III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
-
Upload
christian-julian-forero -
Category
Documents
-
view
220 -
download
0
Transcript of 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
1/25
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
2/25
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
3/25
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
7/25
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
8/25
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
9/25
SOLID STAT E PHYSIC S 8 5
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
10/25
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
12/25
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
14/25
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 - ^idN = -SdT - pdV - Ndfi .
It follows:
'T , /x
J^_ _ / ^ ^ g r a n d \
'T ,V
N = -( ^IH^I^. (A27 )
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
19/25
SOLID STAT E PHYSIC S 9 5
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
20/25
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
21/25
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)
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
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
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
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)
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
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)
-
7/23/2019 III the Sommerfeld Free Electron Theory of Metals 2000 Solid State Physics
25/25
SOLID STAT E PHYSIC S 10 1
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)