane,

the result is

iQ~r ! t t=2

D t = e e f f =2

RP A k k +Q

Qv or 1=

~ ~

iQ~r Q~v t

28 + e f 1 f O

k k +Q

For simplicity, the Blo chwavefunction j j has b een dropp ed. Only free electron results are

given for the remainder of this section. The second term of Eqn. 28 is negligible compared

to the rst at small Q. Therefore, using Eqn. 15, the result for the charge disturbance after

accounting for the long range Coulomb eld is

t=2

~r; t=j j sinQ ~r ! te f f : 29

p k k +Q

This is almost what one would have naively guessed. The amplitude is reduced by 2, and

the frequency is now the plasma frequency, ! , as exp ected, with only a negligible comp onent

left oscillating at the original frequency Q ~v . Comp ensating for what lo oks like an arbitrary

reduction by half of the amplitude is a new e ect, coming from the di erence b etween the

factors f 1 f and f f . In the interacting system, it is p ossible to insert an

k k +Q k k +Q 9 k k+Q

k' k'+Q

~ ~

Figure 4: A typical vacuum uctuation, consisting of two e-h pairs with momentum Q and Q.

electron-hole pair even when the state k is ab ove the Fermi surface f = 0 and the state k + Q

is b elowf = 1. The factor f 1 f of the non-interacting resp onse is zero, but the

k +Q k k +Q

factor f f of the interacting system is -1 the density disturbance has a phase shift of

k k +Q

. Where do es this new term come from? The Coulombinteraction Eqn. 20 continuously

creates \vaccuum uctuations" which are pairs of electron-hole pairs of equal and opp osite

0 0

momentum Q and Q , as shown in Fig. 4. Thus there is the p ossibility that an electron with

k>k and a hole with k + Q

F F

0 0

uctuation which also created a pair k k . Then the \insertion" op erator

F F

0 0

c c removes the already present electron-hole pair, leaving its mate k ,k + Q. This pro cess

k +Q

has equal amplitude as the simpler insertion pro cess, but each is reduced by 2 compared to the

non-interacting case.

Current Density of an Electron-Hole Pair in a Metal

It is also interesting to ask what is the current density asso ciated with an electron-hole pair

excitation. The current density op erator, analogous to Eqn. 4, is

^ ^ ~ ^ ~ ^

~r 30 ~r r ~r r ~r j ~r =

2mi

The ground state carries no current. To rst order in the amplitude , the current densityin

the state j > Eqn. 9 is

k;k+Q

i t=h

k +Q k

~ ~

~r ~r e +c:c: 31 ~r r ~r r j ~r =

k +Q k +Q

k k

2mi

Because the wavevector Q is small, the factor inside the square brackets of Eqn. 31 can b e

written as

iQ~r

[ ]= e j ~r 32

2mi 10

h i

~ ~

j ~r = r r 33

k k k

k k

2mi

where j ~r is the current density asso ciated with the single particle Blo ch state ~r . Since the

k k

corresp onding density j ~r j is time-indep endent, the vector eld j ~r must b e divergenceless.

k k

The spatial average of this current densityis~v = where ~v is the group velo city of the state

k k

and is the volume of the crystal. Thus we get

~ ~

j ~r = 2j jj ~r cos [Q ~r ~v t+ ]f 1 f 34

k k k k +Q

The electron-hole pair conserves charge while it propagates, that is r j + @ =@ t is zero.

However, Eqs. 12 and 34 do not rigorously ob ey this law, since the current j is not equal

everywhere to ~v j j , only on average. The more exact forms 10 and 31 are rigorously

k k

~ ~

charge-conserving. The derivatives rj and @ =@ t are rst order in jQj, whereas j and are

zeroth order. Since Eqs. 12 and 34 throwaway terms rst order in jQj, they also lose some

rst order parts of their derivatives, even though they are correct to the order that they are

written.

It is interesting that the current density is not in general longitudinal. It is p olarized in a

direction whose spatial average is the direction of the group velo city ~v , which is normally not

parallel to Q. The transverse part of the current contains new information which cannot b e

deduced from the density alone.

In uence of CoulombInteraction on Current of the e-h Pair

The same linear resp onse pro ceedure used earlier for the charge can b e generalized for the

current. Analogous to Eqns. 15, 16 we get

n o

j ~r; t = 2Im J t t 35

J t=< 0jT j ~r; tc c j0 > 36

k +q

When this is evaluated without Coulombinteractions, the result is

~ ~

j ~r =2j jj ~r sin[Q ~r ~v t+]f 1 f ; 37

k k k k +Q

analogous to Eqn. 19.

When Coulombinteractions are accounted for by p erturbation theory,we discover that

transverse and longitudinal parts of the current are treated very di erently. Sums over k o ccur

in which there are factors of the typ e

0 0 0 0

~v + ~v F ; : 38

k ? k k +Q

k k

The velo city ~v has b een split into a transverse part p erp endicular to Q and a longitudinal

part. The transverse contribution vanishes, b ecause it is always p ossible, for free electrons

or for band electrons as long as Q has high enough symmetry, to nd two states k which 11

0 0 0 0

, and v and opp osite values of v . Therefore, all Coulomb have identical values of

k +Q k ? k

k k

corrections to the transverse current cancel, and the transverse current continues to oscillate

at its unrenormalized frequency.

The longitudinal part of the current gets screened in RPA the same way the charge do es.

A diagrammatic analysis yields the result

d! !D !

iQ~r i! t

~ ~

Q J t=e e 39

1 v Q Q; !

The nal result for the currentis

2j j

~ ~

j ~r; t = ~v sinQ ~r Q ~v t + f 1 f

k ? k k k +Q

j j!

t=2

^ ~

Q sinQ ~r ! t + e f f 40 +

P k k +Q

This answer was calculated using the Drude resp onse in the same way that Eqn. 29 was

calculated.

Wehavenow learned something interesting ab out e-h pair excitations. Even though the

Coulombinteraction totally alters the charge oscillations of every e-h pair, other prop erties can

remain completely immune to alteration, such as the transverse part of the current oscillation.

This is part of the explanation of the second \apparent" paradox. A more complete explanation

is given in the next section.

What is the Wavefunction of a Plasmon?

Another way of understanding the broad immunity of e-h pairs to Coulombic alteration is

to recognize that it is easy to combine e-h pairs so that the charge is hidden. A trivial way

is to make a random combination of pairs. Rather than a single coherent cosine of charge

oscillation as in Eqn. 12, one now exp ects an incoherent sum of many cosines with di erent

phases, adding up to a disturbance whose charge is smaller than a coherent sum by a factor

1= N where N is the numb er of e-h pairs combined. Then the Coulombinteraction will still

alter the charge oscillation, but this is only a minor asp ect of the complete excitation.

Away of picturing this more elegantly is to consider the space of all single e-h pair states.

The Hamiltonian can couple through the Coulombinteraction only states of the same Q.

Therefore we restrict attention to states of the form

j >= A jk; k + Q> 41

where the sum go es over all k 's such that k is b elow the Fermi surface and k + Q is ab ove. In

this subspace the Hamiltonian matrix is

0 1

+ v Q v Q

k +Q k

1 1

B C

^ B C

v Q + v Q

k +Q k

H = 42

2 2

1eh

@ A

. .

. . 12

The kinetic energy of the pair k; k + Q app ears on the diagonal, and the Coulombinteraction

2 2 0

v Q= 4e =Q couples each pair to every other pair. The exchange term v k k also

couples pairs to each other, provided their spins are the same, but this is a smaller and a

complicating factor which is omitted in rst approximation.

We lo ok for eigenstates of this truncated Hamiltonian. This pro ceedure is called the Tamm-

Danco approximation bynuclear physicists [1], and is an example of what chemists call a

\con guration interaction" calculation. The eigenfunctions ob ey

H jA>= jA> 43

1eh

which is equivalentto

A = v Q A 44

k +Q k k k

Solving for A =const v Q= , we compute A and nd the self-consistency

k k +Q k k

condition

f 1 f

k k +Q

1=v Q 45

k +Q k

This is equivalent to the secular equation detH 1=0:

1eh

The nature of this equation can b e appreciated by considering the form of the right hand

side for a small system with a discrete sp ectrum of e-h pairs of energy . Between every

k +Q k

adjacenttwo pair energies, a ro ot of Eqn. 45 is found, but the largest ro ot =h! is split

o to higher energy as illustrated in Fig. 5. This ro ot is the collective electronic excitation in

Tamm-Danco approximation. Its wavefunction is

CvQ

jk; k + Q> 46 j >=

TD k +Q k

Becauseh! lies ab ove the energy of the pair sp ectrum, all the co ecients in Eqn. 46 are

p ositive. This coherent sum has a large oscillating charge at t 0, and lo oks quite a lot likea

plasmon. The other eigenstates of Eqn. 42 are orthogonal to 46, which means that at t 0 the

charge must largely cancel out. These other states account for the p ersistence of non-interacting

prop erties in the e-h pair sp ectrum of the electron gas with Coulombinteractions.

Unfortunately, the Tamm-Danco approximation, as is well-known in nuclear physics, do es

not yield an accurate answer for the collective mo de sp ectrum. The RPA answer, whichis

apparently surprisingly accurate, is equivalent to a mo di ed secular equation

f 1 f f 1 f

k k +Q k +Q k

47 1=v Q

k +Q k

which is found by setting the real part of the RPA dielectric function to zero. Eqn. 45 is

tantalizingly close to Eqn. 47. However, the additional term which o ccurs in Eqn. 47, having

the factor f 1 f in the numerator, is quite foreign to the Tamm-Danco approximation,

k +Q k

b ecause it seems to refer to states jk; k + Q> where the hole state k is ab ove the Fermi surface

and the electron state k + Q is b elow. 13 5.0

2.5

0.0

-2.5 -30369

Energy

Figure 5: Finding the ro ots of the secular equation. The curve is the right hand side of Eqn.

45. The vertical lines mark the energies of e-h pair states, with a ro ot pinned b etween each

two neighb oring e-h states. One ro ot which b ecomes the plasmon splits o to higher energy.

The pair creation op erator c c op erating on the non-interacting ground state cannot

k +Q

create anything if k is ab ove the Fermi surface and k + Q b elow, but if it acts on the interacting

ground state with vaccuum uctuations included, then the pair creation op erator can destroy

a pair of net momentum Q by destroying a virtual electron in state k and a virtual hole in

0 0

state k + Q. This e ectively releases a pair state in some other state k ;k + Q as shown in

Fig. 4, which had previously b een part of the same vaccuum uctuation as the destroyed pair.

In order to pro duce the mo di ed secular Eqn. 47 in place of Eqn. 45 from our wavefunc-

tion argument, we need an enlarged subspace that includes e-h pairs where the hole is ab ove

and the electron b elow the Fermi level. The kinetic energy part of the Hamiltonian will have

the same form on the diagonal in b oth the previous and the new parts of the subspace,

k +Q k

whereas the p otential energy term must have the form

1 0

v Q v Q v Q v Q

C B

C B v Q v Q v Q v Q

C B

. . . .

C B

. . . .

C B

. . . .

C B ^

48 V =

C B

v Q v Q v Q v Q

C B

v Q v Q v Q v Q

A @

. . . .

where the second set of rows and columns refers to the new states with k ab ove and k + Q

b elow the Fermi level. One way to b egin building this is to rede ne the pair states given in 14

Eqn. 3 by a rst-order p erturbative improvement of the ground state or vaccuum,

j0 >!j0 > + jm> 49

E E

0 m

0 0

~ ~

where jm>is short for any state consisting of two e-h pairs of opp osite momenta Q ,-Q

which can b e created in the non-interacting vaccuum by the Coulombinteraction Eqn. 20.

The original subspace Hamiltonian matrix Eqn. 42 is preserved to zeroth order in v Q, and

to rst order, there are new matrix elements of the kinetic energy op erator which couple an

e-h pair k; k + Q with k b elow the Fermi level to a state with k ab ove. Unfortunately, the

coupling matrix element is not just the term v Q app earing in Eqn. 48, but instead

k +Q k

50 v Q

0 0

k +Q k k +Q k

It will require going to in nite order in p erturbation theory to fully correct the Tamm-Danco

approximation and recover the RPA answer by this route. The Tamm-Danco approximation

is equivalent to a diagrammatic p erturbation theory for the dielectric function whichkeeps only

one e-h pair propagating forward in time, and neglects the backward-in-time propagation which

enters the usual Dyson series when the vaccuum uctuations are included prop erly in the same

order of p erturbation theory. The full RPA treatment can have arbitrarily many additional

~ ~

vaccuum uctuations with wavevectors Q; Q, but our approximate improvement allows at

most one vaccuum uctuation term. It might b e nice to nd explicitly the formula for the

plasmon wavefunction which corresp onds prop erly to the RPA formula, but I have not done

this. The sub ject is treated in texts [1], [3] on the nuclear many-b o dy problem.

Conclusion

The apparent paradoxes now seem largely answered. The answers did not involveany

new physics, but instead required thinking ab out the problem in a way familiar to nuclear

physicists but less familiar to solid state physicists. An e-h pair, even b efore adding the Coulomb

interaction, is seen to carry a charge densitywave, once the interference b etween the pair and

the ground state is considered in the wavefunction. The Coulombinteraction totally alters the

charge-carrying part of this state, but do es not a ect the transverse part of the current. In the

e ort to nd approximate eigenstates of the interacting problem, the Tamm-Danco pro ceedure,

even though ultimately inaccurate, o ers a nice way of seeing how a single plasmon-likemode

can split o from the e-h continuum, leaving the rest of the continuum largely unaltered.

Acknowledgements

I thank G. E. Brown, M. Cardona, V. J. Emery, A. D. Jackson, and esp ecially J. K. Jain for

helpful discussions. This work was supp orted in part by NSF grant no. DMR-9417755. I thank

Prof. R. Car for hospitality at IRRMA in Lausanne, where the rst half of this work was done. 15

References

[1] A particularly complete treatment of the RPA and plasmons is given byA.L.Fetter and J.

D. Walecka, Quantum Theory of Many-Particle Systems McGraw-Hill, New York, 1971.

[2] G. Contreras, A. K. So o d, and M. Cardona, Phys. Rev. B32, 924-9 1985; Phys. Rev.

B32, 930-3 1985.

[3] G. E. Brown, Uni ed Theory of Nuclear Mo dels and Forces, 3rd Ed. North-Holland,

Amsterdam, 1971 p. 42.

[4] L. Tonks and I. Langmuir, Phys. Rev. 33, 195-210 1929.

[5] G. Ruthemann, Ann. Physik 2, 113-34 1948.

[6] W. Lang, Optik 3, 233-246 1948.

[7] J. C. Slater, Insulators, Semiconductors, and Metals McGraw Hill, New York, 1967

pp.119-125.

[8] D. Pines, Rev. Mo d. Phys. 28, 184 1956.

[9] E. D. Isaacs and P. Platzman, Physics Today,February 1996, pp.40-45.

[10] A. vom Felde, J. Sprosser-Prou, and J. Fink, Phys. Rev. B40, 10181 1989.

[11] P. Nozi eres, Theory of Interacting Fermi Systems W. A. Benjamin, New York, 1964. 16