i

NameActinium AluminumAme ri cium

Symbol

NameHafniumHelium

Symbol

Name

SymbolP,

A,AI

Hr

PraseodymiumPromethium

Antimon"Argon Arsenic

-

Am SbM

Holmium Hydrogen inclllJlnJodine

H. HoIi

Protactinium Radium

A.At

I" Ih

RadonRhen ium

..Po

Pm R" R, RhRb

,

Astatme

Iridium

Rhodium

Banum Berkelium Beryllium

B. BkBe

IronKryp ton

F. K,Lo

RubidiumRuthenium

LanthanumLav.'Tcncimn

Bismuth....00

Bromine

Cadmium CalciumCalifornium

CarbonCerium

Cesium

Chlorine Chromium

BI B B, Cd C. Cf C C. C. CI C,Co

L,

Samarium ScandiumSelenium

R.. Sm S. " SI

Lo.JLithiumLutetium

Pb

:-'lagnesiurn\Ianganese \ Ienaele\ jum\1en:urV

~ \lg\ 111

SiliconSilver Soc!iumStrontium

" N. S,S T, T, T. Tb11

\Id

Hg\10

Sulfur TantalumTechne ti um Tellurium

\loIybdenumKeodymium

Nd!'\e

CobaltCopperCurium

Kron "'''ptuniurn

:\p

Tl:rbium ThalliumTII(lriumTIlLl];lJIn Tin

C.. CooDy

Klcke\r\iobiurn

DysprosiumEinstein ium

Erbium EurOIJ ium Fermium F1uorillC Francium Gadolinium Gallium Germanium

E. E,Ell

Cold

." ,

Fm F F, Cd C, C.

"'I trogen ;':obelium Osmium Oxygen PalladIUm Phosphorus Platinum Plutonimn Polonium Potassium

No Nb N:\0

0.

Pd PPI p" Po

"

Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium

Th Tm Sn TtW

UV

X, Vb V Z"Z,

---=- . . . .

- ~--

.. .

.,...~-

... -_.

..

H'I.,

HTPeriodic Table, with the Outer Electron Configurations of Neutral Atoms in Their Ground Siaies

h'

U'0,

Be'",'

,

Na il

Mg "

The nolation u\cd to describe the elc(:lwnic configuration of' ~lt()m' ,md ioo, i' di~cu'~cd in all textbook., of introductory alomH; phy_ie" The letter~ $, fl. d, .. .\ ignify cledron~ having orhital angular momentum D, I, 2, .. . ill unih -t\; the numher to the lei! of the letter denute, the pTlncipaJ quantum numher ot Olle urbit, and the ,uper~cript to the right dcnole~ the nl1mllcr IIi' electroll~ in the orhit

B'2.\.2211

C

N'

0"

F'

Ne'o

2.~~2pl 2s72p~

2s'2p' 2.1:'2,,5 2.,'2p l

AP]

SI"

P"

5"

CJ'1

At "

3,

K"4,\"

'CaN "4.s!

3I.'3p 3s'3p'

~"3p' :1.~'31'

3.,'3p' 3.I:'3p'Brl~

Se"3d4,\,!

Tj irl

V":JrJl4,~l

Ct"3(/5

Mn"

Fe""3d"4.'

Co"

Nili'

Cu"3d'~

Z,'3el l " 4.' Cd"4d"5.~2

Ga"

Ge"

As]1

Se

H

K,'

3d'4.,"

3d'4."

3d'

4.'Mon

4,'

3d"4.~2

4,Agetion of symmetry operations which, applk> [C p cos(2'7Tpda) + Sp sin(211pXia))

,

(3)

II

I

where the p's are positive intcgers and Cpo S" arc real constants, called the Fourier coefficients of the expansion. The factor 2nlo in the argumellts ensures that n(x) has the period tJ:11(.1'

II

+ a) = "0 + :rIC,. cos(211pXia + 2'7Tp) + S" sin(2'7Tpda + 2'7Tp) = no + :rIC" eos(2'7Tpda) + 5" sin(2npxJa) = fI(X) .

(4)

We say that 2'7Tpla is a point ill the rcciprocallaUice or Fourier space of tile crystal. In one dimension these points lie on a line. The reciprocallatticc points tell us the allowed terms in the Fourier series (4) or (5). A term is allowed if it is consistcnt with the periodicity of tllc crystal, as in Fig. 5; othe r points in the reciprocal space are not allo .....ed in the Fourier ex~nsion of a periodic function.

\

,I

lluiprocal Lall ice

31

~

l

I

- --- 'lMAI'--~u.o.ooo~

' 00

....

~ - I

.. A

.. ..

. I 16.l

....

1IlOi . -

..

~

Br-uOOfOlt'

'" -----..

j

-

U...J.....iatcd rompoocnt. of mal~

"""In

figu" J Sketch of a nld by (1.3) . From (9), nCr

+ T) :::

L nc exp(iC' r) exp(iC" T) C

(J6)

But exp(iC' T) ::: 1, because exp(iC . T) ::: cxp[i(v.b,=

+ V2i>2 + V:3b:,) . (u,a, + 1I~2 + 113a3 c,p{i2-n{v,lI, + V2112 + tlJlI.:J]

(17)

The argument of the exponential has the fonn 2m timt..'S an integer, because V,II , + V:2U2 + lIJU3 is all integer. being the sum of products of integers. Thus by (9) we have the desired invariance. nCr + T) ::: nCr). This result PTO\'es that the Fourie r representation of a function periodic in the crystal lattice can contain components nc cxp(iC' r) only at the wciprocal lattice vectors C as defined by (15).

Dif/raction ConditionsT/leorem. The set of reciprocal lattice vt..'Ctors G detelmincs the possible ,..ray reflections. We St..'C in Fig. 6 that the difference in phase factors is exp[i(k - k'), r1 bctwt..'Cn Ix..'ams scattered from volume clcmcnts r apart. The wavevectors of thc incoming and outgoing beams are k and k' o 111C amplitude of the wave scaUert.>d fmm a volume element is proportional to the local electron concentration n(r). The total amplitude of the scatlert..>d wavc in the direction of k' is proportional to the integral over the crystal of n(r) dV limes the phase factor exp[j(k - It) . r J. In other \"'OM, the amplitude of the eit..'Ctric or magnetic field vectors in the scatlen.>d ck-'Ctmmagnetic wave is proportional to the foUowing int egral which defiIlt..S the quantity ,," that we call the scattering am plitude: 'F :::

f

dv nCr) exp[i(k - k') , r) :::

f

dV nCr) exp(- iAk' r)

(18)

where k+Ak ::: k' .(J9)

Here Ak measures the change ill w~l\Ie\'t..'llaUice IlOin ts of the crystal. 111e ~'CC'or k is dmwn in the direction orlhe intident K my hCllm , and the origin is chost:'11 SUL-h that k terminates II.t any rL"dcd beam will be formed Ir lhis .rh're intersect! any other poinl in the n'CillrOl"&l !alike. The ' IJhcreas drawn in tCt"Ct'PIS a ll'ointl"Onneclcd with the M"od ofk by a reciJlfUC'llllallice veclor C . 111e d,If"",lcd ~ray br.-am i. in II..., dirCdicaht.>dro n, as shown in Fig. 13. 111C vectors from the origin to the center of each face are('lTlaX :!: :!: i) ;

(33)

All choices of sign are independent, giving 12 vt.'Cton.

Reciprocal Lattice to f cc LatticeThe plimitive translation vectors of the fcc lattice of Fig. 14 areal :c:

~a(y + i); ~V =

=! a(i + 2);la, . a2 )(

~

"" ia{i +y)

(34)

The volume of the primith-e cell is

aJ ""

in'

(35)

nle primitive tra nslation vectors of the lattice recip rocal to the fcc laUice hi = (2'ITla)(-i

+ y + i); ~ = (21rlaXR - y + i) b:J "" (21TIa)(i + y - i) .

(36)

.,

r 1"(2maX2i) ;

-Figure 14 Primitive basis ," ectors of the fa.:e .,entered cubic I~ttice.

These are p rimitive translation vectors of a bee lattice, so thai the bee lattice is reciprocal to the fcc lattice. -111e volume of the primitive cdl of the rccipl'OC"!.ume of a Brillouin zone is equal to the vOluine lhe primitive parallelepiped in Fourier space. Recall the vector identity (c " a) " (a " b) = (c' a )( b)a .

a

4. Width of diffraction marimum. We suppose that in a linear crystal there are identical point scattering centers at every lattice point p,., = ma, where m is an integer. 8y analogy with (20) the total scattered radiation amplitude will he proportional to F =I expl - ima ' 6kJ. The sum over At lattice points isF = J - CIIJ?!-IM(a . 6It)) J t::qJ[ - I(a . 6.k)]

by the ure of the series

"-1 ~

.. --0

1-:1: '" ,..---- . 1-:1:/>l

(a) The scattered intensity is proportional to IFf Show that

IFlt

PF ... sin iM(a 6k) sintl{a . Ilk)

t

(b) We know that a difTrnction maximum appears when a Ilk - 21Th, where h is an integer. We change Ak slightly and define ~ in a' 6k = 21Th + ~ such that IE gives the

po5ition of the first zero in sin IM(a' l!.k). Show that IE = 2rr1M. 50 that the width cI the diffractiun maximum is proportional to liM and can he extremely narrow lOr macroscopic values ofU. The same resu lt holds true for a three-dimensional crystal.

5. StruC/UTe factor of diamond. The crystal structure of diamond is described in Cilapter I . The basi3( sists of eight atoms if the cell is taken as the conventional cube.

"~ , '" !. i ..,(I ' I )

.~

~ .., -

!

.."

f z

.

-

- o.4S'

0.65" -

I-

P) H -, \-0.70"

.,.Counter .,.,.11000 2f

....=i\::

fo"jgure 2 1 Neutron diITI"..ction pallen. for powtlered diamond. (After G . &con.)

(a) Find the struchll'c factor S rL this basis. (b) Find the zeros of S and show that the allowed reflections of the diamond structure satisfy VI + vt + V3 '" 4n, whtlre all ind ices are even Rnd ,. is any integer, or else all indices arc odd (Fig. 21). (Notice that h, k, I may be writte n for VI , Vi. V:J and this is oRen done.)

6. Form fadQr of atomic hvdrogen. For the hydrogen alom in it5 ground state, thenumbe r de nsity is n(r) = ( ~ - I exp(- 2rlUf). where flO is the Bohr rad ius. Sh(1l.ll that the form f:K"tor is Ie = 161(4 + C2~'l. 7. Diatomic line. Cln~ 'd er a linc of atoms ARAB. . AD, with an A- 8 bond lenbwlule ~.em. The a"..rage f1l1ctu~lion al 0 K a a He atom from iU ~lIilibriUln poSition Is a the onler of 30 10 40 percent of th", neare!;t-Ildghbor dista.nce. The heR\"ier the Rlom_ the less import&llt ~re the :teros (>Oint effects. lf we olTlit :wro-poinl motion, we calculllte a ,nola. "olome 01"9 ern mol- ' ror solid helillm, as compared with theobser"\'ed ,,,]u~of'l:1. 5 &lid 36.8 con.) ,nor' fw liquid Ile~ and liqtUd ~, respecti'"el~. In Ill0, ':" J tlC A. ~ 3&4: .:U~ ~29;, :-2,42. 1.130 4~3-"" 4 O _ 70 .2 72 .3 75.8 55.8 37 .1 1022 428 95.5 93.4

"

-

p,

Np

p,

Am

Cm385 3:gg 92 ,1

5:$.128,

347 . 264 . 456 ;(7j .,.J,60_ -2..13 109. 83 63

-

8k

CI

E,

Fm

Md~

No

~':!~":"::

"

I~

i

"

~

~

Ct

B. 453.7 1562

Table 2

Melting points, in K.

So1814

(After R. H. Lamoreaux, LBL Report 4995)

-

C, 1358Z,692.7

B2365

C

N

63.15 54.36 53.48 24.56

F

N.

N,371.0

M9 922 C,1113

AI

51

933.5 1687

P 5 CI w 317 388.4 172.2 83.81 , 863 A, 5. 1OS9 494 Sb T. B,265.9

..K,

K336.3

TI V 1940 2202

C,2133

M,1520

F.1811

moRh2236

Co

Nt1728

G,

G.

302.9 1211

115.8

Rb312.6

Z, S, To Nb Mo V 1042 1801 2128 2750 2895 2477

R,2527

Pd1827

A91235

Cd

I,

S,

I

Xo

594.3 429.8 505.1

903.9 722.7 386.7 161.4

C,301 .6

0, I, B, C, HI T, PI W R. 1002 1194 2504 3293 3695 3459 336 2720 2045

A, TI H9 1338 234.3 577

Pb

BI

Po

AI

R,

600.7 544.6 527

~

F,

R, 973

Ao1324

I"

C.1072

p,1205

Nd1290

Pm

5m

E,

1340 1091p, 913

oy Gd Tb 1587 1632 1684Cm1613

Ho1745

E,1797

Tm Vb C, 1820 1098 1938

Th2031

p, Np U 1848 1406 910

Am 1449

Bk '1562

CI

E,

Fm

Md

No

lw

H ,.0.002 r~1U

T.able 3 h:othc nnal bulk mod uli! and oompreuibilitlc5 at room te mperatu re

;=U~ -k~t~~1-+---1Na Mg

IBe

Mer K. Gschnl. dner, Jr., Solid state phYSics 16, 275-426 (1964); several ... data are frnm F. Birch, in Handbook of phyrical coNlantl, Geological Society of America Memoir 97, 107-173 (1966). Origin:u refe rences should be consulted when values are nceded for research purposes. Values in parentheses are estimates. Letters in parentheses refer to the crystal form. Let leTS in brackeu refer to the temperature:raj - 77 K; [bl - 273 K:

to miLl00

Held.

~~ -o:-~.

B 1.78

C [41 4.43

&9:: =--I~I 0,010

N 1.1 0 0.012

I, ~

i Ne If)

leI -

1 K: (d ) = 4 K; leI = 81 K.

I~.t~'

Bulk modulus in units 10lJ dynlc~ or 10" Nlm> Compressibility in units 1O" f cm'/dyn or 10' " m'1N

AI p ,w Q.72,2 Q.,9.?8 O}04. 0.178,

lSi

IS "'~

fc;=I;':CI

1.385 l ..ol.2

3.29 ~

5.62-

......

1 7.a.:~

At 1,1 0.013

~ 'l"-S -6:58"b

0.032 0.152 0.435 Ti 1.051 1.619 1.901 0.596 1.683 1.914 1.86 " " Sc~ -~......

IV'''',.IC' 1M IF. ICo IN' 0." IZ" G" " ~G' lAS "-0- -: C, .2.",... -" n.m "..,... .51 == -n-526 -n.,-"-.6;-5~ 'O~S22

1.37

...~~ .~ ~--

' 1~6Z: l"'~~

0.598 0.569 0.772 0.394 0.091~~

,:>o~ '2~5'-

ISe I"d"'"0.Ql8~ --~---

56- -

0.031 0.116 _D.l6.. _D.Bl! 1-702 2.725, (2.~7) 3: 208 2.70~. 32~ J!.li,;... 3-1L ~ ]~ 1).= O~_.~

~

I' ID _INb I~ Ik :no. =~ Ih IWCe (,J 0.239

I~ I~ I~

,1.]) _ O.2~~ :.=' . -.. ASSJ 0 ".9. U-:C :2;R.- ..Q.9tll .2..!ilA ~;35}.007 0.467 0.411 Cd

1~8 08 _

1- I'"

S' "" Sb

X ,T' , 1-e'--1 2-1!J.l "'

Cs '5'1, :; -

Po 0.020 0.103 0.243 1.09 2.00 3.232 3.72 (4.18) 3.55 2.783 1.732 a.3S!. 0.359 0.430 D.lI? (0.26) At ~w.. 3.l?. O.~?; ,O.M[! .QJ~ .Pi-209' (O~ 0.281 Q.,352 O.S7Z, I.~.Q h79 kll... 3,.17,., (3~L ... I Ra l Ac

I"' I'"Q..

g' ~ ~ '" ,~ ,~ HO' ITi

IPb I"'

-=--:O.~ll Lu

I""

Fr ~

I'oth (Il) and (b).

n.e

, ,

,U(El)/w

, , , ,

\

.,

" " "

L

, ,,

FIgUre 6 . 'onn orlhe Lennard-Jone! pote ntial (l O} .. hkh describes the I"ten etio" or lWl,l lncrt gas atom s. '[be ml,,;mum occurs at RIfT "" 1. 12. Notico: how sl CCp the curve i'I" iruide the minimum, and how flat it iii outside the m llumum , The .'IIl ue of U at the m ini mum hi - ~ ; and U - 0 at

2''''.

R - u.

3 Cryslal Binding

65

be obtained from gas-phase data, so that calculations on properties of the solid do not involve disposable pardmeters. O ther empirical forms for the repulsive interdction are widely used. in particular the exponential fonn ). exp(- R1p), where p is a mea~ure of the r.mge of the interaction. This is generally a~ ea~y to handle analytically as the inverse power law form.

EquiUbrium Lattice ConstantsIf we neglect the kinetic energy of the inert gas atoms, the cohesive energy of an inert gas crystal \s given by summing the Lennard-Jones potential (10) over all pairs of atoms in the crystal. If there arc N atoms in the crystal, the total potential energy is

U~ ~ IN(4')[L' (..5:-.)" - L' (..5:-.)']J

p!iR

J

PuR

,

(ll)

where P!lR is the distance between reference atom i and any other atom j, expressed in terms of the nearest neighbor distance R. '111e factor i occurs with the N to compensate for counting twiee each pair of atoms. The summations in (11) have been evaluated, and for the fcc structu re~' -12 .c... P" =

12. 13188 ;

J

... P!I J

'c numbcn Ul thc CQn tOUI'll give the electron concentnttion per primitive cell, with b.tr ,-alene is thell given by

(35)St ress Comporlent8

111e force acting on a unit area in the solid is defined as the stress. There t;.. Z&. The capitallctarc nine stress components: Xr X~. X&. Yro Y~. Y2 ter indicates the direction of the force. and the subscript indicates the nonnal to the plane to which the force is applied. In Fig. 15 the stress component Xr represents a force applied in thc " direction to a unit area of a planc whose nonnallics in thc " direction; thc stress component X~ represents a force applied in thc x direction to a unit area of a planc whose nonnallies in thc y direction. Thc number of indcpendcnt strcss rompooents is reduced from nine to six by applying to an cJemcnt,lry cube (as in Fig. 16) the condition that the angular acceleration vanish;5 and h~ncc that the total tonluc must be 7..ero. It

z...

s-n"s Out.'$ nol mean we umnot 1........1 .,ro/.lc",.!n ... h,ch Ihere Is mt.'1lnS th.tl we ron use Ih" st.ltic SIIUM ..", to de"ne the cbstoc con

,gular ahed in the:t din." ,c>n to I unit area of a plane .... hose 1I0rmal lk'S ill II,,) " di ...."CIion.

,x,--

T' "----+1- '#w + Croe~ + C64eV . Thus the thirty-six clastic stiffness constants are reduced to tv.'enty-one,

(42)

Elastic Stiffness Constanl$ of Cubic CrystalsThe number of independent clastic stiffness constants is reduced further if the crystal possesses symmetry clements, We now show that in cubic crystals there are only three independent stiffness constants. We assert that the elastic energy density of a cubic crystal isU = iC,,{~

+ e!v + e~ + iC.u{c + ~ + ~v) .... + C 12{e.".e= + e=ej(X + e"..ew )

'

(43)

and that no other quadratic terms occur; that is(en-e.... + ...) (44)

do not occur. The minimum symmetry requirement for a cubic structure is the existence offour three-fold rotation a... es. The axes arc in the (Ul] and equivalent directions (Fig. 17). The clrcct of a rotation of 211"13 about these four a... es is to interchange the x, y. z a...es according to the schemes

-x_z_-y_-x; - x-y-z--x,(45)

according to the axis chosen. Under the first of these schemes, for example,

e;.+e:v+t{,-e!v+e~+~" ,

i, ,, ,,Figure 17 Rotation by 2m'3 "bo'" the axis mark.,.! 3 changes r ..... y. y ...... ~ and z_r.

,,3'b ,, ,

86

and similarly for the other terms in parentheses in (43). Thus (13) is invariant under the operations considercO. But each of the tcnns exhibited in (44) is odd in one or more indices. A rotation in the sct (45) can be found which will change the sign of the tCnTI, because e~JI = -eo:(-..). for example. Thus the lenTIli' (44) arc not invariant under the required operations. It remains to verify that the numerical factors in (43) arc (. orrect. By (41)

aU/iJen = Xx

=

e ll en + C I2(e JlJl +

e~J

.

(46)

The appearance ofC1 1 agrees \~;th (38). On further comparison, we sec that e,;x(47)

Further, from (43),aUlae~1I

,.. XII = C44e~1I ;

(48)

on comparisOn ",;th (38) we have

C 61

:;;

C62 = C63

=

C64

= eM ""

0

(49)

Thus from (43) we find that the array of values of the clastic stiffness con-

stants is reduced for a cubic crystal to the matrix

X. Y, Z. Y. Z. X,

C" C" C" C" C" C" C" C" C"0 0 0 0 0 0 0 0 0

0 0 0

0 0 0 0

0 0 0 0 0 (SO)

C..0 0

C..0

C..

For cubic crystals the stiffness and compliance constants are related byC 44

= 11544

;

CII - C I2

= (5 11 -

51

v-

1

;

CII

+ 2C I2 = (5 11 + 25,J- 1

(51)

These relations follow on cva1uating the iuverse matru to (SO). Bulk Modulus and Compressibility Consider the unifonn dilation en .... ew:O: eLl: = ia. For this defonnationthe energy density (43) of a cubie crystal is U = i(C

I

+ 2C I"~

.

(52)

We may define the bulL: modulus B by the relation U ~ IB8' (53)

3

In.ti.: Co... t4 ..,.

81

Figure 18 Cu"," of volume lu, 4y fJ.:t acted on by a stress - X.(%) O il the face at %, and

Volu me /J.x t:.y /J.=.

X.{%

+ fJ.x) ill X.(%) + i1X. Axat%

face

+ fJ.%.

"

on the parallel

TIle

IIet

forceJ

u

(i1~~

fJ.x) fJ.y fJ..:. Other forres in the

di~

lion arise from the ..... rlation across the clIl.>er 7T/a . The re are three polarizations p fcr each value of K; in one dimension two of these arc transverse and Olle longitudinal. In three dimensions the polarizations arc this simple only for wavt.,\'eclors in certain special crystal directions. Another device for enumerating modes is orten used that is C(lually valid. \Ve consider the medium as unbounded, bul require thai the solutions be periodico\'cr a large distance L, so that u(sa) = u(sa + L). TIle method of periodic boundary conditions (Figs. 4 and 5) docs not change the physics of the problem in all y essen tial respect for a large system. In the running wave solution u. = u(O) exp[i(sKa - wxt)] the allowed values of K areK = O,

.. . ,

N~

L

(14)

TIlis method of enume ration gives the same number of modes (one per mobile atom) as given by (12), but we have now both plus and m inus values d" K, with the intcrval llK = 2'1t1L bctween suC('Cssivc valucs of K. For periodic boundary conditions the number of modes per unit range of K is U21r for -1rla :s K ::s TTla, and 0 otherwise. The situation in a two--dimensional lattice is portrayed in Fig. 6. We need to know D(w), the number of modes per unit frequency range. The number of modes D(w) d w in dw at w is given in one dimension by

LdK L dw D(w)dw=--dw =- - - .TT dw1r cbldK

(15)

We can obtai n the group velocity dwldK from the dispersion relation w versus K. There is a singularity in D (w) whcnever the dispersion relation w(K) is horizontal; that is, whenever the group velocity is zero.

Density af Stl/fCS in Three DimensionsWe apply periodic boundary conditions O"er N 3 primitive cells within a cube of side L, so that K is determined by the condition exp[i(Krc whenccK, ,~ , K:. = 0;

+ K~y + KLZ))

exp{i{K.{x

+ L) + ~(I) + L) + K:.(z + L))} + -L 4~

(16) (1 7)

2~ +- .

N~

-

L

L

TherefOre there is one allowed value of K per ,olume (2TT1L'j1 in K space, or

L)' V (2TT = 8r.

(18)

S

f'hlllU,)M II. Thenncll'roptrfin

121

FIgure 4 CuISidept.'1l~ ,mlle of K per IlI"l:3 (2mIOa)l- (2mL"f, so that "'i !hin the Qrc:ie of ar~a .. Ks the ~moolhed number of allcJ.,.-ed points is ","'U2'ftf.

'22allowed values of K per unit volume of K space, for each polarization andfOT

each branch. 'Inc volume of the specimen is V = L3. TIle tolal number of modes with wavcvcctor less than K is found from (18) to be (U27T)3 times the volume of a sphere of radius K. Thus(19)

for each poiari7.3tion type. The density of states for each polarization isD{w) = dNltlwDel}!JC Model for Density of States In the Ocbye approximation the velocity of sound is taken as constant for each polari:.r.alion type, as it would be for a classical elas tic continuum. 'Inc dispersion relation is written as::::I

(VK2/2-zr)(dKJdw)

(20)

w = oK .wit h0

(21)

the constant velocity of sound.D(w) = Vw'l/2.,rv 3

The density of states (20) becomes(22)

If there are N primili\'c cells in the specimen, the total number of acoustic phonon modes is N. A cutoS' frequency Wv is determined by (19) as

Wb =

6-rr'lrlNlV

(23)

To this frequency thcre corresponds a cutoff wavevector in K space:Kv = wdv :::: (67fNIV)1I3 .

(24)

On the Dcbye modcl wc do not allow modes of w3\'c\'eclor largcr than KI) . The num ber of modes with K :s; Kv exhausts the number of dcgrees offrecdom of a monatomic lattice. The thennal encrgy (9) is given by

u::::

f

dw D{wXn(w)liw -

r'"

dw

(2";:3) (ij:~w_ 1)(26)

for each polari7.3tion type. For brevity we assumc that the phonon vclocity is independcnt of the polarization, so that we multiply by the factor 3 to obtain

whcre x

"wi.,

>II

" wlkBT andXo ~ "wu1kBTiiiili

81T .

(27)

This defincs tile Dcbye temperature 0 in terms of WI) dcfined by (23). We m ay express 0 as

o ~ ~. (&n'N)'~kll V

(

(28)

5

Phonom II. Thernwll'ropert;"1

123

/

V

, ,rogure 1 lIeat capaon

II

l

!I I

so thai the total phonon energy isU = 9NkB

T(~)' LXV dx ~8 e" - I

",..here N is tbe number of atoms in the specimen and Xv = BIT. The heat capacity is found most easily by differentiating tbe middle expressiOn of (26) with respect to temperature. ThenC"=

v

27ilti'kBT2

3Vh

2

I"b dw (e"w"'....0

e"wlT

T

1'1

= 9NkB _

(T)' LX" dx00

X4

(e" -

e"1)2

(30)

The Debye heal capacity is plotted in Fig. 7. At T J> 8 tbe heal capacity approaches the classical \laJue of 3Nk B Measured values for silicon and germaig. 8. nium are plotted i

..,Debye T3 LawAt very low temperatures we may approximate (29) by letting the upper limit go to infinity. We have

" 1 ' " - = d:u L L liw,n;a. A substantial proportion of all phonon (;ollisions will then be U processes, with the attendant high momentum erumge in the collision. In this regime we can estimate the thermal resisti\ity without p..1rticular distinction between Nand U pn)(.'csses; by the earlier argumcnt ahout nonlinear effects we cxpect to find a lattice thermal resistivity IX T at high temperatures. The energy of phonons KJ, K2 suitable lOr umklapp to occur is of the order of ikBfJ, because each of the phOlLOOS 1 mLd 2 must have wavcvcuors of the order of iG io order fOf" the collision (47) to be possible. If both phollons ha\e low K, and therefOre low energy, there is 00 \\.'ay to gel from their collision a phonon of wavevcetor outside the first wILe. 'me um)(tapp process must conserve energy, just as for the normal process. At low temperatures the number of suitable phonons of the high energy ik/!6 rCfJuired may be expected to vary roughly as exp(-612T). according to the Boltzmann rador. 111c exponential fonn is in good agreement with experiment. IlL summary, the phOllon mean

,

:;

Phot,o", II. TMnnai Proptrlin

131

i

" .3

10050

,

,Fig\IrE 18 n.crmal ronductlvity of a highly purir.cd crystal of SQd; um fluoriodc , after II. E. Jackson. C. T. Walkc.-, and T. F. McNdly.

,

"

" Xl

free path which enters (42) is the mean free path for umklapp c.'O lIisiolls belwccn phooolls and not for all collisions behveen phonons.Impe''/ecfio,1S

Ceometrical effccts may also be important in limiting the mean free path. We must consider scattering by crystal boundaries, the distributioll of isotopiC masses in natural chemical clements, chemical impurities, lattice imperfections, and amorphous structures. When at low tem peratures the mean free path f becomes comparable with the width of the test specimen, the value of f is limited by the width, and the thennal conuuctivity bcc.'Omes a function of the dimensions of the specimen. This effect was discovered by de Haas and Bierma5z. The abrupt decrease in thermal conductivity of pure crystals at low temperatures is caused by the sizecITed .

At low temperatures the umklapp pTOCCSS hecomes ineffec.'tive in limiting the thcrmal conductivity, and the size effect becomes dominant, as shown in Fig. 18. One would expect then that the phonon mean frcc path would be constant and of the order of the diameter D of the specime n. 50 thatK -CvD.(48)

The only tempcraturc-dependent term on the right is C, the heat capac.ity, which varics as r 3 at low temperatures. We expect the thennal c.'Onductivity to vary as r 3 at low temperatures. The size effect enters whenevcr the phollon mean free path becomes comparable with the diameter of thc specimen.

7

l13600~

K - O.06T' /

~

~n nd\t ~. Ile rc the density of modes is di.scontinuous. 2. Hrn.f thermal dilation 0 cnJ$tol cell. (a) Es ti m.ate for 300 K tllC root mean square / thcrmal dilation .6.VI Vfar a primitive cell of $Odium. Take the bulk modulus as 7 x 10 '0 erg an - 3. Note that the Ocbre temperature 158 K Is less than 300 K. so that the thermal energy is of the vnl.CI" of kilT. (b) Use this rCiul1 to estimate the root mean square thcrnlal fluctuation c.a1a of the lattice parameter .

3. 7Aro poin t lattice (li311i.acem,," t and . trai". (a) In the Oebye approximation, showthat the mean square displacement of an atom at ah salute wro is (R~ = I) is the velocity of sound. Start from the result (4.29) summcd O\'e r the independent la tt ice modes: (n~ - (tll2pV )}:w - I. We have included a factor of i to go from mcan squllre am plit udc to mean S(juare (lisplllCt'men t . (h) Show that };w- I amI (R~ diverge for a one-dimcnsionallattiee, bu t tl,at the mean square strain is fl nite. Consider (aruaxf) = IIKtllS as the mean square 5train, and show that it is L-qual to IIWbU4A1No 3 for a line of N atoms each of mass AI, counting longitudinal modes only, The divCI"gencc of nt is not Significant for any ph)1'icaJ measuremen t.3Ii~-n'pv3, where

4. Ileat copacjly o/layer lattice. (a) Conside r a d ielectric crystal made UI) of layers of atoms, with rigid coupli ng bctv."CCn laycrs so that the motion of the atoms is restricted to the plane of the Ia)'er. Show that tile phonon hmt capacity in the Oc hre approximation in the low temperature limit is proportional to T i , (b) Suppose ill~tcad. :u in many laye r structu res, that adjacen t layers are very weakly bound to each 0111(.'1", What form would )"011 expect Ihe pllOllon heat capacity to approach at extrL'rncly low temperatures?

-5. Gnmdsen constan'. (a) ShowtlUlt the free cnergyof a phonon modeclfrequcncy wis ksT In 1 sinh (1Iw12kIlT )j. It is nl'C'Cssary to retai n the zero-point energy illw to 2 obtain this result. (b) If C. is the fractional volume c1Jange, then ti le fi-ce ene rgy of the crystal may be written as

'"where B is the hulk modulu.. Assume II,at tI,e volume dependence of W)( is liwlw = - yd, where y is known as the Griinciscn constant. If'Y is taken as independent orthe mode K, SIIOW that F is a min imum with respect to l!. wi,en BIJ.. "" y}:i1iw (.'Otl! (hwl2k lJ T). and show that th is may be written in terms of ti le thermal energy dens ity as IJ. = yU(1')JB. (e) Show that on the Debyc model y "" -a In fJliJ In V. Note: Many approximations arc involved in this theory: the result (a) is val id only if w is independent of tempcrature; ymay Ix: qUite different lOr differentmodCli.

ReferencesR A. Cowley, Anharmonic cry.tals, Repl. Prog. Phys. 31 , pt . I, 123- 166 (1968).M

C. Leibrriral lraMilfom and me/tlng. NoordholJ, 1964. R. S. Krishnan, Thermal expcmsion of crysIol$, Plenum, 1980. A. D. Broce and H. A. CowlC)', Sfrucfurll/ pilau frlln.t/fioIlS. Taylor and Francis, 1981. M . Toda, Theory of nonlinear lGlticell, Springer , 1981.M

THERMAL CONDUCflVITY

Parrott and A. D. StnclcC$. TI,ernw/ cvndllcfh,-;ty of ~olids. Academic" Pre.s, 1975. P. C. K!cmellll, '"Thcrm."ll conducti'i ly and lattice vibration mode.:' Solid slate physks 7,1 -98 ( 1958); we also E"cvdo. of ,Jiysit;8 14, 198 (1956). C. Y. Ho, R W. Pov."ell and P. E. Liley, Thermal co"dllctivitv oflhe eumlfm l~; A CO/llprehensioo . review. J. of J'h.~'$. and a,,:m. Rd. Data, \ '01 3, SUJlPlemcnt I. H. r. Tre, cd., Thermal comluctivily, Acadcmk Pre". 1969. j. M . Ziman , Electrons and phoOIOnB. Oxford, 1960, Cllllptc r 8. H. Sennan, Thenual CO"dllction In solids, Oxford , U176. C. M. Bhandari and D . M. Howe, ThenlWIl oo"J"clion if, $emlco,ul"cl~, W iley, 1958.

J. E.

6Free Electron Fermi GasENERGY LEVELS IN ONE DIMENSION EFFECT OF TEMPERATURE ON TIlE FERMIDI.RACDISTRIBUTION FREE ELEcrRON GAS IN THREE DIMENSIONS 146 146

144

HEAT CAPACITY OF TIm ELECTRON CAS Experimental heat capacity of metals Heavy fermions ELECfRICAL CONDUCllVrry AND OHM'S LAW Experimelltal electriClll resistivity of metals Umklapp scattering MOTION IN MAGNETIC FIELDS Hall effectTIIERMAL CONDUcnVlIT OF METAI. S

151 155 156 1S6 159 162 163 164166

Ratio of thermal to eled:rical conductivity

166

NANosrnucruREsPROBLEMSL. Khlelic energy of e ltttron gas 2. Pressure and bulk modull15 of an electron gas 3. Chemical potential in two dimensions

168IG9

169 169 169169 170 170 170 170

4. 5. 6. 7. 8.

Fermi gases in aslroph}"sics Liquid lIe' Frequency dependence of the clectrieal conductivity Dynamic magnetoconductivity t('llsor for free electrons Coheshe energy of (ree eleclron Fermi gas

9. Static magnelooondudivity tensor 10. Maximum surface rfiistance 11. SmaIl metal spheres 12:. Density of states-nanometric wire 13. QuantiUltion of ronductanceREFERENCES

171 111 171 172 172172

1' ;&11 .11' I Schematic model of a crystaJ. of sodium metal. 1lIc atomic cores are Na ioll!; they are Immersed in a rea of c:onductlon clcd:rons. TIle (~o",:ludion electrons nrC deri\'ro from the 3s ,'tdcnce " lefCli na:upy a rdalj\'dy small plitt (- 15 IX'rccnt) of the total \-olumc " rthe cr}'SbI., but in a ooblc metal (Cu, AI:. Au) the ~omic rotC>; arC relat ively \uscI' and rna)' be in contact w ilh each oth. The < :ammon cry.tal structu,( ' n)(Mn Icmpcr.Jfllre is b&: {or the alkali metals and fcc {or the noble mcblb.

''' ---

CHAPTER

6:

FREE ELECTRON FERMI CAS

In a theory which Iws gicen n .. ,lt.l like tl,ese, there mlt3f certainlll be a great deal cf trull,. H. A. Lorentz

We can understand many physical properties of metals, all(lnot only of the simple metros, in terms of the free electron model. Acoordillg to this model, the valence electrons of the constituent atoms become conduction electrons and move about freely through the m lume of the metal. Even in metals for which the free electron model .,\lorks best, tile charge distribution of the (.'o ll(ludion electrons relleds the strong electrostatic potelltial of the ion cores. The utility of the free electron model is greatest ror properties that depend essentially on the kinetic propcrtk'S of the conduction elC(.,trollS. The interaction of the conductiolL electrons with the ions of the lattice is treated in Chapter 7. The simplest metals are the alkali metals-lithium, sodium, pot ssium, cesium, and rubidium. In a free atom of sodium the \alell(.'C ciectroll is in a as state; in the metal this electron becomes a conduction electron. We speak d the as conduction band. A monovalent crystal wh ich contains N atoms will have N (.'onduetion electrons and N positive ion cores. n le Na+ ion core COIltains to electrons that occupy the 1$, 21, and 2p shells of the frec ion , with a spatial distribution that is essentially the same whell in the metal as in the frec ion. The ion cores fill only about 15 percent of the volume of a sodium crystal, as in Fig. 1. The radius of the free Na+ ion is 0.98 A, whereas one-half of the ncarest-neighbor distan(.'C of the metal is 1.83 A. The hlterpretation of metallic properties ill terms of the motion of frcc electrons was developed long before the invention of quantum mechanics. TIlC classical theory had se\'eral conspicuous su(.'(.'Csses. notably the deri\'ation of the fo nn of Ohm's law and the relation between the electrical and thennal conducthity. 'Ine classical theory fails to explain the heat capacity and the magnetic susceptibility of the condudion electrons. (These arc not fuilures of the frcc electron model, but fai lures of the Maxwell distribution function.) There is a further difficulty. From many types of cxperiments it is clear that a conduction electron in a metal can mo\'e freely in a straight path over many atomie distances, ulldeflccted by collisions with other conduction electrons or by collisions with the atom cores. In a vcry pure specimen at low temperatures the mean frec path may be as long as lOS inte ratom ic spaeings (more than 1 em). \Vhy is condensed matter so trallsparent to conduction electrons? 111e answer to the q. ion contains two parts: (a) A conduction electron is not

...deflected by ion COTes arranged on a lJeriodic lattice because matter waves prop.1gah: fr'-'ely in a periodic structure. {b} A conduction e lectron is scattered only infrequently by other conduction electrons. This property is a consequence ofthe Pauli exclusion principle. Bya free electron Fermi gas, we mean a gas of free electrons subject to the Pauli principle.

ENERGY LEVELS IN 01'\E OU.IENSION

Consider a free electron gas in one dimension , taking account of quantum theory and of the Pauli principle. An electron of mass m is confined 10 a length L by infinite b.'lrriers (Fig. 2). TIle wavefunction ",,,(x) of the electron is a solution of the SchrOdinger equation 11/1 = ttl/!; witll the neglect ofpotentiaJ energy we have 1 = ,,212m, where p is the momentum. In quantum theory 1) may be re p." csented by - ill (lIdx, so that(I )

where E" is the energy of the electron in the orbital . \Ve use the term orbital to denote a solution of the wave equation for a system of only one electron. Tbe term allows us to distinguish between an exact quantum state of the wave equation of a system of N electrons and an approximate quantum state wbich we construct b)' assigning the N electrons to N different orbitals, where each orbital is a solution of a wave equation for one electron. lhe orbital model is t!xact onl), if the re are no interactions between electrons. TIle boundar)' conditions are 1/1 .. (0) = 0; I/I,,(L) = 0, as imposed b)' the infinite potential energy barriers. They are satisfied if the wavefunction is sinelike with an integral number" of half-wa\'elengths between 0 and L:

"'~ = A sin(~7T.r)"d~. _ - - - A - - "" (nw) . - - r cb: L L 'whence the e nergyE"

;

inA" = L ,

(2)

where A is a constant. We see that (2) is a solution of (I), becnuse

(nr.)

d 1/1" = _ A(nw)2 sin

2

dr

L

(~l")L

is given byE ..

=

!!:....( nw)' L 2m

(3)

We want to accommodate N electrons on the line. According to the Pauli exclusion principle no two electrons can ha\'e all their quantum numbers idcn-

~ - -E",, is defined by

(38)

This fOTm arises in a natur'dl way because ~I' is im'crsely proportional to the mass of the electron, whence;, ex m. Values of the mHo are given i,\ Tablc 2. The departure from unity iuvolves three scparalt: effects; The interaction of the conduction electrons with the periodic potential oflhe rigid crystal lattice. The effective mass of an electron in this potential is called the band effective mass and is treated later. The inlcmction of the conduction e lectrons with phonons. An electron tends to polarize or distort the Janice in its neighborhood. so that the moving e1~ tron tries to dnlg nearby ions along, thereby increasing the effective mass of the electron. The inler-dction of the conduction electrons with themselves. A moving elechun causes an inertia] reaction in the surrounding electron gas, thereby increasing the effective mass of the electron.

Heavy Jiermiom. Several metallic compounds have been discovered that have enormous values, two or three orders of magnitude higher thlm usuul, of the electrollic heat capacity. The heavy fcnnion compounds include UDe I 3. CcAI:J, and CeCu2Si2' 11 has been suggested that f electrons in these compounds may have inertial masses as high as 1000 m, because of the weak o\'erlap of wavefunctions off electrons on neighboring ions (see Chapter 9, "tight binding'/. References are given by Z, Fisk. J. L. Smith, and H. R. Ott. Physics Today, 38, S-2O Uanuary, 1985). The heavy fermion compounds form a class of superconductors k"11o ...m as "exotic superconductors."ELECTRICAL CONDUCTIVIlY AND OHM'S LAW

TIle momentum ofa free e lectron is related to the wavevcctor by "I\' = hk. In an e lectric field E and magnetic field D the force F on an electron of charge -e is - e[E + (llc)\' x DI. so that Newton's seoond law of motion becomes (CCS)

F:: m~ = h~ :: -e(E + ~\, D) . dl dl cX

(39)

In the absence of collisions the Fermi sphere (Fig. lO) in k space is displaced at a ullifol"ln ratc by a constant applied electric field . We integrate with D = 0 to obtaink(l) - k(O) "" - eEIII! .(40)

rLI163 0.749

B.0.17

Table 2 (From

E:lperimenlal and free electron values of electronic: heat capacity constant 'Y of metals kindly fll rni'hcd by N. Philli!,S !

SUrting liP.

.

:.

~. ~J', ,

++

scdion i. placro ill a magnetic field B

Figure 14 '111C ~Ialldard geometry for the n ail effect: a rod-shaped Ipccimcn of n!dangulal' CI"QIO'll$ in (a). An clo.'dric field E. applied across t he end elect rodes aiOu:lCS ..., electric current d ensity J. to flow down the rod. The drift ,""Iocit y of the negali,-cly-charged elect...,,,. immed iately aft; applied as !.L.own in (h). The deflection ill the Ii direction ii I;lIUSCd by the magllclic field. E~mlli accumulate OIl o nC,' bee of the rod and. posit ive iol1 ('Keen i~ cstablid>ed Oil the opposite ~ u nlil , as in (el, the Irru15\-CI"SC elect r iC fiekl (ll all Ileld) just cancel. the Loren tz ror~ dne to the magnetic fleld.

This is negative for free electrons. for e is positive by definition . The loy,>er the carrier concentration, the greater the magnitude of the Hall coefficient. Measuring H, / is an important way or measuring the carrier concentration. The symbol RII denotes the Hall coefficient (54). but it is sometimes used \\i th a d ifferent meaning. that of Hall rcsi5tance in two-dimensional problems. When we treat such problems in Chapter 19. we shall instead letPII:=

BRit ;. E,jlx

(55a)

denote the Hall resistance, where j~ is the surface current density. TIle simple result (55) follows from the assumption that all rclaxation times are equal, independent of the velocity of the electron. A numerical factor of order unity entcrs ifthc rclaxation time is a function ofthc velocity. The expression becomes somewhat morc complicated if both electrons and holes contribute to the conductjvity. 111e theory of the Hall effect again becomes simple in high magnetic fi elds such that WeT" 1, where We is the cyclotron freq ue ncy and T the relaxation time. (See Q1"S. pp. 24 1- 244.)

..

".In Table 4 observed valnes ohhe Hall coefficient arc compared with values calculated from the carrier concentration. nlC most accurate measurements are made by the method of helicon resonance which is treated as a problem in Chapter 10. In the table "cony." stands for 'conventional." 'Inc accurate values for sodium and pota~sium arc in excellent agreement with values calculated for onc conduction electron per atom, using (55). Notice, however. the experimental values for the trivalent elements aluminum and indium: these agrre \\ith values calculated for onc positive charge carrier per atom and thus disagree in magnitude and sign with values calculated for the expected three negative charge carriers. TIle problem of an apparent positive sign for the charge l'3rricrs arises also for Be and As, as seen in the table. The anomaly of tile sign was explained by I'eierls (1928). 11lC motion of carriers of apparenl positive sign, which lIeisenberg later calletl "holes," cannot be cxplainoo by a free electron gas, hut finds a IMlural explanation ill terms of tile (;nel"gy bllld theory developed in Chapters 7- 9. Hand theory also accounts for the occurrellce of \'ery large values of the Hall cocfHcient, as for As. Sb, ami Hi.

THERMAL CONDUCflVITY OF METALS

In Chapter 5 \"C found an expression K "" led for tile thermal conductivity of particles of velocity v, heat capacity C per unit volume. and mean free path t. 'nle thermal conductivity of a Fermi gas follows from (36) for the heat capacity. and with t:F = im4:(56)

Here l = VF"T; the electron concentration is n, and "T is the collision time. Do the electrons or the phonons carry the greater part of the heat current in a metal? In pure metals the electronic contribution is dominant at all temperatures. In impure metals or in disordered alloys. the electron mean free path is reduced by collision ,\oith impurities. and the phonon contribution may be comparable with the electronic contribution.Rotio of Thermol to Elect,.ical Cooductivity

The Wiedemann-Franz law states that for metals at not too low temperatures the ratio of the thermal conductivity to the electrical conductivity is directly proportional to the temperature, with the value ofthcoonstant ofpl"oportionality independent of the particular metal. This result wa~ important in the

(

6

Frre t:krtron Fu-mi C...

16

Table 4,

Comparison of observed Hall

~mcients

with free eled-ron theory

{The experimental v"lues of RII as obtained by OOIlventional methods are Summarized trom data at room temperature presented in the Landolt-Bomstein tables. The values obtained by the helicon wa ... e metll(l(l at 4 K are by J. M. Goodman. The values of the carrier concentration n are from Table 1.4 except for Na. K, AI, In, where Goodman'li values arc used. To l.'OIlvert the lI1IIuc of IIH in CCS units to the value in voItcmlampgauss, multiply by 9 X 1011; to con\'('rt fill in CCS to m'/cou:omb, multiply by 9 X

JOI3. J

Metal

Method

"., in 10- 1-1 CCS ,olits- J.89 - 2.619 - 2.3- 4.~

uperirr.."'la\

A~umt:d

C..lt:ulaloo- J/n.

carrie" per ~ t om 1 e lectron 1 elt!(;trOIl 1 electl'Oll 1 electron 1 e lectron I electron 1 electron

in 10- '" CCS units - 1.48

U

con .... helicon conv. helicon conv. conv. conv.~" ,,",W,

N.K

-2.603 - 4.9- 6.04- 0.82 - 1.19

- 4.7

RhCuAg

- 5.6

- 0.6- 1.0 - O.S +2.7- 0.92

AuBeMg

- US

conv.~" .

Al10

helicon helicon COny.~".

+1.136 +1.774

1 holc 1 holc

+1.135 +1 .780

'" Shm

+50.- 22.

~" .

- 6000.

J..

Table 5L X 10" WlIu-ohmldci" Metal

Experimental LoI'Cm: numbersL )( If/' Wlltt-ohmltlcg"-

O

Uc cos ex =

AaL 8(x - sa) ,

(33)

whcre A is a constant and a the lattice spacing. Thc sum is over aJl integers s between 0 and 1/(1. The boundary conditions are periodic over a ring of unit length, which means over lIa atoms. 1110S the Fourier coefficients of thc potential arcUc =

LI

dx U(x) cos ex

=

A(I~

LI

dx 8(x - sa) cos ex(34)

"" AaL cos Gsa"" A2-J"his tn... tment"-all

suggested by Surjit Singh, Am.

J.

pil)'I . ~I, 179 (1983).

188

We write the central equation with k as the Bloch Index. lllUs (31) becomes(Ak - E)C(k)

+ iL C(k -

21m/a) :: 0

(35)

"where Ak fl 2k'lf2m and the sum is over all integers n. We want to solve (35) for E(k). We define (36) I(k) C(k - 2mJ.) ,

L"

so that (35) becomes

C(k) = _ (2mAlh~I(k) k'1- - (2f)lE/fr'lJBecause the sum (36) is over all coefficients C,\o\-"e

(37) have. for any n,(38)

I(k) - I(k - 2mJ.) .-nils relation leis us write

C(k - 2mJ.) ~ -(2mAlh')I(k)[(k - 2~o1.f - (2m"h~I-'

(39)

We sum both sides over all n to obtain, using (36) and cancellingf(k) from both sides.

(h'/2mA) - 'OIC

L [(k "

2mJ.)' - (2m"h~r ' .

(40)

sum can be carried out with the help of the standard relationctnx""L..

1017"+%

.

(4 1)

After trigonometric manipulations in which \\'e use relations for the difference of two cotangents and the product of two sines, the sum in (40) becomes

if sin Ko.4Ka{cos ka - cos Ka) where \.\.avevector!; happen to be given outside the first zone, they are carried back into the first zone by subtracting a

7 E"crgylland$

169

suitable rcciprocallattice vector. SlIch a translation can always be found . 'Ille operation is helpful in visualization and economical of graph paper. When band e nergies are approximated fairly v.ell by free electron energies lk :: h~k2l2m, It is advisable to start a calculation by carrying the free electron energies back into the first wne. l 11c procedure is simple enough once you get the hang of it. \ Ve look for a G such that a k' in the first wne satisfies

k'+G = k .where k is unrestricted and is the true free eledron wavevector in the empty lattice. (Once the plane wave is modulated by the lattice, there is no single "true" wavevector (iw the state l/I.) If we drop the prime on k' as unnecessary baggage. the free electron energy can always be written asl(k~,kl/ .kz) = (h 2 I2m)(k

+ G)2

= (h2I2m) [(k... + G...)2 + (kl/ + GIl + (Ie" + Gz)2)

with k in the first zone and G aH ov.ed to ron over the appropriate reciprocal lattice points. We consider as an example the !OY.-Iying free electron hands of a simple cubic lattice. Suppose we want to exhibit the energy as a function of k in the (looJ direction . For convenience, choose units such that ft212f1l == I. We show several low-lying bands in this empty lattice approximation with their energies l(OOO) at k = 0 and l(k...oo) along the k... axis in the first zone:

...I

~

Cal21T000 100,100 O1o,oio,OO I ,OOI 11 0,101, 110,101 JIO,Io l ,TIo,101 OIl,OII,OIi,OTI 0

( (000)

t (k,OO)

2,3 4,5,6,7 8,9, 10, 11 12, 13, 14, 15 16,17, 18, 19

(2,",0)1

(27Tla)lI 2I..27T10f 2I..21Tlo)1 2(21'Tlo)t

" 1..;+

(k... :t 2mof (2mo)lI (k... + 2maf + (27TJO)' (k. - 2ma)1 + (27T1a'f I..~ + 2(21'T/o)t

1l1 free elcctron bands are plotted in Fig. 8. It is a good exercise to plot the ese same bands for k parallel to the [II J) direction of wavevcctor space.Approximate Solution Near a Zone BOIItlMry

\ Ve suppose Ihat the Fourier components Uc of the potential energy are small in comparison with the kinetic energy of a free electron at the zone boundary, We first consider a wavevector exactly at the zone boundary at ie, that is, at wIn . Here ,(k -

cl' =

(IC - C)' = (IC)' ,

100

L __

--=:::,_-;,-_""'---__-,!0 ..

i._

"i"

Jo'jgure 8 Low-Iying free electron energy I~ the emjlty Ie laUice. "5 translOrmed to the fln! Brillouin ZOne ~nd plotted n. {k.OO). The rree electron energy is "'(Ie + C),f2m, where the C'I are gi"eJl in the second ro/umn of the table. The bold curves are in the fin;t Brillouin zone, wilh -frla:iS Ie. :s mao Energy bands drllwn In this way llTe uK! to be in the reducec:l :rone $Cherne.

a

so that at the zone boundary the kinetic energy of the two component waves k = ::t Ie arc equal. If C(jG) is an important coefftcient in the orbital (29) at the zone bound~ ary. then C(- iG) is also an Important coefficient. This result also follows from the discusSion of(5). We retain only those equations in the central equation that contain both coefficients C(IC) and G(-iC), and neglect all other coefficients. One equation of (31) becomes, with k = IG and ,\ - li~tG)2/2m,(> - .)C(IG)

+ uC(- IG) + Uc(IG)

= 0

(44)

Another equation of (31) becomes(> - .)C(-IG)

= 0

(45)

These two equations have nontrivial solutions for the two coefficients if the ener'b'Y E satisfiesA- E

U

I =0

(46)

7

Energy Bond.

191

whence,

E = A U = _(!G)2 U .

h'

2m

(47)

The energy has two roots, one lower than the free electron kinetic energy by U. and one higher by U. Thus the potential energy 2U cos Gx has created an energy gap 2U at the zone boundary. The ratio of the C's may be found from cither (44) or (45):

-"C~(-",I'O'Gcc ~ ) C(IG)

_E_-_,

U

~ + 1 -,

(48)

where the last step uses (47). Thus the Fourier expansion of I./{x) at the zone boundary has the two solutionsI{I{x) = exp(iGx/2) exp( - iG112) .

These orbitals are identical to (5). One solution gives the wavcfunction at the bottom of the energy gap; the other gives the wave function at the top of the gap. Which solution has the lower energy depends on the sign of U. We now solve for orhitals with wavevector k near the zone boundary iG. We use the same two-component approximation, now with a wa,efunction of the form I./{x) = C(k) e lh + C(k - G) ei(k - G)~ . (49) As directed by the central equation (31), we solve the pair of equations

('. - E)C(k)(Ak_C -

+ UC(k - G) ~ 0 E)C(k - G) + UC(k) =

;0E

with Ak defined as h 2J(l12m. These equations have a solution if the energy satisfies

Ak _ C

U

-

E

I

- 0

whence ~ - E(Ak- G + A.v + Ak_CAk - U 2 = The energy has two roots:

o.(SO)

and each root describes an energy band, plotted in Fig. 9. It is convenient to expand the energy in terms of a quantity j( (the mark over the j( is called a tilde), which measures the difference k !!!!!! k - iC in wavevector between k and the zone boundary:EI;:=(h2/2m)(lG 2 + j(2'j [4A(h 2k 2/2m) ""' (h2/2m)(lG 2 +

+ lPl'f2 k 2) U[l + 2(AllP)(1i2j(2/2ml]

(51)

in the region 1i2Cltf2m onS ofr-;o) In the

"'Im ..

Writing the two zone boundary roots of (47) as E{:t), we may write (51) as(52)-nlCSC

are the roots for the energy when the y,oavevcctor i5 very close to the zone boundary at iG. Note the quadratic dependence ofthe energy on the wavevedor iand can also be ell:actly filled .

Metal. and ,,,,ulatoTiIf the valence electrons exactly fi1l one or more bands, leaving others empty, the crystal will be an insulator. An ell:temal electric field will not cause current flow in an insulator. (We suppose that the electric field is not strong enough to disrupt the electronic structure.) Provided that a filled band is separated by an energy gap from the next higher band, there is no continuous way to change the total momentum of the electrons if every accessible state is filled. Nothing changes when the field is applied. 111is is quite unlike the situation for free electrons for which k increases uniformly in a field (Chapter 6).

'94

o

1(0)

-,;

H

o

1- -

,b ,

Figure II Oo::\'pied stales and band slruclun,s giving (II) An lruuw.t..... , (bJ. mdal or Asemimctal bt.>' part or Ip[;(1< + G) d c

(8)

The expectation value of the momentum of an electron in the state k isPd = (Ie to the hole if we describe the \U..-.ce hand a.'I occupied hy one hole. 'rhus k. - - k.; the wlw",..,.,torofthe hole is the """'"' as the wa\l~-ectoror lhc electron which remains a' C. For the enlire s>'Slem the tola1 WlI\'evector-afier the absorption of the photon is k. + Ie., - 0, so thai Ihe lotal wavcv, Is :r.ero F because dtldl:, - O. (hI An electric field E, is applied in the ~:r. dirt-dion. The on the electrons Is in the - It, direction and all electrons make transitions together in the - k.t\iredipn, ffiOYing the hole the stale E. (el After a further ,n'ervallhe electrons mO\'c tanher along in k spaee lind the hole ;s now at D.

ron:e

a,

Figure 10 Molioo of e"-'droos In the conduclioll oond and holesin the valence I.,>d in the eklime seal." for Icmpt'r .ltu.-c k"T d a dono.- because "h..,., ionired it donates an elcrtron to the coodUCIioll band.

The application to germanium and silicon is complicated by the anisotropic dfective mass of the conduction electrons. But the dielectric constant has the more important effect Q the donor energy because it enters as the square, n whereas the effecti ve ma~s enters only as the first power. To obtain a general impression of the impurity levels we usc m~ - 0.1 m for electrons in gennanium and m. "., 0.2 m in silicon. TIle static dielectric constant is given in Table 4. The ionization energy of the free hydrogen atom is 13.6 eV. For germanium the donor ionization energy Ed on our mode l is 5 meV. reduced with respect to hydrogen by the factor mJml "" 4 x 10- 4 TIle colTesponding result for silicon is 20 meV. Calculations using the correct anisotropic mass tensor predict 9.05 mcV for gCl1113nlum and 29.8 meV for silicon. Observed values of donor ionization e nergies in Si and Gc arc givcn in Table 5. Rec".I11 that I meV iii! 10- 3 eV. In eaAs donors have Ed - 6 meV. l 1le radius of the first Bollf orbit is increa~cd by ~mJIlIc ovcr the valueTable"Crystal

StatU:: relative dieleCIric COn!ltant

or semiconductorsCrySlll1

11 .7 15.8

15.69 13. 13 10. 1 10,3 10.2 7.1

Diamond 51 Cc InSbI"", I,P

c..sb

17.8814.55 12.37

.'"AISb SiC

GaA,

-

CUtO

Table 5 Donor ionizatm energies Ed of pcnta"alent impurities in germanium and silicon, in mcVp

Sb39. 9.6

s;Co

45. 12.0

0.53 A for the free hydrogen atom. The rorresponding radius is (l60XO.53) "'" 80 Ain germanium and (60XO.53) "'" 30 Ain siliron. 'IneS! are large radii. so that donor orbits overlap at relatively low donor concentrations, compared to the number of host atoms. With appreciable overlap. an "impurity band" is fonned from the donor states: see the discussion of the metal-insulator transition in Chapter 10. 'Ine semiconductor can conduct in the impurity band by electrons hopping from donor to donor. The p rocess of impurity band ronduction sets in at lower donor concentration levels if there are also some acceptor atoms present, so that some of the donors are always ionized. It is easier for a donor electron to hop to an ionized (unoccupied) donor than to an occupied donor atom, so t hat two electrons wiU not have to occupy the same site during charge transport. Acceptor States. A hole may be bound to a trivalent impurity in germanium or silicon (Fig. 20). just as an electron is bound to a pentavalent impurity. Trivalent impurities such a.~ B. AI , Ga, and In are caJled acceptors because they accept electrons from the valence band in order to complete the covalent bonds nith neighbor atoms, leaving holes in the band. When an acceptor is ioni7.ed a hole is f~, which requires an input of energy. On the usual energy band diagram, an electron rises wh en it gains energy, wflereas a hole sinks in gaining energy. Experimental ionization e nergies of acceptors in germanium amI silicon are given in Table 6. The Bohr model applies Qualitatively for holes jU6t as for electrons, but the degeneracy at the top of the valence band complkates the effective mass problem. The tables show' that donor and acreptor ioni7.ation energies in 5i are comparable with kBT at room temperature (26 meV), so that the thermal ioniz."1tion of donors and acceptors is important in the e lectrical conductivit}' of silicon at room temperature. If donor atoms are present in considerably greater numbers than acceptors, the thennal ioni7.ation of donors will release electrons into the conduction band. The conductivity of the specimen Ihen will be controlled by electrons (ne~}1tive charges), and the material is said to be n type. If acceptors are dominant, holes will be released into the valence band and the conductivity will be controlled by holes (positive charges): the materia] is 1 }

8

Snnicond.. cto~ Cr!flllaU

,Table 6 Acceptor ionization cncrgiC$ E. of trivalent impul'ities in gcnnanium and silicon, in mcV

Si Go45. lOA

AI

G65. 10.8

"157. 11.2

57. 10.2

Att"4'pIor b...."d .......L

- --- ---- E..j

0 ,

Figure 20 Boron h as only three ~ el

= (}.t -

Eo> + ikB'l)'e

(58)

and is positive. Equations (56) and (58) are the result or our simple drift velocity theory; a treatment by the BoIt7.Jnann transport equation gives minor nume ri cal differences. 4 The absolute thermoelectric power Q is defined from the open circuit . electric field created by a temperature gradient:E~Qll"'dT

.

(59)

The Peltier < :oefficient n is related to the thermoelectric power Q by

l1 =QT.

(60)

This is the famous Kelvin relation of irreversible thermodynamics.S A measurement of the sign of the voltage across a semiconductor specimen, one cnd of which is heated. is a rough and ready wa)' to tell if the specimen is n type o r p type (Fig. 23).SE"'UMETALS

In semimetals the conduction band edge is very slightly lower in energy than the valence band edge. A small overlap in energy of the conduction and valence bands leads to small concentmtion of holes in the valence band and of electrons in the conduction band O'able 7). Three of the sem imetais, arsenic, antimony, and bismuth, are in group V of the periodic table. 'Their atoms associate in pairs in the crystal lattice, with two ions and ten valence electrons per primitive cell. 'The even number of valence electrons would allow these elements to be insulators , Like sem iconductors, the semi metals may be doped with suitable impurities to vary the relative numbel'S of holes and electrons. TIleir concentrations may also be varied with pressure, for the band edge overlap varies \\lith pressure,SVPERI.ATIICES

Consider a multilayer crystal of alternating thin layers of different compositions. Coherent layers on a nanometer thickness scale may be deposited byOR, A. Smith, Stmiconducton, 211d ed.; Cambr idge, 1978; .... Frltzscbl", Solid State Coro ln ull . 9, 1813 (197l). A simple discussioo or Boltzmann t ransport theory is gh'ell in Appr.ndi~ F. 511 . B. Callen , Thermc" 1965.

(

,,,..d

"'\

,, I , "." , ""']" ,

..\iJ"(\n~)

-Q

-Q

~ .J.

-,

...

""

.., ,.,

T.... "pe .... u...,. in K

.. ....""(2. 12 0.01) X JcPO (5.49 a.m) X JOIIl3.00 X 1017

Figure 23 P~hi~r oocfficien t of n and p silicoo .5 II rUllc l ion of temperature. Above 600 K the ...,ecimcns ad as intrinsic SropIe lcoulel define a ,,,etal a, Ma ,oIUllCjlh a Fermi Iturface." 1'hi" may Iln;.erlhetelill be Ille m~1 mtllllj"gful lIefiJli/ivJI of a me/al olle C(l1I si~e loday; il represell" a profoulI(i (lClt'(lllce ill liIe uullenta.Klillg of ICilY melals /HiIa!:e IiII ,lley do. 1'Ile OO'lCejl / of Ihe Fermi :mrface, fI& (le.;eloped by quun'"f11 physic" 1Jroddes a preciu errJlanalwm of Ih e maln p/lysicai propel'Ijes of melals.

A. R. Mackintosh

1110 Fenni surface is the surface of constant ellerb'Y t' ill k sp....t'e. TIle Fermi surr., ce sepnrates the unfilled orbitals from the filled orbitals, at absolute zero. TIle electrical properties of the metal are determined by the shape of the Fermi surf.'l.Ce, becausc the current is due to changes in the occupancy of states near thc Fermi surface. 11)c shape may be very intricate ill a reduced zone scheme and yet h:lVe a simple interpretation when reconstructed to lie near the surface of a sphere. We exllibit in Fig. 1 the free electron Fermi surfat'est'()nstructed ror two metals that have the race-centered cubic crystal structure: coppcr, with one valence electron, MId aluminum, with three. The rree e lectron Fenni surfaces were developed rrom sphe res of radius kt' detennined by the valence electron concentration. How do we constnlct these sllrL'l.Ces rrom a sphere? The t"Onstructiolls require the reduced and pt:riodic zone schemes.Reducetl Zmle SchemeIt is always possible to select the wa"evector index k or any Bloch runction to lie within the first Brillouin zone. 'nle procedure is known as mapping the balld in the reduced zone scheme. If we encounter a Bloch function written as tf.\.(r) = e1 k'I/k(r), with k' outside the first zone, as in Fig. 2. we may always find a suitable reciprocal lattice "ector G ~uch that k = 10;' + G lies within the first Brillouin zone. ThenI/Ii,.(r) = e'k"'udr) = ti....(e- 'cu".(r iil

e""'uk(r) ... I/Ii,(r) ,

(I)

where "k(r) .. e- IG'uk.(r). Both e- .c :md u".(r) are periodic in the crystalla.ttice, so u,,(r) is also, whence "'J..~r) is or the Bloch rorm . Even with rl'ee electrons it is useful to work in the reduced 7..one M:heme, as in Fig. 3. Any 'rgy k' for k' outside the fil"51 'Wile is equal to an " in the

(f

flrst wn.,

Figure 2 First Bril100in zolle d a square lattice or side d . 'l1le ""lI\", ..n . e(or 10:' can be cwried into the IJy formillg k' + C . The W3'"C\'------C~--ok--or((.- 10- :1' eV, 01' 8 > tOOO G.

E :.

SUMMARY

A Fermi surface is the surface in k space of constant energy equal to t:,... The Fermi surface separatL'S filled states from empty states at absolute zero. The form of the Fenni surface is usually exhibited ~t in the reduced zone scheme, but the connectivity of the surfaces is clearest in the periodic zone scheme. An energy band is a Single branch of the q, venus k surfat-e. 11le cohesion of simple metals is accounted for by the lowering of energy of the k = 0 conduction band orbital when Ule boundary conditions on the wavefunction arc changed from Schrlklingel' to Wigner-Seit-. The periodicity in the de Ha:ls-van Alphen effect mcasurL"S the extremal crO:'is-section area S in k spat.'C of tJle Fermi surfat'C, the cross section being taken perpendicular to 8;,,(-'-) ~ 2w, .B lIeS

I R W. Slark and '- M. Falicov, Hp.,bgnc.1.ic Breakdown In Metals. Hin low tm.pn-oIu~ physIcs, Vol. V, ,,"orth Holland, 1967. pp. 23&-256.

'"ProblemsI. Brillouin WI'IU of rrdllngulor lattice. Make a plot of Ihe fin! two Brillouin zones of a IJrim itivc J'(.'(.1angular twu-dhncnsiooal luttice with ruces G, b - 36.

2. Brillouin ume, rtangular lattice. A two-dimensional metal has one atom of \111.Icney onc in a simple rt!(;tangular prilllith,.;: cell a - 2 A. 1 - 4 A. (0) Ornw the first , Brillouin zone. Give its dimensions, in em - I , (1)) Calculate tIle nldius of the free clcd:ron Feml! sphere, in em - I, (c) Omw this sillierI.' to SC'.uc on a drawing of the first Brillouin 7Al11C. Make another sketch to show the first few periotls of the free crt.:clron band in tl}c periodic zone scheme. fur both thc first and sl'COnd ene rgy bands. Assume there is a small cnCIb'Y gap al the zone boundary.

3. Ile:tagonal clore-packed structure. C,.onsidcr the fin! Brillouin 7.onC of a cr}'lital with a silnplc hexagonal lattice in thrL'C dimensions wilh IlIltice constant.I~m111

.

J

f"(O)

fj.O)

(40)

The mass at tll'fX'Ior is pure imaginary of mlgIlitude g1~~ D)' the brok.eo!me in lhe figun:. In Ihe gap the",.,.., I Uenuates as eJp(-IK\.t). and ".., see from lhe plot that ~ allmnalien is much stl'Orlgel" near ..... than ne.."'r... The d"tr.II:te' ort},., bmncIoes wrie$ with K. there il;. region of mixl':d "le('lric-mechanicalaspeds Ilear the nomhUlI crosson', Note , fonally, II Is Intuitively OO"ious that th., group ve locity oI' light in th" medIum ill always Ciden' electromagnetic WII\ "et oHrequencies """ < OIl < "'r. will nol p~te in the medium, but " "ill be n:(]eed at the bou ndary.

' ,", .. , ,

,

,

,- I

0

,

- I0

-.,

..

~.~.W

-

t-""!"""")" in lit

. . ..

..

'~~

,~

" Igure 13b I);ekclric function (rea1 IN-rt) ofSrF. me~uud O'o"er a wide frequency 1"lIlge. exhibiting the dOXTea5e of the ionic polanu.billl), at high rreq...:nc~ {A. 11011 lIippd.}

(

E

.- ,L,ro phc....,

f-

............

OlIia,

I I I

,I I I I

I

.,

I

r=I."ngil.,.]ina! V opt;...!

I Tr..l'I'~

,

,hmoo

I , , I

12..5

'"

~. 6

.,

"

Fizure 16 R"flectarK'e ,.,nus wa".,kngth cli UF film bacl The ohject of the theory is to give a unified account oCthe effect ofintcrnctions. A Fermi gas is a system of no ninteracting fe nnions; the same system with interactions is a Fermi liq uid. lAndau's theory gives a good account of the low-lying single particle excitations of the system of interacting electrons. These Single particle excitations are called quasiparticles; they have a one-to-one correspondellce with the siug1e particle excitations of the free electron gas, A quasiparticle may be t hought of as a Single particle accompanied by a distortion cloud in the electron gas. One effect of the coulomb interactions between electrons is to change the effective mass of the electron; in the alkali metals the increase is roughly of the onler of 25 percent. Electron-Electron CoUision!. It is an astonishing property of meta ls that cond uction e lectrons, although crowded togethe r only 2 A apart, trave l long d istanecs octween collisions with each othe r. 111e mean free paths for e1cctronelectron collisiollS are longer than 104 A at room temperature and longer than 1OcmatlK. Two fuctors are responsible for these long mean free paths, without which the free electron model of metals would have little value. 'The most powerful factor is the exclusion princi ple (Fig. 17), and the second factor is the scr eening of the coulomb interaction between two electrons. We show how the exclusion principle reduces the collision freq uency of an electron that has a low excitation e ne rgy E I outside a filled Fermi sphe re (F ig. 18). We esti mate the effect of the exclusion principle on the two-booy collision 1 + 2 _ 3 + 4 he tween an electron in the excited orbital 1 and an~ L Landau, '-I'he, m .. 10 I'J g. the ",elocit y is - lOS on s I . Alf"e ll w:wes h.we been observetl in scmimetals anti in clectrol1-llOle (Irops in germaniullI (Chapter II).4. lie/icon IOOVU. (a) Employ the method

Problem 3 to t reat a specimen with ooly one carrie r tytle, SIlY hole s in concentr.ttion p .and in the li mit I 0) = I . 1fie system is a shee t of copper; the fieldis applied normal to the sheet. Include the damping. Sol,'e the differential t,'Quation by elementary methods. -II. Cop pill.7mOrl$lmd tile carl del" Wooh inleracliorl. Consider twosemi-inflnite media with pla ne surfaces.t '" O,d. 'nle dielectric function of the identical media is E(W). Show that for surface plasmons symmetrkal with respect to the gap the frt,'Quency n)ust satisfy t:{w) = -tanh (Kd/2), where K2 - Po + ~. The electric potential will have the form 'P - f (%) exp(ik,x + ik.}J - lu) .

Look lOr nonretarded solutions-that is, solutions of the Laplace t,'Quation rather than of the wave equatioo. The sum of the zero-point energy of all gap modes is the nonretllrded ~t of the van der Waals attraction Letween the two specimens- see N. C. van Kampen , B. R. A. Nijboer, and K. Schram, Physics Letters 26A , 3ffl (l968).

ReferencesJ. N. 1I0dp0n, OptictJ aWorpl iotlllnd d~ In 1Ol;d,s, Chal'm"l1 "lid Hall . wand & w do not contribute mILch because the fUllction In 1 + w)J(& - w~ is small in these regions. (& /IIlltlwma fical Note. To obtain the Cauchy integral (10) we take the integral f a (s)(s - wt l ds over the contom l in Fig. 2. 111e function a(s) is analytic in the upper halfphme, so that the ,a]ue of the integral is zero. The contribution of segment 4 to the integral ,'anishcs if the integrand a(s)J& _ 0 faster than 1sl- 1 as 11- w. For the response function (9) the integrand_ 0 as 181- 3 ; and for the 8

'&-e E. T. \\1'ittaker

~nt.l C . N. Watson,

Modern lInnlyril. C

'ridge, 1935, p. 117,

\

JJ

Oll/iall I'r'Oa!ue,

,,,,d &cit0nl

31 ,

figure 2 Contuur Xl' the Caochy principal v.alue integral.

,

conductivity o(s) the integrand _ 0 as 1.~ -2. 111C sl."gmcnt 2 contributes, in the limit as II _ 0, o(s) (0 iu elf! dO - - ds_ cr(w) ). If! '" -7Ticr{w) (2) s - w .. lie

J

to thc intl."gral, wherc s = w + II e 19 11,c segmcnts 1 and 3 are by definition thc principal part of the intl.ogral between --:r;I atHI 00. Because the integral over 1 + 2 + 3 + 4 must vanish.

as in (10).

III

J+ J Je p(3)

- _ - d.~ = 7Tia{w)W

0(,)

(15)

_ ", S

EXAMPLE: Cortduc,ivity ufCollisiortless Ekclnm Cas. Consider a ps offrec e1eth , dec.tric fi eld, te mperature, pressure, or unJaxial stress, for exam ple. The spectroscopy of de ri\'utivcs is called modulation spcctrosoop)'. The rellition (21) docs not exclude spectral structure in a crystal, because transitions accumulate at frequcncies for whidl thc bands c, v are parallclthat is, at frequenCies where

V' ['dIee large in com lJRO!oOll "'" a 'Ih lalt;,oe cons tant.

,00\'"

F;~re 4b A tighdy bound os ,,'re.,kcl exciton show" 1ocaIOd on onc alom in an alkali halide CT)'>"lal. All i.kal ,"....,,,kel cllCiloll ",11 t"wc! as a ''1'3.>" t1,roughoullhc crystal, bulthc clcdron IS alwlI)'S ck>Je 10 doc hole.

Table I

Binding energy or excitons, in mcV

SiC. CoA, C,P

14.710

3. Rljlecticm (II normal j,jciflcJlCf). fi('1l\ ('Oll1poncnb uf the fonn

Con~i(ler 'Ill

d ...'CtromJgnetic wave in

WICUUnI,

with

,(inc)'" B~( inc) ~ A~L-- ~ .

Let the \va,'c be inciUent UI)oII a medium of ,1.e1ectric constant f: anll permeability IJ. - 1 tl!.lt fllls the Iwlf-sp.~ x> O. Show tbat tile rl,nectivity coemdcnt r{w) as deBnt:d by (rell) = r(w)";(juc) Is givclJ byII

+ il( -

1

dw)= .. +iK+ Iwhere n

+ il(

-

tY2, with " anc.! K real. Show further til:" the rdlC('lance isR(w) = (n -

If + 1..>'1

(n+ lf+Kl*4. Con(fllcticity '"m rille Ilml ~" ,)(Jroo/lfrllctidty. \Ve write the elcdrirnl ("OlIdlldivity as u{w) .. u(w) + jol'(w). wlllTC u', U' are real. (a) Show by.t Klmucrs.Kronig relation tklt

-But s t rn.:quencieli 0

lim w d'(w)

~

-2~

L -

0'(,) d, .

l11is restllt is used ill th e theory of superconducti\ity. If at ''ery IJigh fTt."(llIencics (sudl :u; :>.r.. y frequencies) dali and .I!or$, ~ Solid 11:IIr. phYIic'S 29. 139 (19'74). T. M. Rice. ~E1eclron-ho1e liquid in semirondUC"lon: thMreticalasptttli. ~ SolId Jlalr. pll)'5ics. 32. I ( 19'Tl). M . Cardona. ed . Liy)Il _ ' teri"fl In -elida. 3 vo .... Springe>". 1 982~83.

'''''.

12SuperconductivityEXPERIMENTAL SURVE\' 33:S Oa:urrcnce of superwoouctivity 337 Destruction of superrondUCCivity by nlagnetic faelds 338 Meissner effect 338 Heal cap3city 342 Encl"gy gllp 344 Microv.'lwc and infra red properties 3415 Isotope effect 346

THEORl-:11CAL SURVEY Tllcnnocl),namics of the ~upen:Ollducting t l'1lnsirion London equulioll Coherence lengd! BCS thoory of supercollducth'ily BCS ground stale Flu.l quantization in a superconducting ring l>..,,"lIlioo of persistent currclllS Trpc II superconductors Vortel stale lry n:s;slanre used was t 72. 7 ohms in tl,e liquid ~-ondII'On at O"c; e.lrnpolation rrom the melting poillt to O"C hy meanS " r the tempernture coefficienl or solid me~ury Il:".~ a relisiance conespondillg to Ihis or 39.7 ohms in the solid slate. At 4.3 K tlus had sunk to 0 0&1 oI,ms, thai il. 10 0 0021 times the res.islance which the wild mcrrury \\ouW have al We. At 3 K Ihe msistance was round 1 have rallcn below 3 x 10 8 ohms, tt,at is to One ten-m,lIionth of the .-alue 0 which ,t wouW have at O"C. As tI,e temper-liure san k rurther to I .S K Ihis value remah,oo tI,e upper limit of the resistance." Hislorical rcre..,nces are give" by e. J. Cortcr. Rev. Mod Ploys. J6, 109&1).

zJ. "ile and R.

.hils, Phys. Rev. Lell. 10,93 (t963).

3:35

U

B,0.026

Table 1

SUllerconuuclivily purumct('(. oi til(' ci('m('nls

~BC N

An ast('risk denotes an el('ment sup('roonducting only in thin flIms or under high pres~ure in a crystal modification not normally stable. Data courtesy of " B. T. Matthias, revised hy T, Gebllll('.

_lr5i-

N,,-

M.Transition temperature in K Critical magnetic field at absolute zero in gauss (10 'tesla)

AIl.f40 i05-

P'

K

C,

5,

Ti0.39 100

V5.38 1420

C,'

M,

F,

Mo

Co

Ni

C,

Z,0.875 53

G,

G,'

As'

- .- - -- - ~

a

F

N,

-

5'

CI

A,

5,'

B,

T,W

~

- R, Rh

_ 1.09r51

--- - - 5n (.. ) Sb'309

Rb

5,

V'

Z,0.54647

K,

~

Nb9.SO 1980

"C,'B,'~.-~

-0.92- 1.77 - 0.51 95 l4l 0 70

.0003.049

la

Icc

HI0.12

-

~

-

_.,

6~OO 1100

- -- -

F,

R,

A,

"

C,'

"Th

- - -- .- -' -- - - -- - - - -- -- - - -- - - - - , --To

-- Pt

Pd

A.

Cd0.""30

I,293(~)

340> 'rt22TI Pb

-

R,

0,65

I,

A,

- --- - 1~

T,'

X,

Hg

Bit

Po

At

R,

4.483

830

0.012 1.07

1.4 198

0.655

0.1419

...-:"". eo

-

~

4.15J412

21]' J.193171803

~

p,

Nd

Pm

Sm

E,

Gd

Tb

D,

Ho

E,

Tm

Vb

L,

,Fm

~-

-

~

;.C

p,

Ut (a)

J'W : rc. 1.62

-

~

-- -- Np~

p,

Am

Cm

~

~

--

--,

-_ . .

-

- Bk

Cf

-

E,

Md

No

-

L,

Figure 2 M~I55ner cffect in a 5uperconducting spht'l"e rooled In a constant applied -magn~~ field, on passing below the transition temperature: the hne l d induction B an!: ejected from the .phe re.

It is an eKperimental fact that a bulk superconductor in a v.'eak magnetic

field will act as a perfect diamaJ,(net, with zero magnetic induction in the interior. When a specimcn is placed in a magne tic field and is then cooled through the transition temperature for superconductivity, the magnetic flux originally present is ejected from the specimen. 111is is called the Meissner effect. 111e Setluence of events is shown in Fig. 2. 'nle unique magnetic properties of superconductors are central to the characterization of thc superconducting state. 'me superconducting state is an ordered state of the conduction electrons of the metal . 11le oruer is in the iOnnatioll of loosely associated pairs of elec.trons. The electrons are ordered at temperatures below the transition temperature, and the)' arc disordercrties arc conSL,ctor k and l>l lin up is occupied, then the orbital with wavevector - k and spin down js also occupied. lfklt -i! vacant, then - kit is also vacant. The pairs are ealiL'(] Cooper pairs, treated in Appendix J-I. They have spin zero and have many attributes of basons. 'Nle boson condensation temperature (TP, Chapte r i) calculated for metallic concentrations is of the order.of the Fermi temperature (J(yt - l dl K). 'nle superco nducting transition temperature is much lower and takes place when the electron pairs break up into two fen nions. "n le model of a supercond uctor as composed of noninte racting bosons cannot be taken absolutely literally , for there are about }(f electrons in the volume occupiL'(] by a single Cooper pair.

Flux Quantization in a Superconducliflg RingWe prove that the total magnetic Ilux that pa55(''5 th rough a superconducting ri ng may assume only quanti:a.>d \'alu(.'S, int(."gral multiples of the Ilu.\ quantum 2Trlrd q , where by experi ment q = 2e, the charge of an electron pair. Flux quantization is a beautiful example of a long-range quantum df(."(:t in which the coh erence of the supcrc;ond ucting state extends over a ring or solenoid. Let us first consider the eiL'Ctromagnetic field as an example of a similar boson fiel d. The electric field intensity ";(r) acts qualitatively as a probability fiel d amplitude. When the total number of photons is large, the energy density may be written as E*(r)E( r)/47T a n(r)/tw , where n(r) is the number de nsity of photons offrcquency w. Then we may write the electric field in a sem iclassical app roximation as

where 6(r) is the phase of the field . A similar probability amplitude describes Cooper pairs. " 111e arguments that follow appl)' to a boson gas with a large number of bosons in the same orbital. We then can trcat the boson probability amplitude as a classical quantity, j ust as the electromagnetic field is used for photons. Both amplitude and phase are then meaningful and observable. The arguments do not apply to a metal in the normal state because an electron in the nonnal state acts as a single unpaired fennion that cannot be treatL.od class ically. We first show that a eharged boson gas obeys the London equation. Let

(

"

"

.Jl'igu~ IS (a) Probability P tlull an orbital cJl.:;nc.:tic e nergy If is ot'O.:I.Ipicd In t he ground st~te oft},e " oninteracting t-"e nni gas; (b) the BCS ground state differs rrum lill,! f."enn; state in a "--gilIn of width of the order 0( the energy gap Et 80th "'''Vel are ror abscl.ule urn.

.p(r) be the particle probability amplitude. We suppose that the pair concentration n = ",*!JI = constant. AI absolute zero II is one-half of the concentration or electrons ill the conduction band, for II refers to pairs. '1l1cn we may write(19)

111c phase 9{r) is important for what follows. (0 SI units, sd e e l in the equations that follow. "l'lIc velocity of a particle is, from the Il amilion equations of mechanics,(CGS)

nlC particle nux is given by(20)

so that the electric current density isj = q4t'*"'IjI = -

nd at all. Nothing fundamental has bCt."ll done to the e1ectmnic stn-ctUTe of lead by this amount of alloying. but the magnclie behavior as a supe rconductor has changed drastically. The theory of type II SupcTd magnetic field is in-

12

S"~r('Ooufuctit'ily

361

(

Ill! 111 '1111111111111I'"7.

I7.

(bJ

Filure l7 (II) Magl'll.:tic field pmctrntion into a thin film of lhiI."'JIal equal to the pctlCtration tleptll A. "Ole arrows indiCRte the intCll$ily of the rnagntlic flCld. (h) Magnetic flCld.penetMllion in a homogeneOlls bulk strocture in Ihe mixed or vorlex slate. with ahemale laye" in nor mal and supen.'Qrldooing st,ltt."$. 'me SUllerronduding byers are thill in C(lfnpari.otl with A. "Ole laminar .trocIure is ~hO'wn fur ro.wcnicnce; the .auaI structure l'OIlliisls of rods of the 001'11131 state "" r_ rounded by t he su perron,Iucting slate. (TIre 11' regions in the vortex slate are not exactly normal, but are dClCribetl by low ~'3lues of the stabil ization energy ,\crull)'.)

creased. A superconductor is t)1lC I if tile surface energy is ahvays posilh'e as the magnetic field is increased, and type II if the surface energy lx .'c omes n(.>gative as the magnetic field is increased . 'fhe sign of the surface energy has no importance lOr the transition temperature. '111c free e nergy of a bulk supcrconductor is incre:tscd wh cn the magnetic field is expelled. lIov.'C\er, a parnUel field can pcnetratc a vcry thin film nearly uniformly (Fig. 17), only a part of the lIux is expe lled , and the energy of thc supcrronducling film will increase only slowly as the external magnetic field is increased . This causes a large incn.':lSC in the field intensity required for the destruction of superconductivity. 111e (jIm has the usual cnergy gap and will be rcsistanccless. A thin film is not a type II superconductor, but the fil m res ults show that under suitable conditions superconductivity can exist in high magnetic fields. Vortex Stale. The results for thin mms suggest the qu(."Stion: Arc there stable configurations of a superconductor in a magnetic field with regions (in the form of thin rods or plates) in the nonnal state, each normal region surrounded hy a superconducting rt.->gion? In such 11 mix(.>rl state, called the vortex state, the external magnclic field will penetrate the thin normal regions uni fonnly, and the field will also penetrate somewhat into the surrounding superconduding material, as in Fig. lB. The term vortex state descri bes the circulation or supcrronducting cur rents in \'orticcs throughout the bulk sp(.'Cimen, as in Fig. 19 below. There is no chemical or crystallographic dificrencc between the normal and the superconducling regions ill the vortex statc. The vortex state is stahle when the pcnetra-

,~

'.

,-

~"

'" - ,,

1') .... II ",,,,'K /'c '

(00)

TIle total current is the sum of}" OIndl b_Thccurrcnt through each junction is of the fonn (47), so thilt

hob.!

= ) o{sin

(50+

:c

tl + sin

(~ $

:c )}

= 2(Josin 80)

(:OS :

l11C Clirrent \la rit..'S with .Ilid h.1S maxima when

cl flc =

S1T ,

= integer

(61)

'n.c periodi(;it}' of the current is shown ill Fi~. 26. The ~hort l)Criod variation is (lrtK.\lICl..o by inlcnere llcc from the two junctions, as pre dicted hy (61). "I11C longer period variation is II diffrllCtion effl.'Ct alld ariscs (mill the finite dimc nsiol"ls of cach junction - tl.is causes (I> to dCI)clld on the pcn:onducting state fonned frulll pairs of electrons k t and - k t. 'nICSC pairs ali as i.losons.

we ~ < A. Thc lTitkill ficlds are rclatctl b)' Hel .... Type II supefCOndudors h.. (lA)H" and lId "" (AJ{)HC' 111e Cinzburg-L.1Illlau parameter K is defin(.'tl as Alf, Values of lld arc as high as 5(X) kC = 50 T.

ProblemsL Magl1dic field pctJCir(ltiol1 in (I plate, loc pcnctration IXjuation may be written il$ A2V2n ... n, whcre A i5 tllC penctmtion depth. (a) Show that n(r) in5ide a supercol\dueliog plate pcrpt.'l'ldicular to the r axis nod of thieko(''$$ [; is giVClI byB(x) "" 8 cosh (VA) cmh (mA)

where n. is thc field outsidc the plate a'ld parallel to it; here x == 0 is:lt the ccnter of thc plate. (b) The effectjvc magncti:t.atioo M(x) in the plate i~ defined by B(f) - B ... 4'11'I\I(x). show that. in CCS, 4'17M(r) _ -B.,( II8Atx~ - ~. for [; 4 A. In Sl we (('place the 4'17 by ~.

2. Critical field

of tliin fiim&. (a) Using the result of Problem lb, show that the frt'C energy demity at T ,., 0 K within a supcrO'" = O.

The correct local field is just equal to the applied field, E"-I = Eo. for an atom site with a cubic environment in a spherical specimen. nlU S the local field is not the same as the macroscopic average fie ld EWe now develop an expression for the local field at a gene ral lattice site, not necessarily of cubic symmetry. The local field at an atom is the sum of the electric field Eo from extemal sources and of the field from the dipoles w ithin the sp{:cime n. Jt is convenient to decompose the dipole field so that part of the summation over dipoles may be replaced by integration. \ Ve write

(14)~A tom site, in acubicCl)1ta1 do not n~-ccssarily have cubic symmetry: th"s the 0"- sites in the barium titanate ' tructure of Fig. 10 do not have a cubic environment. Howev T,

Fiaure 11 Landau free energy f"nClio

J'J =

I

+ 4'1TPI

"" J

+ 4rr1){T

- 'J . O)

(44)

of the form of (36), 111e result applics whcther thc transition is of the first or sceond order, but if second order we have To ::;: '/~~ iffirst ordcr, thcn To < Te. Equation (39) defines 'fo, but Te is thc transition temperature.

Antiferroelectricity

A fcrroelectric displacemcnt is not thc only t)1>C of instabilily that may develop in a dielectric Crystal. Othcr deformations oa."ur, as in Fig. 19. "mcse dcrormations, evcn if thc)' do not give a spontancous polarization, may be accompanied by changes in the dielectric (."onstant . Onc type of defOrmation is callcd antiferroelectric and bas neighboring lines of ions displM."cd in opposite scnses, Thc pcrovskite structure appears to be su!>(."cptiblc to many types of deformation , often with littlc differcn T,

T < T,

T < T,

EBEEJ EBEn

-

I @ @1@ 1@ I 1 1@ I I 1' @ I I I @ @ 1@ 1@ I I @1 @1 I @ @ 1@ I @1 I @1 I I 1@ I @ @ 1@ I @1 I @1 @1 I IFigme In Schemat ic representation of fu nd.o.me ota\ trIleS (If ~tnJCtura' pha5e tr,m,; ;hons from a cenlJ"OSrmmdric prototypp. (ARer Uncs and C'as~.)

".Table 3 Antifcn'OCll:ctric cl"},slalsTran"(1QnI~mpel':ll"r"

ID

Crysloll \V03

anlif"rrocl1"10,

Figt're 20 F'cuock'dnc F, KntifcrroclcctricA, anclllanIC'~'C.tr;" P phases ofthc 1t.'"MI 7.ir,

Figure 21 (a) $dlc:malic dnowing of atomic dl.placements on ",ther ~idc a boundary Ix:twt.'t!'n domains pobri:r.t.-d in QIl~itc dirertions in I k rrock:dric crystal; (h) "jew of a domain SIru('b'"" ~hll....ing 1110" m'lld... ico between domains polari7.cd in O(Ipt..sllc directions.

F~..roel~'ctric doona'ns O th" rae:.: ri a ~ngl" crystal or !Jar."," "Iaolal". 11." rare is n normal 10 Ih" Idragmal or C Ill rcpr ncnt dipole 'TIOfr>ents, cooch sci ~Ihree arrow5 rept'C1iel

It,.

' ..

Consider the paramagnetic phase: an applied fiela v" \viil cause a fin ite magnetization and this in turn will cause a finite exchallge field Bf ;. If X" is the paramagnetic susceptibility, (CGS)M = X,.(B.. +

Bd ;

lSI) """': X~B. + Bd .

(2)

111c maglletization is equal to a col/stant susceptibility times a field only if the fractional alignmcnt is small: this is where thc assumptiOIl elltcrs that the speciIncn is ill the paramaguc tic phase. The paramagnetic susceptibility is given b y tllc Curie law X" :: CIT, whe re C is the Curie collslallt. Substitute (I) in (2); we fin d M T = C(B" + " M)

'nd(CCS)X

" C = -8. = -=c--"-=,,(T - CAl

(3)

The susceptibility (3) has a singulmi ty at 'f = CA. At th is tcmpc.-ature (and below) there exists a spolltancous magnetization. because if X is infi uile we can have a finite !II for zero R". From (3) we