iii the sommerfeld free electron theory of metals 2000 solid state physics

7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics

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The Sommerfel d free-electro n theor y o f metals

1. Q u an tum th eory of the free-electrongas 7 7

2.

Ferm i-D irac d is tr ibu tio function and chem ica p o ten tia 8 2

3.

E lectro ni specific he at i n m et als and ther m od yn am ifunctions 8 6

4.Therm ioni emission from m eta ls 8 8

A pp en di A . O utline of st a tis tic a ph ysics and th er m o d yn am ire la tions 8 9

A l .

M icrocanonicaensembl and therm od yn am iq uantit ies 8 9

A2.Ca non ica ensem bl and the rm od yn am iq u an tit ies 9 1

A3.G ra nd cano nica ens em bl and th er m o dy n am iq uan tit ies 9 3

Ap pen di B . Ferm i-Dirac and Bo se -Ein steis ta tis tic for ind ep en de n pa rtic les . 9 5

Ap pe nd i C . Modified Fe rm i-D irac sta tis tic i n a m od e of cor relatio effects . . . 9 8

Fu rth e read ing 10 0

In th is ch apte w e dis cu s the free-elec tronth eo ry o f m eta ls origin ally develope b y

Som merfel and oth er s T he free-electronm od el w ith its pa ra bo li en erg y-w avevec to

disp ersio cu rve prov ides a re as ona blde sc rip tio for con du ctio ele ctr on i n sim ple

m et als i t als o m ay giv e use fu gu ide lines fo r o th er m eta ls w it h m ore co m pU cated

co nd uc tio b an d s Be caus of the sim plic ity of t he m od e and its de ns ity-o f-st ate sw e

can work out exphcitly thermodynam ipro p ert ie s and i n p art ic u la the spec ific heat

and the th er m io ni em iss ion I n the Appe ndice w e sim om ariz for a gen er a q u an tum

sy stem the th erm od yn am ifu nc tions o f m ore frequentuse i n sta tis tic a ph ys ics T h e

Ferm i-D irac and Bo se-E in st e i d is tr ib ution func tions fo r ind epen den ferm ions an d

bo so n are disc ussedfinally we ob ta in the mod ified Fe rm i-D ir ac dis tr ib u tio fun ctio n

in a m od e case of c or re la tio am ong elect ro n i n loc alized st a te s

1 Q u an tu m theor y o f th e free -elec tro n ga s

An ele ct ron i n a cr ys ta feels th e p o te nti a en ergy due t o al l the nu clei and al l th e

o th er elec tro ns T he de te rm in a tio o f t he cr ys tal lin p o te n tia i n specific m ateria ls i s

a ra th er de m an din pr ob lem (see C h ap te r I V and V ) . How ever i n seve ra m et als i t

tu rns out reas on ab l t o ass ume th at th e co nductio elec tro ns feel a p o te n ti a which

is co ns tan th ro u g h ou th e sa m ple th is m od el sug ge ste b y So mmerfel i n 1928, i s

77


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78

II I TH E SOMMBRFEL D FREE-ELECTRO N THEOR Y O F METAL S

Ef

^vac

Ep+

^c-^

vacuum

vacuum

' ' ' ' ' . . ' . ' . ' . ' ' . ' . ' .

meta

S^^^m^^t

W

^ F ^ c

E ( k ) = E c + ^

2m

-k.

(a)

(b)

kp k z

F ig. 1 (a ) Th e Sommerfel model for a meta l Th e energy Ec denote the bo ttom o f the

conductio ban d ^vac denote the energy of an elec tron at rest in the vacuum the electron

affinity i s x = ^vac

Ec Th e Fermi energy i s denoted b y EF, and the work function W

equals x ~ ^F - (b ) Free-electro energy dispersio curve along a direc tion say /c^, in the

reciproca space At T = 0 all the states with k < kp are occupie by two electron of e ith er

spin

still o f value fo r a n orienta tive und er st an d in o f a nu m ber o f p ro pe rtie o f sim ple

m eta ls

I n the Sommerfel mo del t he Sc hrod ing e equ at ion for co nduc tio elec tro n inside

the me tal takes the sim ple form

p + F

2^^^'^

^{T) = E^{V),

(1)

whe re Ec de no tes a n ap p ro pr ia t co n st an specific o f t he me tal im der inv es tiga tio

(see Fig. 1) . Th e nu m be o f free elec tron s o f the m et al i s as sm ne t o co rres po n

to th e nu m ber o f ele ctro ns i n th e m ost ex te rn a she ll o f t he co m po sin ato m s Fo r

in sta nc i n alka li m eta ls w e ex pe c on e free elec tron per at o m, since an alkali a tom

has ju st on e ele ctr on i n the m ost ex te rn a sh ell. I n alum in um (atom ic co nfigu ratio

l5^,25^2p^,3s^3p^w e ex pec th ree free elec tro n per ato m.

At first s ight i t co uld app ear surp ris ing th at th e co nd uc tion elec tro ns i n actu al

met a ls (o r i n any ty pe o f solid) feel a n ap pr o xim at el co ns tan po te n tia l a s st ro ng

sp atia variations of t he cr yst al lin po te n ti a occur near the nu clei Ho wever the co n-

du ction ele ctrons are hard ly sens itiv to the region close to the nuc lei becaus of the

orthogonalizat ioeffects due to the co re e le ctron s ta tes th us the effective p o te n tia l

or pseudopotential(se e Sec tion V-4 ) fo r co nductio e lectro ns m ay ind eed beco me

a sm oo th l vary ing qu an tity eve ntua ll ap pr ox im ate w ith a c on sta ntthese are the

un de rly in rea so ns fo r th e succes o f t he free-ele ctro m od el (o r o f th e nearly-free

ele ctr on im ple m en tat ion si n a nu m be of a ctu a m eta ls

T he eigen functio n o f t he Schro dinge equati on (1 ) , no rm aliz e to one in the volu me


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SOLID STAT E PHYSIC S 7 9

V o f the m eta l are the plane waves

V;{k,r) = - ^ e * ' ;(2 )

the p la ne waves are ju st a p art ic u la case of Bloch fu nctio ns w hen the period ic mod-

u la ting p a rt i s co ns ta ntT h e eigen value of E q. (1 ) a re

For sim plic ity th ro u g h o u th is chapte r w e set the ze ro of ener gy at the bo t tom Ec of

the con duct io band and th us take Ec = 0 ; wheneve necessarywe can re in s ta t the

ac tu a va lue of Ec by an app ro pria t shift i n the en ergy scale [I n some sit ua tio n sfor

in sta nc i n the discus sio of p hoto e lectro em iss ion i t can be mo re co nv en ien t o set

the zero of energy at the vacuum level, i.e. Evac = 0 ; in oth er pro ble m s for in sta nc i n

the discu ss io of m any-b od effects th e ze ro of en er gy is ge ne ra ll taken at t he Fer mi

level i.e. Ep = 0 ; of c our se any choice i s lawful, pro vided on e keeps i n m ind which

choice has b eendone]

Co ns ide no w a n elec tron gas w ith N fre e elec tro ns i n a vo lume V , and elec tron

de ns it n = N/V. T o specif the electron de ns it of a me ta it is cu st o m ar to co ns ide

the dim ens ionlesp ar am et e r ^ co nn ec te t o n b y the re lat ion

w he re as = 0.529 A i s the Bohr rad iu s Vsasrepr es en t the ra d ius of t he sph ere that

conta in i n ave ra ge one electron

We eva luate r ^ fo r alkali m eta ls (fo r insta nc e) Li , Na , K , R b have bcc st ru ct u re

w ith cube edge a equa t o 3.4 9 A , 4.23 A , 5.2 3 A an d 5.5 9 A , respectively I n th e

vo lume a^ we have two at o ms and tw o conductio e lectrons From E q. (4 ) w e ob ta in

(4/3 )7rrf a| = a^/2 an d th us

as 2 yirj

the va lues for r ^ ar e 3 .25, 3.94, 4.87 and 5.2 0 for Li , Na , K and R b, resp ec tive ly A

sim ila re as onin can be do ne for o th er crys ta ls For insta nce the cry st a s tr u c tu r of

Al i s fee, w ith la tt ice co nsta n a = 4 .0 5 A. I n the vo lume a^ we ha ve four a to ms and

twelve co ndu ct io elec tro ns from Eq. (4 ) we ha ve {4:/3 )Trr^a %= a^ /12 and

Vg

= 2.07.

M eta llic de ns itie of c on du ctio ele ctro n occin mo stly i n the ra nge 2 ksT th e

dis tribu tio function app roach e un ity; i f f j - / i > fc^T, f{E) fall s exp one ntiall t o

ze ro w ith a B oltz m ann ta il. A t T = 0 the F erm i-D irac d is tr ib u tio fun ction be co m es

the st ep fu nctionQ{IJLQ

E), wh ere /i o de no tes the Fermi energy a t T = 0 ; at zero

t emp er a t u r eal l t he s ta tes w ith ener gy lower t h an/X Qa re oc cu pie a nd al l t he s ta tes

w ith en er gy hig her th an /i o are em pty A t finite te m p e ra tu r T , f{E) de viat e s fro m

the st ep function on ly i n the th erm a energy range of ord er fcjgT aro und /x(T) .

We conside now th e prop ert ies o f the function {df/dE). A t T = 0 w e have

{df/dE)= S{E / io) . A t f in it e tem pe ratu r T < ^ Tp , i t holds approxim atel

that{df/dE) 6{E

/i) (see Fig. 4) . I n fact, a t finite te m p er a tu r th e func tion

{df/dE)i s very steep w ith its m axim um at E = / i a nd differs sig nif ica ntl from ze ro

in the energy ra nge of the ord er of fc^ T ar ou nd /i . I t i s easy to verify t h at{df/dE)

is an ev en fun ction o f E aro und / i a nd va nis he ex po nen tia llfo rIE *

/ i | fc^T; w e


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SOLID STAT E PHYSIC S

83

(iU=*'=* '

J

i i

. i l l

H

9E/T;^ i i

Fig. 4 Th e Ferm i-Dirac distr ibu tion function f{E) an d energy derivative{-df/dE)a t

T = 0 and at a finite tem pe raturT , with T


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8 4

II I

T H E S OM ME RF EL D F RE E-E LE CT RO N T HE OR Y O F M ETAL S

inse rtin th is expressio into Eq. (14), and using Eq. (13 ), we ob ta in

Odd powersof{E-/x )inthe expansio have been om itted becaus they give zero

contributionThe coefficien o f G {^ )can be written as

1 C^ ^ 8f 1

/*

JE-fj,)/kBT 1

-2L ( ^ - ^ - a l ' ^ - L ..,ii-ST ^

^0

r+00

(e^+1)2'

(in the last passag

we

have denote by x the dimensionlesvariabl

x

= {E ^i)/kBT).

To perform the integra in thexvariable we notice that

0J o

-2x

\dx

_

( 1 1 1 \ _ 7r2

This explains the numerica coefficien i n front of G {^)i n expressio (15). Successiv

terms in expansio (15) can be calcu late i n

a

sim ila way.

Equation (15 ) i s known as the Som m erfeld

expansion.

T o appreciat the fact th at

it isarapidl convergen expansio forT


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Temperature dependence of the chemical potential

T he ch em ica p o te nt ia /i d ep en d (slightly on the te m p era tu rein the stu dy of se vera

t ransporpro pert ie s the te m p era tu r dep en de nc of the ch em ica po te n tia (w ha tev e

sm al i t m ig ht app ear a t first sigh t) has qu ite im p o rt an co ns eq ue nc esa nd now w e

wo rk out ex pl ic itly the beh av iou o fIJL{T).

Let D{E) b e the (single-pa rticle den sity-of-sta tefo r bo th sp in d irections for th e

metallic sam ple of volu me V; no p ar tic u la as su m pt io on D{E) o r on the co nd uc tio

ba nd en ergy jB(k ) i s do ne a t thi s st ag e Le t N b e th e to tal n u m ber o f co nd uc tion

electron o f th e sam ple A t an y tem pe ratu r T , w e have th at / i (T ) i s determ ined

(im phci tly en forc ing the equal ity

/

+ 0 0

D{E) f{E) dE .

-oo

Such an in te gra has in gene ra a ra th e co m plex s tr u ctu re Ho wever i f T


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86 II I TH E SOMMERFEL D FREE-ELECTRO N THEOR Y O F METAL S

3 E le ctroni c specifi c hea t i n m e ta l s a n d t he rm odynam i c func tion s

T he hea t cap acityat con stantvolumeof a sam ple i s def ined b y

w he re 8Q i s the am ou n o f he at tran sfe rre fro m th e ex ter na wo rld t o the syste m

and dT i s the co rre sp on din ch an ge i n te m p e ra tu r o f the sys tem ke pt a t co ns ta n

vo lume V. T h e h eat ca pa city i s an exte ns iv qu an tity i.e. a q u an tity pr op or tio na t o

the volu me of t he sam ple for th is re ason depend in on the nat inre of the system u nder

investigat ioni t may beco me preferabl t o in tr o du c the specificheat per m ole, or th e

specifi heat per un it vo lu m e or per u n it ce ll, or p er co m po sin a to ms or elec trons

W e can ex pr ess the he at capacity (22 ) i n co nv en ien al te rn at ive forms. From th e

first la w of therm ody nam icsw e kn ow th at th e ch an ge dU o f t he in te rn a en ergy of a

sy ste m i n a tra nsfo rm at io i n which an infin ites im a q u an ti ty o f h eat 6Q i s received

by the system and 8L i s the work do ne on t he system by exte rna forces i s given by

dU = 6Q^6L.

I n the case the exte rn a forces are only m echanic a forces exert ing a pre ssur p on the

syste m w e ha ve 8L =

pdV\

i f t he vo lume V o f t he sy st em i s ke pt co nst an during

the tra ns fo rm at io th en 6L = 0; it follows dU = 6Q and

C . = ( f ) ^ . (23 a )

I n th e case o f reversibl tra ns for m atio nst h e second la w o f therm od yn am ics ta tes

t h at 6Q = T dS ,w he re S i s the entropy o f t he syst em W e ha ve th us

C=T{

(23b )

W e now calcula t the h eat capacit of an ele ctron gas using Eq. (2 3 a). The in te rn a

energy o f t he Fermi gas i s

/

+ 00

ED{E)f{E)dE

-oo

= ED{E) dE + ^klT^[Difx) + tiD'{^)] + 0{T^) , (24 )

w he re the s ta n d ar So mmerfe l ex pa nsio (17 ) has been us ed B y diffe ren tiatin w it h

res pe c t o the tem p er a tu r b o th m em be rs o f Eq . (24) , and keep ing only th e m ost

re lev an am ong the te rms co nt ainin (d // /d T ), we ha ve


11/25

S O LI D S T A T E PH Y SIC S

87

Fig. 5 Electronic con trib utio Cv{T) t o the heat capacit of a meta at cons tan volume as

a function o f tem pe raturethe same expressio holds for the electroni contributio t o the

entropy

Using Eq. (19) , and re placingD{IJL)w ith JD(/xo)j we o b ta in

CviT) = -klTD{^o)

(25)

Prom Eq. (2 3 b), i t can be no ticed th at ex pres sio (25 ) repre sen talso the entr opy of

the electron syste m

The specific h eat p er u n it vo lume cy = Cy/V b ecom e s

cv{T)= ^klT

DJtM))

V

iT,

where

1

KQ

V

(26a)

(26b)

T he co n trib ut io t o the specific he at from co nd uc tio ele ctr ons i s pr o po rtion at o T

for al l te m p era tu re o f in te re st since T i s alw ays m uch less th an Tp- T h e ele ctr on ic

con trib utio bec om e the lead ing one at sufficientl low te m p era tu re sw here it preva ils

over the T^ D eb y e co n tributio orig in at ed b y the lat tice vib ra tions (see Se ction IX -

5).

W e also no tice th at th e kno wledge o f th e de ns ity-o f-sta teD{f io) a t th e Fermi

en ergy /i o co m ple tel de te rm ine Cy ; th is gives a first ins ight o n the imp or ta nc o f

the electro ni s ta tes a t o r near th e Fe rmi surface i n me ta ls I n disc ussin tr an sp o r

phenomenaim p u ri ty screen ing e tc. w e will fu rth er see the bas ic role play ed b y th e

electron i s ta tes lyin g i n th e th er m a interv al o f th e ord er ksT ar ou n d th e Ferm i

en ergy

I t i s inte res tin t o specify Eq . (25 ) fo r th e case o f t h e free-electrongas , where

D{fjLo

= (3 /2 ) N/fio ac co rding to E q. (11 ) . W e have

7r2 T

Cv{T) = -jkBN.

(27)

For com pa ris on w e rec al th at th e cla ss ica s ta tis tic a m ec ha nic wou ld give the ex-

press io Cv = iS/2)kBNfo r the he at cap ac it of a gas of

TV

no n- in te ra ct in pa rtic les


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88 II I -TH E SOMMERFEL D FREE-ELECTRO N THEOR Y O F METAL S

T he co rre c resu l (27 ) c an be in te rp re te not ic ing th at on ly t he elec tro n in t he ther-

m al in te rv a ksT aro und the Fe rmi en er gy/X Q= ksTp ca n va ry th e ir energ y and th us

the effective nu m ber o f elec tron w ith classica be ha vio ur i s not N b u t ra th er th e

fraction T/Tp o f N.

4 Th erm io ni c em issio n fro m m eta l s

W e can ap ply th e Ferm i-D irac s ta tis tics t o st u dy un der ve ry simp lified con ditio ns

the ther m ion ic em ission from m et als i.e. th e em ission o f e lec tron from a m etal i n

the vac uum bec au se o f t he effect o f a finite te m p e ra tu re Fo r our sem i-q uan tit a tiv

considerat ionsw e do n ot co ns ide i n det a i po ss ib l reflection o f ele ct ro n im ping ing

at the su rfac e w e sim ply as sume th at al l t he e le ctron th at arr ive at the su rface wit h

an en er gy sufficien t o overcome th e surface b arr ier ar e tra nsfer re t o th e vac uum

(a nd sw ept aw ay by some sm al app hed elect ric field w ithout accim iu la tio o f space

charge)T he m od e elect ro ni s tr u c tu r of t he m eta l with e lectron affinity x s^nd work

fun ction W , i s illu str at e i n Fig. 1 ; we wish to ob ta in the num be o f elec tro n which

es cap from the m eta l ke pt a t te m pe ra tu r T .

I n the m eta l the elec tro n are d is tr ib u te i n en er gy ac co rd ing to the F erm i-D ir ac

stat ist icsLet us in dic at w ith z the d irection n o rm a to the su rfa ce the curre n density

of e sc ap in ele ctr on i s given by

w he re Vz = hkz /m. Not ic e th at ex pres sio (28) , i n the case Vz is ju st rep laced b y a

drift velo city v in dep en den o n k (an d th e int egra tion over kz extends from oo t o

-hoc , wo uld give the s ta nd ar de ns ity Js = {e)nv.Th e Um itation i n Eq. (28 ) fo r

the int eg ra tion o n kz i s ju st t o m ake sure th at th e es ca ping elect ro ns have enough

kinetic en ergy fi?kl/2m> x i n ^^z dir ec tion to leave the m et a l

I n ord er t o pe rfo rm th e in te gra (28) , let us notice th at

Sin ce i n gen era the work funct ion V F > fc^T , we can safely neglec the im ity i n the

Fe rm d is tri bu tio fun ction i n Eq. (28) . W e th us o b ta in

_ ( - e ) h T ^ / kzdkz / dkx / dkyexD

oo o o

W e now use the stan d ar re su lt for G au ss ia fun ct ions

hHkl +k l+kp

ksT2mkBT

L

+ 0 0

dkxexp

2mkBT

y/2'KmkBT


13/25

SOLID STATE PHYSICS 8 9

T ab le 1 Experimen ta values of the work function W for some meta ls

meta

Li

N a

K

Cs

Ag

A u

W(eV)

2.49

2.28

2.24

1.81

4.3

5.2

meta

Al

Ga

Sn

Pb

Pt

W

W(eV)

4.2

4.1

4.4

4.0

5.6

4.5

and obtain

* 5 = ^ 2 T~ / k^dk^exp /^

ft2fc2

ksT2mkBT

The integra can be easily pe rforme wit h the ch an ge of va ria b le (ft^fc^/2m

/x = a:,

and w e ar rive at th e R icha rd so ex pressio

T he abso lu t va lue of the nu m er ica fac tor i n the R ichard so law (29 ) i s

^ ^ = 120.4 a mp

c m '

.

K '^ , (30 )

in many m eta ls m easure va lues are in de ed i n the ra n ge 5 0

120 amp

cm~^

K~^.

A q u an ti ta ti v tr ea tm en o f t he elec tron em iss io from m eta ls requ ires a num be of

ref inementof t he sim plified m ode here co nsider edI n the m od e of Fig. 1 , the m e t a l-

va cu um bo u n d ar i s re pr es en te by an a b ru p disc ont in ui t i n the po te n tia lI n re al ity

an elect ron ou ts ide the m etal feels a n a tt racti ve im age p o te n ti a (t o b e dete rm in ed

in prin cip le q u an tiun m ech an ically w it h con sequ en ceo n th e reflec tion coefficient

of es ca pin ele ct ro ns T h e im age po te n tia also lea ds t o a re d uction o f t he ap pare n

work fun ction i n th e pres en c o f ap plied ele ctr ic fields (S ch ot tky effect). T h e work

fu nction i s also sensit iv to various effects fo r in stance surface im purit ies and charge

m od ification a t th e su rface Obs erved va lues o f t he work fun ction fo r some m eta ls

are rep o rted i n Ta ble 1 ; for a m ore co m ple te Us t see for insta nce H . B . Mich ae lson

H an db oo of C he m is tr and Ph ys ics ed ited by R. C . We as (CR C Pr es s Cleve land

1962); D . E. E a s tm a n P h ys Re v. B2, 1 (1970) and refe ren ce q uoted th ere in

A P P E N D IX A . Outlin e o f sta tis tic a l physic s a nd the rm odynam i c

relations

A l.

Microcanonical ensemble and the rmody nam i q uan tities

T he ba sic pr in ciples o f c la ss ic a and quantimr s ta tis tics ca n b e foim d i n st an d ard

te x tb o o k on sta tis tic a ph ys ics the pu rp o s of t his ap p en di i s sim ply to sum m ariz

the rec ipes for the co nn ec tio be tw een st at is tic and th erm od yn am ic s


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90 II I TH E SOMMERFELD FREE-ELECTRON THEOR Y OF METAL S

Let

US

co nsid e a p hysica system co m pose by N identica pa rt ic les confin ed w it h in

a volu me V. Q u an tum mechanic pro vid es for th is co nfined system discre tiz e energy

levels

we labe w ith an integ e nu m b e m al l the d is tin c eige ns tate of t he sy stem i n

increasin energy order Em {--- < Em < Em-\-i . {A23 )

rrifS

Fo llow ing m u ta tis m u ta n d is th e rea so nin do ne i n the A pp en dix A2 , we can easi ly

est ab lis th at the gra nd ca non ica p o te n ti a eq ua ls

g r a nd = C / - T 5 - i V / i = - f c ^ T h i Z g , a n d ( T F , / i) {A24)

wh ere U i s th e m ean inte rnal energy an d N i s th e m ean particle nu m ber i n th e

gra nd canonica d is tr ib ution R ela tion (^424) i s the basic re sul of the gra nd canonic a

ap p ar a t u sall the oth er the rm od yn am irelat io nsh ip follow from it .

From the first and se co nd prin ciples of th erm o dyn am icw e have

dU = TdS-p dV -]'fidN.

By diffe rentia tin grand and using the above ex press ionwe obtain

dfigrand = dC/ ~ TdS - SdT - Wdfi - îdN = -SdT - pdV - Ndfi .

It follows:

'T , /x

J^_ _ / ^ ^ g r a n d \

'T ,V

N = -( ÎHÎ^. (A27 )


19/25


R ela tion (^425) p ro vid e the en tr op of the sy ste m re la tion (^426) pro vide the eq uation

of Sta te; rela tion (^427) p ro vides th e m ean n um ber o f part ic le s al l oth er therm ody

n amic fun ctio ns can b e eas ily o bta in ed from the eq uation so far establis hed

A P P E N D IX B . Fe rm i -D i ra c an d Bose -E inste i n s ta tis tic s fo r

independent particle s

Fermi-Diracstatistics

W e co ns ide a ph ys ica sy st em com po se b y N ide nt ical p artic les confined wit h in a

vo lm ne V. T h e part ic les of the system are re g ard e as no n-in teracting(exce pt for a n

ar bi tra r sm all intera ctio n t o en su re th er m od yn am iequ ilib riu m ) W e th us dis cu ss

the energy levels of the m any-b od system in te rms of in dependenone-partic lesta tes.

Q u an tum m ec hanics pro vides fo r a sin gle pa rt ic le confined with in th e vo lume V

disc retize en ergy leve ls we labe w ith an in tege num be i all the d is tin c eigen st ates

of en ergies e^, o f t he sin gle-par tic l q u an tum pro blem Since the part ic les are non-

in te ra ctin g th e to tal en ergy o f t he many-b o d s ys tem i s the sum o f t he en erg ies of

the in dividu a parti c les

i

w here n^ de note the num be of p art ic les w ith en ergy e ; the to tal n um be of p art ic les

is

Ar = ^ n i. {B 2)

i

W e now c onside specificall a system of id entic a Fermi p art ic le s as a co ns eq uen c

of t he Pa u li pr inc iple the possibl values o f t he oc cu pa tio nu m be rs Ui are eit her 0

or 1 . T he mo st gen era accessib l s ta te fo r th e sy stem o f ind istin gu ish ab lpa rtic les

is defined b y a set o f n um bers {n^ } (i.e. any sequ enc of in tege n um be rs equa to 0

or 1) . Th e g ra nd part iti on fun ction {A21) fo r a q u an tum sys tem o f n on -in ter ac tin

fe rm io ns becom es

T h is sum can be ca rr ied o ut exactly and giv es

i

T he pa ss ag from E q. (S 3 ) t o E q. (B 4 ) can be do ne for in stance w ith the following

reasoningC onside first the part ic u la s it u ati on of a sys tem w ith a singleon e-elec tro

level say 1 ; in th is case the sum i n Eq. {B3) prov ides Z i = 1 -h exp[ ^(e

/x)].

Consid e th en the pa rt ic u la s it uati on of a sy stem with only two levels say1a nd 2;

the sum defined i n Eq. ( S 3) pro vid es Z = Z1 Z2 . Sim ilarly for a sy stem w ith n levels


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96

II I T H E SOMMERFEL D FR EE-ELECTRO N THEOR Y O F METAL S

i,

2,

.

,

^n, the sum dej&ned in Eq. ( 5 3 ) pro vid es Z = Z1Z 2 . .. ^n and for a s ystem

w ith any n m nbe o f levels w e o b ta in E q. {B4),

Now t h at the gra nd canonic a p ar ti t ion fu nction i s know n we can calc ula t w hate ve

desired the rm od yn am iq uan tit y For ins ta nc we can ca lcu la t the av erage oc cup atio

num be f{ei) o f a given st a te {. W e ha ve

fisi) =

{rii

= J2 ^i

-PE^ji^j-t^)

gra nd

{nj}

T he sum over th e co nfig ura tion {rij} ca n b e easily ca rried ou t following, m u ta tis

m u ta n dis the proced ur do ne for the ca lcu la tio o f Zgrand W e ha ve

- / 3 ( i - M )

^ e r a nd . . ^ L J 1

'g ra nd

Ji^i)

^ g - ^ ( e i - / i )

We th us recove im m ed iate l the F erm i-D irac s ta tis tics

f{ei) =

1

el3(ei-n)+ 1

(B5)

W e can ob ta in the ex press io o f the free ene rgy and o f the entr opy of a system of

non-in te ra ctin ferm ions W e st a r from the ex pres sio of the grand ca nonica p o te n ti a

a

grand = ^keTIn Zgran d =

^ B T

J] I n [1+

e-^(^^-^

] .

F rom E q. {A24 ) w e have for t he free en er gy

-P{ei-ti)

(S6)

(S7)

F rom Eq. ( ^ 2 5 ), w e have for the en tropy

i i

W e use the id enti ty

/3(i ~ /x ) = I n

where fi denotes by b rev ityf{ei).W e o b t a in

S=kBY^

- l n ( l - / i ) + / i l n

l-fi

fi


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S OL ID S T AT E P H Y S IC S

97

amd finally

S = -kB Yl [fi I n /i + ( 1 - fi) l n ( l - fi)]

(58)

which i s the de sired exp ress io for t he en tr opy of non-in te ra ctin fermion s

A noth e im p o rt an ex pre ss io can be prov ed for the to ta nu m be of p arti cl es Fr om

Eq. (i427), we have

N

^ _ f d n ^\

^ ^ B T T - ^

ln [ l +

e-^^^-'^)

g - / 3 ( e i - / x )

2^

1 -f- e-/5(et-M)i

Fin al ly for t he equation of s ta te (^ 2 6) w e have

E/^-

p =

T he en ergy eigenvalue o f a parti cle confined i n a cu bic box of v olu me V L^ ar e

given by

^ = ^ ( l ) ('^ x + 'iy + ^ z) n ^,n j n ^ = 1 ,2 ,3 , . . .

W e ha ve th us de/dL = 2e/L and also

ae _ a e a L _ _ 2

dV~dLdV~ZV

Exp ress io (B 9 ) th us be co m es

(BIO)

and then

P V = -U .

(511)

Bo se-E insteinstatistics

I n th is ca se diflFerently fro m th e pre vio us tr e a tm e n tth e sequ enc { n j ca n con tain

any integ er from zero to infin ity. T h e gra nd pa rti ti on func tion fo r a system o f non-

in te ra ctin bosons i s st ill given by ex pressio ( 5 3 ) , keep ing i n m ind th at rii can now

take any in tege va lue from ze ro to infin ity. T h e sum (J53) can be carr ied out exactly

and gives for b osons

2grand(T, F, / i) = J J ^ _ -0{e,-n) '

(B12)


22/25

98

II I TH E SOMMERFEL D FREE-ELECTRO N THEOR Y O F METAL S

T he pas sa g from Eq. ( 5 3 ) t o Eq. {B12 ) ca n be do ne (as before co ns ide rin first th e

p art ic u la ca se of a sin gle on e- par tic l leve l, say ei. I n th is case the sum i n Eq. ( 5 3 )

become

n i = 0

-/3(ei-M)

1 _ e - / 3 ( 6 : i - M )

For a s ys tem wi th a rb it ra ry nu m be o f on e- pa rtic l levels we th us o b ta in E q. ( 5 1 2 ) .

T he g ra nd ca no nic a p o te n tia for a sy st em of ind ep en denbosons i s

a

rand = ^ B T l n Z g ra n d = fc^ T^ ^ln [ l - e'^^^^^) ]

(513)

W e can now ob tain all the th er m od yn am iqu an tit ies o f in te res fo r a system o f non

in te ra ctin bos ons T he av erage occupatio num be of a given st a te i s given by

m)

1

e0(^i-fi) - 1

Sim ila rly the entr opy o f a system of n on-in te ra ctin bosons i s given by

S = -kBY ^[fi Infi - ( 1 + fi) hi(l -f fi)]

(514)

(515)

A P P E N D I X C . M o d ifie d Fe rm i -D i ra c statistic s i n a m o d e l o f

correlation effect s

I n the pre vious A pp en dix we have co ns ider e a p hy sica system co m pos e by N in dis-

t inguishabln on-in te ra ctin ferm ion s con fined wit h in a vo lume V. Sup po se th at th e

on e-elec tro H am ilt o n ian do es not rem ove the spin de ge ne racy i n the ind ep enden

part icle app roxim ationt he occu patio prob ab iUt of the spaceorbital (of en ergy e^),

regardlessof the spin deg eneracy ,i s th en given by

w here the factor 2 account for t he spin de ge ner ac of t he orb ita leve l.

In Ch ap te X II I , i n the stu dy of d op ed sem icond uctorsw e ne ed to kn ow n ot on ly

the oc cu pa tio pro ba bility o f valence and co ndu ct io st a tes (de sc rib ed b y s ta n d ard

deloca lize Bloch wave func tions)b ut also the oc cu pa tio p ro b ab ih t o f im pu rity lo-

ca hzed st a tes i n the en ergy gap. I n se ve ra s ituation (fo r insta nce fo r do nor levels)

the Coulo mb repuls ion betw een electro ns m ay preventdouble occupationof a given

loca lizedorbital. W e now dis cu ss how th is effect (whic h i s th e sim ples exam ple o f

cor re la tio be yo nd th e on e-e lec tro ap pro x im atio n modifies th e F erm i-D irac d is tr i-

b u tion fun ctio n

For sake of c larity consid e a b and sta te i n an allowed energy region of the cry sta

and de sc rib e b y a Bloch wavefu nc tion a b and level can b e em pty o r oc cu pied b y


23/25

S OL ID ST A T E PH Y S IC S

99

E=0

N=0

4-

N=l

E = E i

N = l

E = 2 e j

N=2

case (a)

E=0

N=0

4 -

E=ei

N=l

E=ei

N=l

case (b)

4

E=ei

N=l

N=l

44^

N=2

case (c)

Fig. 6 Schemati illustratio of possibl occupatio of a given spa tia orb ita of energy {.

(a) The given level can accep tw o electron of eith e sp in (this i s the common situation for

band states (b ) T he given level can accep only one electro of either spin (th is situ ation

is common for donor impurity levels i n sem iconductors(c ) T h e given level can accom

modat one or two electro ns hut not ze ro (this situation i s common for accepto levels in

semiconductors)

one electron o f e ith er sp in, or by tw o electro n of opposit spin; the four poss ib ilit ie

are illu stra te i n Fig. 6a. Co ns ide now an imp urity s ta te wi th in the ene rgy gap and

de sc ribe b y a loca lized wavefu nc tion a do nor leve l, for in sta n ce can b e em pty, o r

occu pie b y one ele ctron o f eit her sp in, bu t no t b y tw o elec tro ns o f opposit spin,

be ca us of the penal ty in the ele ctro sta tirep uls io en ergy the si tu ation i s ill ustra te

in Fig. 6b.

We calcu late fo r bo th s it ua tio ns th e av erage num ber o f elec tro ns i n th e st a te {

(re ga rd les of the sp in dir ection) the average nu m be i s given by

{rii)

(3{Em,N-fiN)

(C2)

In the ca se of F ig. 6a, Eq. (C 2 ) gives

(rii)

M ) ]

[i4-e-/3( i-M)]^'^7^^

1 + e-/5(^

as ex pe cte d the res ul ( C I ) i s reco vere

/^)-hl

(C3)


24/25

100

II I T H E SO MM ERFEL D FR EE-ELEC TRO N THEOR Y O F METAL S

4f(e)

F ig. 7 Average occup atio nu mbe (dashed line) for an orbita of energy e th at can accep

up to two electron of eithe spin (sta nd ar Fermi-D irac statis tic s)T he average occu pa tio

num be (so lid line) for an o rb ita of en ergy e that can accep only one elec tron of eithe spin

is also reported For e

fi ^ ks T th e two curves coincide

In the case of F ig. 6b, w e have inste ad

g- /3 (e i- / x) _| _ g-/3 (et - /x)

{rii)

I -| _ g -/ 3 (e i- /x) _| _ ^-f3{ei-tM)

le/3(ei-M) + 1

(C4)

T he oc cu rre nc of the fac tor 1/ 2 i n E q. (C 4 ) , can be eas ily un d ers to o qu ali tat ive ly

in the lim it ing case of a B o lt zm ann ta il (see Fig. 7 ) .

For sake of com plet en es swe consid e also the st a ti s ti c of the ac cepto leve ls wh ose

electro ni s tr u c tu r i s st ud ied i n C h a p te X III . A n ac ce pto level can be oc cu pie by

two pa ir ed electrons or one of ei th e sp in b ut ca n no be em p ty be ca us of the pe na lty

in electrostatirep uls ion ener gy betw een the two holes the s it u ation i s sc hem atic al l

ind ica te i n Fig. 6c. T he ap plic atio of E q. (C 2 ) gives

1 g-^(e,-M + 1

If we indica te by {pi) 2

(rii) the m ean nm nb e o f ho les we o b ta in

1

(Pi) =

l e / 3 ( M - i) - f 1

(C5)

(C6)

Further readin g

N.W. As hcrof and N . D. Me rm in Solid S ta te Phys ics (Holt, Ri n eh ar and W in sto n

New Y ork 1976)


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H. B . Callen Th erm od yn am ic and a n Int ro du cti o t o Th erm os tat ics(Wiley , New

York 1985, seco n ed itio n)

K . Hua ng S ta tistica M ec han ics (W iley , Ne w Yo rk 1987, seco nd edition)

C. Kit tel Ele m en tar Sta tis tic a Ph ys ics (Wiley , New York 1958)

R. K ubo S ta tis tic a M ec ha nic s (N or th -H oll an d A m ste rd am 1988, seve nt ed ition

A. Miinster Sta tistic a Th er m od yn am ic sVols . I and H (Spr ing er Berl in 1969)

R. K . Pa thr ia S ta tis tic a Mec ha nic s (P er gam on Pre ss Oxford 1972 )

L. E . Re ich A M od ern Coin-se in S ta tis tica Phy sics (A rn old, Lo nd on 1980)

A . H . W ilson Th e T h eory o f M eta ls ( Cam brid ge Unive rs ity P ress 1954)

iii the sommerfeld free electron theory of metals 2000 solid state physics

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