Introduction To Solid State Physics - C Kittel [PDF]

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"arne



Symbol



"arne



Symbol



Name



Symbol



Actinium AIUlnmum An1CnClum Antimolw



A, AI Am Sb



Hafnium Helium Holmium Hychogen IndIum Iodine Iridium J= Krypton Lanthanum La....TellClum



11£



P, Pm P.



Lnod



Pb



Lithium Lutetium



~



Praseodymium Promethium Protactinium Radium Ibdon Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver SodIum Strontlum Sulfur Tantalum Technetium Tellurium Tt:rbium Thallium TI!orium TIlulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium



""00 Arse.nlC



,



Astatine Banum Berkelium Beryllmffi BIsmuth Ilo a2. 33 is called a primitive cell (Fig. 5b). A primitive cell is a type ofrell aT unit cell. (The adjective unit is supcrlluQlls and not needed.) A cell will fill all space by the repetition of suitable crystal translation operations. A primitive cell is a minimum-volume cell. There are many ways of choosing the primitive a.xes and primi.tive cell for a given lattice. The lIumber of atoms in a primitive cell or primitive basis is always the same for a given crystal structure.



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CryltDI Sh·wc'....



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•



•



•



• ••



"



••



"



k::::k::::



.,



••



• -"•



•• •• ,,'



•



•• ••



••



fig...-e s.. Uttice point:l of a spaes in tlnee dimensions ";"mlJer



l\eslridious on conventional ~'dl a:o:cs ~nd 311/1:Ies



of S~'l;lem



Ialllces



Tridinic



a\;.o!'a2';"1aJ



ayi.pv.y Monoclinic



2



al ;.o!'



Ol1!JorllOlllbic



,



a=y-OO'v"p



a'.!. "" aJ



al~a2~a3



a-p-y=OCJO



Tctr;\b'Onal



2



al '"



a'.!. '"' O.l



""'P"'y=9O" Cubic



3



111- 112 '" 03



a-{J"'y=9O" Trigonal



o,=a2=o3



Hexagonal



0,-02,",03



a - p - y < 120". '" a-p-9(f y - 12Cr



9(f



J



•



·f]



• •



• •



•



Crpl.al Sfr.-:,urc



•



•



•



•



• (aJ Sq...." lattn



f·,l loaG ., .. llO"



r~EJ •



"



•



\\



•



" ~



•



•



•



•



"dstaJ AgBr rbS KCI 1Ul'



1.08 1. 420



-I.·n 563



•



5.17 1. 692 629 659



Figure 19 is a photograph of crystals of lead su16de (PbS) from Joplin, Missouri. The Joplin specimens form in beautiful cubes.



Cesium Chloride Structure 111e cesium chloride stTUcture is shown in Fig. 20. There is one molecule per primitive cell, with atoms at the comers 000 and body-ccntcred positions iii of the simple cubic space lattice. Each atom may be viewed as at the ccnter ofa cube of atoms of the opposite kind, so that the number of nearest neighbors or coordination number is eight.



Crystal BLoCu A1NI



CuZo C/l-bruu) CuPd Ag:\tg



•



2701. 2.88 294 299 3.28



Crystal Lillg NH..CI



no,



esCI



nt



•



3291. 3.67 3.97 411 420



Hexagonal Close·packed Structure (hcp) There are an infinite number of ways of arranging identical spheres in a regular array that maximizes the packing fraction (Fig. 2L). One is the facecentered cubic stnlcture; another is the hexagonal dose-packed structure (Fig. 22). The fraction of the total volume occupied by the spheres is 0.74 for both structures. No structure, rCl'(ular or not. has dense.- packing. Spheres are arranged in a single closest-packed layer A by placing cach sphere in contact with six othen. This layer may serve as either the basal plane eX an hcp structure or the (111) plane of the fcc structure. A second similar layer B may be added by placing each sphere orB in contact \\ith three spheres of the bottom layer. as in Fig. 21. A third layer C may be added in two ways. We obtain the fcc structure jfthe spheres ofthe third layer arc arkled over the holes in the fint layer that are not occupied by B. We obtain the hcp structure whcn the spheres in the third layer arc placed directly O\'cr the centcrs of the spheres in the first layer.



17



IS



" fo'igure 21 A close-P"K'kcd la)'erol" ~phcres is shO\vn, with (;enters alpoints marh..! A. A s.,coud and identical layer spheres UlO1 be ploct:d on lop of this, above :lnd parallel to the plane of the dmwing, with centers o>'cr the points marh>tl B. TIler" are two choices 'Or .. third tarcr. It can go



or



in over A or over C. If il goes in over A the SCQucn,-e;s ABA8AB. . and the Sindure is hexagonal close-pal·ked. If the thin! J.1)'er gol.'S in O\'CT C the sequence is tlBCABCtlBC . .. and the sIn.. is r~ntcred olLic.



,t",,,



, ,



a



.



•,



.. a



• :



,



::;,



,



"



I



II II



II



I'igurc 22 The l,cxagonal cIO!>C-l'acL.:d sinK" lure. Th" atom positions in Ih~ slru ~ tJ3arc integers and ab a2. 113 arc thc crystal



"co. • To form a crystal we attach to every lattice point an identical basis composed of' atoms at the positions rJ := rjll + Yjlz + Zjl3' .....:ithj "" 1. 2, . . • • s. Here x, y, Z may he selected to have values between 0 and I. axes Ill' 112, aJarc primitive for the minimum cell volume tal • a2 l< aJl for which the crystal can be construch..-d from a !Jttice translation operator T and a basis at every lattice point.



• 111C



Problems I. Telrol.edmlongla. The angles betwecn the tctraJ'cdral bonds of diamond are the



same as the angles between th(' bod)' diagonals of a cube, as in Fig. 12. Use dcm('ntary ..·ector analysis to fim the value of the angle.



2. Indiur of plllna. ComiJaIplc phfJria of 4"iroIropic lOlids. BenjamIn, 1914 B K Vains!\!cin. Mcxkrn cryltallogrwphy. Springer, t981. J. F. "ye, I'hfJdoal propr"ws 0/ ~Ial.l'. ~hO"'n as line bL1Ck lines. 'nle lines in while"", perpendicular bi~1ors of die ..,..



sro..""



ripn::lC'll1 L,ulC:'e ..,eton ~ centra! sq,we is the smallN ."Ofume about the Clf"'Igin whidlls bounded elltire!)~ ..."'-lle lines. The ~ is the W-~-Srib. pri.niliH~ cell of the reSCribed by the h" ~. l':) defined by (30). The volume of this cell in reciprocal space is b l • ~. b;, = 2/..2,"10'1. The eell contains one reciprocal lattice point. because each of the eight comer points is shared among eight parallelepipeds. Each



p:ar.dlelcpipcd contains one-eighth of each of eight comer points. In solid state physics we take the central (Wigne.--Seit-J.:) cell oCthe reciprocallaUice as the first Brillouin zone. Each such cell contains one lattice point at the central point of the cell. This zone (for the bee lattice) is bounded by the



planes normal to the 12 vectors of Eq. (32) at their midpoints. -Inc ZOlle is a regular l2-faced solid, a rhombic dodt.'C8.hedron, as shown in Fig. 13. "Ole \uetors from the OI'igin 10 the center of each face arc



(1Tla)(:!:y ± i) ;



(33)



All choices of sign arc independent, giving 12 v(:cton.



ReciproctlllAtlice to fcc lAttice The primitive translation vectors of the fee lattice of Fig. 14 are 3.::::



!a(S' +



i);



lit



=! tl(i + t);



33::::



!a(i + y)



(34)



1bc volume of the primitive cell is V = 1301 . It" x



a3I :::: ~a3



(35)



1lIc primiti\'e translation vectors of the lattice reciprocal to the fcc lattice b l = (2'Jl"/aK-i + 5' + i) ;



ht =



b:J :::: (2'Jl"/tlXi + Y -



(2m'aXi i) .



y + i)



(36)



r 1 o



./



Figure 14 Primitive basis ,"eclors of lhe f;M:e



Vi. VJ are integers or zero.



• "O-Je scattered amplitude in the direction Ie' "" k



tional to the geometrical structure factor:



+ .6.k = k + G is propor-



so Sc • "i..



£ t:~p(-irJ' G) = "i:.Jj cxpf -i21T(xJv. + YjV2 + %:iVa}]



•



where) Tuns over the saloms ortbe basis, alldJj is the atomic form factor (49) of the jill atom of the basis. 1be ~"pression on the right+hand side is "'Titten for a reflection (VIV~. for which G "" v1b. + 1'21>,; + v:>b:J. • Any function inv:.lriant under a lattice translation T may be el'punded in a Fourier series of the fonn



nCr) =



L ttc exp(iC' r) c



• TIle first Brillouin zone is the Wigner-Seit-.t primitive cell of the reciprocal lattice. Only waves whose wan~...ector k drav.ll from the origin tenninates on a surface of the Brillouin zone can be diffracted by the crystal



• e'ysiallattice



"iTst BriUouln wne



Simple cubic



Cube



Body-centered cubic



IDKlmbic dodecahedron (Fig. 13)



Face-centered cubic



Truncated octahedron (Fig. 15)



.



, ~'-



,



'" ';0'



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·



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• • • • • •• • • •• • •• • • • • • • •• • • • • • • •• • • •• •• • •• • • • • • •• • • • • ••



., ~



•



.' Figure 20 j'hotograph of the calculatL-a I'ouri"r t..ansrann (diffraction pattern) or an icosahedral ,......icrystal alallg olle or the fi....,rold IUC!I, iUuslraling ll.-fold .ymmclT)". TIle transform is cakul~lcd from. lheoretical roml)ljle,..~nentcd model, b)( !woel JaLVb"



tI,,,



Problems I. Interplanar sepamtion. Consider a plane hkl in a crystal lattice. (a) Prove that the



reciprocal lattice vector G = hb, + k~ + 1b:l is perpendicular to this plane. (b) Prove that the distance between two adjacent parallel planes of the lattice i$ d(hkl) .. 21d1C1 . (c) Show fOr a simple cobic lattice that d t = al/(h l + Ie'- + P) . 2. H ~ &pOint molion, we eaIt'uIlIte a mobr .-oIumr ol9 em" mol-I lOr did helium, lIS C'Ofn~ "'lIh theobsen>::d \...J.-01"27.5 and 36.8 an" 1001-' b liquid lIe4 and liquid He"• .-..5pecti\-e1y In the- ........ nd state ofbclium v.-e must w.e ~t oI"thc :zrro.point ,notiool of point



the



alOffil.



~



figure 2 Cubic close-pad,e. .~ ~ -Z6P~ )..sz,t .2.""-



'"



P, Hp P. U (0.761 0.987 -



At



"



~



R.



0.397 0.133 0.411 2.52J :Z.52... .2.§3



-- --- -.,. .. - • -- - Cm



..-



Kr lal 0.Dl8



I



,



JPO"": Ar III



8,



....



He ('1 0.010



Md



Ho



"



{ ,i:.



•



Lr



~



~



"--



~



. - - - . - - - j f - " -1 FigurCl 3 Coocd'llates of the two oscmooton.



As a model we consider two identical linear hannonic oscillators I and 2 separated by R. Each oscillator bears charges ±t: with separations %1 and X2. as in Fig. 3. The particles oscillate along the r axis. Let PI and 1'2 denote the momenta, The force constant is C. 111el1 the hamiltonian orlhe unperturbed system is 1



2



2



"0"" -PI + lexi 2m



1



2



2



+ -1'2 + iCXi 2m



(1)



Each uncoupled oscillator is assumed to have the frequency Wo of the strongest optical absorption line of the atom. Tbus C :> ~ Let ". be the coulomb interaction energy of the two OSCillators. The geom· elry is shown in the figure. TIle internuclear CQOnlinate is R. TIleD e2



tl tl ------;



el



?t, --+ R R + Xl



ICCS)



-



X2



R + %1



R-



(2)



Xl



in the approximation Ix,l. Irsd ~ R we expand (2) to obtain in lowest order:



1/,



Table 4



a.: -



~X,XI R3



(3)



t'mperties or inert gas crystals



(Extrapolal:cO loOK and zero pressure)



_..... JOfli·



t 9S ~"



· O 0 O II



11



'"



'"



tle



I'igu~ 5 lbc ~ect of I\luli principle on the ~bl\'c energy, in an extreme example, two h)'drogclI aloms are pushed together unlil the protons arc almost in COfltact. n.e energy of the electron S)~em aJonc can be taken from observations on atomic He, ..hich has two electrons. In (a> the flectrons hnc antiparallel "Pills and the r>.uJi principle has no effect: the electrons U'e bound by -78.98 e\'. 111 (b) Inc spin' are plIrallel: the Pauli principle IOrc-es the promotion of an clcclrQI' from a 18 f orbllal of H to II 2s t orbobll of I-Ie. The electrons now Ilre bound by -59.38 eV, leI.s than (a) by 19.00 eN. l1lis is the amount by which the Pauli pcinciple has iI~ the repulsion. We ha..., omitt~ the repuhh-e coulomb energy 01' the tYoV protons, ,,-hidl i$ the "me ill Leth (ll) "leI (b).



, , • ,



•



, , ,



••



.. ,, .. l'l..... _



\



" "



Figure 6 Form of the LeIlnanl-JOOes potential (10) "hkh dcsc'ibc5 the inlcl"Ild:ion oflwo Inert gllS atoms. The minimum ooeun at AlIT _ 2'.... 1_12. Notjc,e how sleep the etJn'C is Inside the miniIT''um, 81M! bow fb,t it is o,,,tsodc the mlmmum. The ,..J.ue of U at the minimum is and U '" 0 at



-0:,



R"



fT.



be obtained from gas-phase data, so t.hat calculations on propertics of thc solid do not invoh'e disposable parameters. Othcr empirical form.s for the repulsive int.eraction are widely used, in particular t.he exponential fonn A. exp( -Rip), where p is a measure of t.he range of the interaction. This is generally as ea.~y to handle analytically as t.he inverse po"..er law form. Equilibrium Lallice Constants



Jf we neglect the kinetic energy oflhc inert gas atoms, thc cohcsive encrgy of all incrt gas crystal i.s given by summing t.he Lennard-Joncs pot.entiaJ (10) ovcr all pairs of atoms in the cryst.al. If there arc N atoms in the crystal, the total potential energy is



U_~IN(4')[L'J (--"-r -L' (--"-r]· pvR J puR



(11)



",-here pyR is the dist.ance between rcference atom i and any other atom j, expressed in t.erms of t.he nearest neighbor distancc R. The factor t occurs 'with



the N t.o compensat.e for counting twice each pair of at.oms. The summations in (11) have been e"aluated, and for the fcc structure



L'p;12 =



"" -, = 14.45392 .



12..13188 ;



£.. PIJ



)



(12)



}



1bcre arc 12 nearest-neighbor sites in the fex: structure; we sec that the series are rapidly converging and l13.\·e \'alues not far from 12. The ncarest neighbors contribute most of the interaction energy of inert gas crystals. The corresponding sums for the hcp structure arc 12.13229 and 14.45489. If We take UIOI in (11) as the total ene'l,')' of the crystal, the Cfjuilihrium valuc Ro is given by requiring tI13.1 UIOI be a minimum with respcct to variations in thc ncarest. neighbor distance fl:



dU,. dR"" =



if] '



u"



[



0 = -2N£ (12XI2.13) RI3 - (6X I4 .45)"'Rf



(13)



"..hencc



RJu =



(14)



1.09 ,



the same ror all elements with an fcc structure. The observed \'alllcs of RJu, using the independcntly determined \'allies of u given in Table 4, are: Nc



1.14



M 1.11



Xc 1.10



1.09



The agreement with (14) is remarkable. TIle slight departure of RJu tOr the Iightcr atoms from the IIlliVeTSal value 1.09 predicted for inert gases can be



66



explained by zero-point quantum effects. from measurements on the gas phase we have predicted the lattice COIlstant of the crystal.



Cohesive Energy The cohesive energy of inel1 gas crystals at absolute zero and at zero pressure is obtained by substituting (12) and (14) in (11):



U...c(R) = and. at R =



2lV~[(12.13)(:Y2 -



(l4.45)(;fl



(15)



Ro. (16)



the same for all inert gases. This is the calculated cohesive energy when the atoms arc at rest. Quantum-mechanical corrections act to reduce the binding b)' 28, 10, 6. and 4 percent of E~1. (16) for Nc, AT, Kr, and Xc., respccth-c1y. The heavier the atom, the smaller the quantum correction. \Ve call understand the origin of the quantum corrt:ction by consideration of a simple model in which an atom is COnfined by fixed boundaries. rf the particle has the quantum wavelength A. where A is ~termined by the boundaries, then the particle has kinetic energy pi/2M = (hlA'f/2lof with the de Broglie relation p = hlA fOr the connection betv.'een the momentum and the w~welength of a particle. On tins model the quantum 7.ero-point correction to the energy is inversely proportional to the ma~s. '111e final calculated cohesive eucrgies agree with the ClCpcrimental value.~ of Table 4 within 7 to 1 percent. One conSC(juence of the quantum kinetic energy is that a crystal of the isotope Nel!D is observed to have a larger lattice constant tllan a crystal of Ne 22 . The higher quantum kinetic energy of the lighter isotope expands the lattice., because the kinetic energy is reduced by expansion. The obscrvcd lattice constant~ (exh~dpolated to absolute zero from 2.5 K) arc NeW, 4.4644 A; Nc 22, 4.4559 A. IONIC CRYSTALS



Ionic crystals are made up of posith'e and negative iOlls. nle ionic bond results from the electrostatic interaction of oppositely charged iOIl~. Two common crystal structures found for ionic crystals, the sodium chloride and the cesium chloride structures, were shown in Chapter 1. 11te electronic configurations of all ions of a simple ionic crystal OOrTespond to closed electronic shclls. as in the incrt gas atoms. In lithium fluoridc the configuration of the ncutral atoms arc. according to the periodic table in the front endpapers oftbis book, Li:ls~. F:l~2.02ps. '(11e singly charged ions havc the configurations Li+: 1$2. F-: 1s2zs~p6, as for helium and ncon. respeeti\'c1y. Inert gas atoms ha\·c closed shells. and the charge distributions are spherically S)'mmetric, We expect that the charge ill.!, utiom on each ion in



3



Cryslal Dinding



61



)-"ig",.., 1 E::lcclron density dislribulion ill the base plane of NaCl • ..ncr x-ny sludies br C. Sehokneehl. 11le "umbers Oil the conlours gin, lhe rcl~th'e eleeIron COflCClll..... lion.



+



l""i7.atl""



c.



G..! Figure 8 The e ..ergy per molecule unit of a eryslal of sodium chlOride I, (7.9 - 5.1 + 3.6)" 6.4 eV 10\.... lhan Ihe e"ergy of scparaled neutral atoms. 11le !alike energy wilh respect 10 separated jo"s is 1.9 eV per molecule unil. All ,-alues on tl,e rtgure ~re c.-per[me.. laI. Values of tl,., ionization energy are given in Table 5, a"d ,-al""5 of the electro" affi"ity are given in Table 6.



. ""



5.14 c\'



e"",'1O'



+



01



-a -



""



+



e.,



3.61.N



''''''"' ~ffinily



""



-



+



G..!



e.,



+



79.·\



C,hn, l.nergy. 8. Young's morn.lus and Poisson's raliv. A cubic crystal is subjt..'Cl to tcnsiun in thc 1100) dirLdioo. Find expression~ in teom uf the clastic stiffncsscs for Young's modulu~ and Puis~on's ratio as dcfined in Fig. 21.



9. umgitudinalwave velocity. Show that the velocity of a longitudinal wave in the [111] dirtdiOlI of a cubic crystal is given by v• ., Ii(C II + 2C 12 + 4C44 )1PPII!. Hint: For such a wave II = V = w. Let " = uoe l K(X+I/+a)lV3e-t..... and use Eq. (57a).



93



. '+,61_1



~ r



-



UnderO"~bQdY7



,



,



,,, ,I ,,



L,



T~_



-



L



.1. __ •. \



L



J



I'lgure 22 This debmation is compounded from the: two shalrs,,~ - -II!....



.-.,...., 21 Young's moclulus is ddiJM,d as streW dram lOr a temile ltreP acting in (Inc direction. "'ith the sides r# specimt.'tl M:I\ £n.-c. PoWem'$ 1':01>0 is dt... fl, ..., ] all (&du:)f(Wl) b this situation.



10. Tran.wene lrove velocity. Show that the ~'e[ocily of transverse wa\"es in the [llli direction of a cubic crystal is given by v. - [j{C'1 - C llt + C. 4)1p]I/ll. Hill!: See I'roblcm 9. 11. £ffcdive shea,. comtllol. Show that the shear constant j{C II - G,:z) in a cubic CT)'5' tal is defined by setting en co -ew "" k and all other strains equal to 7.ero, as in Fig. 22.. Hint; Consider the energy deruoity (43); look for a C' such that U - +e'e'. 12. lhterminantal approtJeh. It is known thai an R-dimensional. square matriJ: ....id, all demenu equal to unity has roots RandO, \\;th the R occurring once and the zero occurring R - I limes- If ail elements kwc the \'8lue P. then the roots are R" and O. (a) Show thai if the dja~ma1 elements are (I and all other elements are 1'. then there is one root equal to (R - l)p + (, and R - I roots equal to q - p. (b) Show &om the elastic equation (57) for a wa.~·e in the [Ill) direction of a crJbk crystal that the determinantal equation which gives ",} as a function of K is



(/ - W-p



,J



lJ



q-6}p



•



I'



• Jl



.. 0 ,



q- cJp



where q • i~CII + 2C~ and " . iK'{c 11 + C.J. This exp~ the oonditm that three linea.- homogeneous a1geb.-aK:equations lOr the three displacement U)fflponents U, 11, W ha~'e a SOIUUO'l. Use the resllit of part (a) to find the three roots of



cJ; check



with dle results gh-en lOr Problems 9 and 10.



13. Genual propagation dir«lion. (a) By suhstilutKln in (57) find tile dctcrminantal equaUon whKfl expres!ieS the ll'1l and physic! of melals and alfoy!, Wiky-Intcr.;cicncc, 1912. F. A. Cotton aod C. Wilk,nson, Adx;attced Inorganic chemi.>ltry, 4lh L'rt'sentatkm by lemot'S and matrices, o,.,ford, 1951. Il. H. Hunlington. "Elastic conSlanls of ''''' K. for mo::wkI m rlg. e boundary the groop ,'ekJal) is



«



= Long Wllvelengll. Umit



When Ka 41: I we expand cos Kn tion (7)



I - !(Kaf, so thai the dispersion rela-



iii



bCl.'OInC$



(15) The result that the frequency is dircct:ly proportional to the wavcvcetor in the long wavelength limit is cqulvalenllo the statement that the velocity of sound is independent of frequency in this limit. Thus v.: wlK. exactly a.~ in thccontin· uum theory of elastic waves-in the continuum limit a ::::: 0 and thus KG = O. Derivation



0/ Force COllstantl from



Erperinumt



In metals the effective forces may be of quite long range, carried from ion to ion through the conduction electro.l sea (Chapter 10). Interactions have been found between planes of atoms separated by as many as 20 planes. We can make a statement about the rallgc of the forces from the observed dispersion relation for w. 111C generalization of the dispersion relation (7) to p nearest planes is easily round to be w2 = (21M)



L.



C,Al -



(.'OS



pKu)



(l6a)



,,>0



We solve for the interplanar rorce COnstants C" by multiplying both sides



by cos rKo, where r is an integer, and integrating over the range or indepen. dent values or K;



M



L::



dK



wI: cos rKa



= 2 =



p~ C



p



L:,:



-2m:/a .



The integral vanishes except ror p



::c



7T



(1Gb)



r. Thus



G" = - -MaI~' 2 dK gives the rOrce OOj



dK (l - cos pKa) cos rKa



- ..In



wl COS 1'Ko



(11)



nt at range 1m, lOr a structure with a monatomic basis.



,0
-, .. till] dlft'dion Fit;ure 8b D.spC'rsion l'UM.'S In II~ [ II tI d....-ctoon in KBr II 90 K. afler A. D. B. Woods. B :'\ BrocI(h,w/7T,,)----'1



Normal Mode Enumeration "Inc energy of a collection of oscillators of frequencies equilibrium is found from (1) and (2):



u ~ 2: 2: I



hWK.,



wKp



in thermal



(8)



cxp(liwl 71'0. There are three polarizations p for COlen value of K: in one dimension two of these are transverse and one longitudinal. In three dimensions the polarizations arc this simple only for wavevectors in certain special crystal directions. Another dc\"ice for enumerating modes is often used that is equally valid. \Ve consider the medium as unbounded. bul require thai the solutions be periodic over a large distance L, so that u(sa) = «(sa + L). The method of peri. odic boundary conditions (Figs. 4 and 5) docs not change the physics of the problem in any essential respect for a large ~1'stem. In the running wave solution u. = u(O) exp[i(sKa - WKI)] the allowed ,'alues of K are K=O,



2'17



4'17



+ + -T' -T'



+



6'17



~L'



N7r L



(14)



This method of enumcration givcs the same number of modes (onc per mobile atom) as given b)· (12), but we ha"e now both plus and minus values of K, with the intcrvall1K = 27TfL bctwecn successivc values of K. For periodic boundary ctw~"Cn K.... and the DeLye cutoff Ko are not excited 3t all. Of Ihe 3N possible modes, lhe rrdction exciled is (K-,lKDt' .. (T/Uf'. l>t:cause lhis is the ratio of lhe "olume of the inner sphere 10 ll,e oul.,r sph.,rc. The energy is U ... kilT' 3N(TI9'f', and Ihe heal capacily is C., .. aU/aT'" 12Xk8(T/9)'.



)25



U



- - -



Table 1 Debye temperature and thermal conductivity"



e.



344- 'clence 181. 999 (1973),



~



ph~



S



11, T'-mal E'rfllJUfin



, ,



,.



•



~-



,.



, , ,



0



- --



•



.1/



./



/ o



0.1



0.2



o.~



0.3



05



06



0.1



o.tI



011



10



n" Figure II Comparison of .,xpt:rimcntal.....Jue. of Ihe h~t ca\*:ity of dilltnOnd ",Ih Vll!ue. calc... Iak.JOflthe L'nrhe.1 qu;lllium (EinsteIn) motlc:l, 'lloing tlw chillllcterislic lemperature 81'. - hlAlllce be-



I......." '" 1"'0 8.



l29



130



In rcal crystals nOne of these consequences is satisfied a(,'Curately. The deviations may be attributed to the neglect of anharmonic (higher than quadratic) terms in the interatomic displacements. We discuss some ofthe simpler aspects of anharmonic effects. Beautiful demonstrations ofanharmonic effccts are the experiments on the interaction of two phonons to produce a third phonon at a frequency w;; ;: WI +~. Shiren described an experiment in which a beam of longitudinal phonons of frequency 9.20 Cllz interacts in an MgO cI)'stal with a parallel beam of longitudinal phonons at 9.18 Cllz. The interaction the two beams produced a third beam of longitudinal phollons at 9.20 + 9.18 = 18.38 GHz. Three-phonon processes are roused by third-order terms in tbe lattice po~ntial energy. A typic-oil term might be U3 = Ae•...ewec, where the e's are stram components and A is a constant. The A's have the same dimensions as elastic stiffness constants but may have values perhaps an order of magnitude larger. The physics of the phonon interaction can be stated simply; the presence of One phonon cuuses a periodic clastic strain which (through the anharmonic interaction) modulates in space and time the elastic constant of the crystul. A second phonon perceives the mooulution of the e1ustic constant and thereupon is scuttered 10 produce a third phonon, just as from a moving threedimensional grating.



or



Thermal Expansion We may understand thermal expansion by considering for a classical oscillator the effect of anharmonic tenns in the potential energy on the mean separnliOn of a pair of atoms at a temperature T. \\'e take the potential energy of the atoms at a displacement x from their equilibrium sepamtion at absolute zero as (38)



with c, g, and f all positive. lbe term in ~? represents the asymmetry of the mutual repulsion of theatoms and the tenn in x4 represents the softening of the vibration at large amplitudes. 111e minimum at x = 0 is not an absolute minimum, but for small Oscillations the form is an adequate representation of an interatomic potential. We calculate the average displacement by using the Boltzmann distribution function, which weights the poSSible values of x according to their thennodynamic probability:



f" d:c



/



x exp[-I3U(x)]



(r) ~ -'=;--'0-----



L:



dx exp[ -I3U(x)]



,



1'''-oN II. T'--'l Prcrprrlia



T....



'"



.., I ". .,.



?



,;



~



.I



~



'j



,,,,,---.,.--rl&'1f'e IS J...atlft constant oholid argon a. a lU..,..



0



/



/



-'



/ K



f dx x ..p( -{JU) '" f dx [e.p( -Ilcr~l(x + pgx' + /3fi/') ~ (3"P'/4Y.1!Jc""1P'''' , f dx e,",,-pU) '" f dx e.p(-Ilcr~: ("'flc)'~ (39) whence the thermal expansion is (40)



in the classical region. Note that in (39) we have left 0: 2 in the exponential, but we have expanded exP(Pb>x 3 + f3ft4) a 1 + pgx3 + 13ft· + .. " Measurements of the lattice constant of solid argon are shown in Fig. 15. The slope of the curve is proportional to the thermal expansion coefficient. lne expansion coefficient vanishes a5 T - 0, as we expect from Problem 5. In lowest order the thermal expansion does not involve the symmetric term fx4 in U(x), but only the anlisymmetric term gx3. THERMAL CONDVCnVI1Y



The thermal conductivity coefficient K of a solid is defined with respect to the steady.state flow of heat do....'Il a long rod with a temperature gradient



f!w,na>.' A substantial proportion of all phonon t'OlIisions will then be U processes, with the attendant high momentum change in the t'OlIision. In this regimc wc can cstimatc the thennal rcsistivity without p.1rlicular distinction bctwccn Nand U pnx.'Csses; by the earlicr argumcnt ahout nonlinear cffccts we cxpect to find a lattice thcrmal resistivity IX T at high temperatures. The cnergy of phonons K 1 , K2 suitable for umklapp to occur is ofthc order or ikfjO, because each of the phonons I and 2 must ha\'e wa\'c\'cctors or thc order of iG in order for the collision (47) to be possiblc. Ifboth phollons have low K, and therefore low energy, there is no ....':1y to get from their collision a phonon of "':1\'c\'cctor outside the first ZOILC. 'rne umkbpll process must conserve energy, just as for thc normal process. At low temperdturcs the number of suitable phonons of the high energy ikfj8 required may be expected to vary roughly as cxp(-812T), according to the Bolt-.nnann factor. 111e cxJXlnential IOnn is in good agrccment with experiment In summary, the phonon mean



s



PItor.ctN II. T""-l Propmin



.. , '00



•



'!



•,



18 n.c:rm...1 conductivit)' of a h;ghly purifocd crystal of :!Odium fluoride, aflCt'



00



,



Fiplr~



H. E. Jackson, C. T. Wallru, ....d T. F.



',,-L-:---:---f.,-;;;---,J;-~ 2 S 10 20 SO 100 T~..." ..



MeNcD)'.



free path which enters (42) is the mean free path for umklapp collisions between phonons and not for all collisions between Ilhonons. Imperfediort3



Geometrical effects may also be important in limiting the mean free path. We must consider scattering by crystal boundaries. the distribution of isotopic masses in nalural chemical clements, chemical impurities, lattice imperfections, and amorphous structures. When at low temperatures the mean frcc path t becomes comparable with the width of Ihe tesl specimen. the value of t is limited by tbe \\;dth, and the tbennal conductivity be /.I.tI. Ilere the den5ity of modes is discontinuous. 2. 11:,", t~mol dilalion 0/ crystal cell. (a) Estimate for 300 K the root mean square thermal dilation t1V1V'for a primith'e cdl of §Odium. Take the bulk modulus as 7 x to LO erg el.1l-3. Note that the Deb)'t" temperature 158 K is less than JOO K, llO that the themw energy is of the on:Ier of k.T. (b) Usc this result to estimate lbe root mean square thernlal nuctuatioo .1aJa of the lattice parameter.



3. Zero point lattice rlit]J/aCfmlcnl atld stroin. hysit;8 14, 198 (1956). C. Y. Ho, R W. Pov.'ell and P. E. Liley, Then/wi co"dllctivitlJ of the eUllIumfs, A comprehensioo review. J. of J'h~,. and a'em. Rd. Data, \'01- 3, SUllPlemcnt I. It P. T~'C, cd., Thermal comillctivily, AC'lidemic Pre", 1969. J. M. Ziman, Electrons and pho..ons, Oxford, 1960, C1\llptcr 8. It Bcnnan, Thermal co"dllctit:Jn In solids, Oxford, U176.



C. M. Bhandari and D. M. Howe, ThenlWl conductio" if' 6emlco,u1uctOfll, Wiley, 1958.



(



6 Free Electron Fermi Gas ENERGY LEVELS IN ONE DIMENSION



144



EFFECT OF TEMPERATIJRE ON TIlE FERMI-DIRAC DISTRIBtnlON



146



FREE ELECfRON GAS IN THREE DIMENSIONS



146



HEAT CAPACITY OF TIlE ELECl1\ON GAS



151



E:Ilperimental heat capacity of metals



1M



Heavy fermions



156



ELECffiICAL CONDUCI1vrrr AND DUM'S LAW Experimental electrical resistivity of metals



Umklapp scattering MOTION IN MAGNETIC FlEWS



HalJ



e(f~t



J56 159



162 163 164



TIIERMAL CONDVC11VI'IY Of METALS Ratio of thermal to dectrK-al rooductivity



166 166



NANOSTRUcrURES



HiS



PROBLEMS



Joo



t. Kinetic



~ of e1ed.roo ps Pressure and bulk modulus of an electron gas Chemical potential in two dimensions Fermi gases in astrophysics Uquid lie' Frequency dependence c:J the electrical condUdhity o,.·namk magnetocooouctivity tensor for free eledl'"(llls Cohesi~'e energy of (ree eled:ron Fermi gas Sialic magnetooonductivity tensor Maximum surface resistance n. Small metal spl~res 12. Density of states-nanomelric wire 13. Quantiultion d condudance



2. 3. 4. 5. 6. 7. 8. 9. 10.



REFERENCES



16&



169 169 1GB 170 170 170 170 111 171 171 172 172 172



+



+



+



i"igure I Sch~matic model rJ a crystal of sodium meW. 1hc alomic cord AI"(! Na+ loons; they are imm".,;cd In a sea of ('Oflduction dCdrons. The conduction electrons arc derived from the 3s ,·aIc"ct! electrons of the free aloms. The alomlc COR'S contain 10 c1cdrous in the COflI1j;lurolion l~pa In an alkali metal the alomic ~""'C1 llC.'l'\lP)'. relali\"dy small pIIrt (-15 percent) of the total \'Olumc of the cO)..tal, but in a nobk mdal (Cu. Ag. Au) the a10mic core> arc rclRth'c1y larger and mIl\" lJc in contact "';Ih cadi othrr. The mmmCll l.T)"Stal struc.1u ' room Icrnpemlure iJ bl-c .... the alkali metY



'.



Figure 4 In the ground slate of a system of N fn'C e1ecIrons the occupied orbitals of the S)'litcm IlII II. sphere of radim k". where ~ •. - h 21.-f../2m is the energy of an ck-ctwll having a wa,'C\'cctor k._



1,



".)



1, FigureS DCllsity of single-particle slalt.'5 , Ef ·, nnd the second integral gives the energy neeTAL \\'Inl I:»OIREC1" CAP Ahoorplion



Figure 4. Opticaillbsorplionin pure in"llIaIOf$ _, ab..,l"t" orero. in (3) Iht, Ihrbhold d"'"rmlllt'l ,"" e.Jergy gllP as E. - Aw.. In (hI the: opticoal absorplioo il we;J;e, nar the threshokl: at h", ~ £." + flO. photon is absorbed "ilh 1M crnIion ollhrce.particles; • free declron•• rift! hole....d. phonon of ~ "n. In (b) rho: energy E..... marks the ~ b- ~ ew: ploo do not ~how absor-ption lines that ~Ddintel are seen lyingjusllo the Io.-·energy lide ollhe threshold. such linn"" due 10 the cn"lIIion of. bDuod



electron·hole plir, called all e>ociron.



•J filUl"e S In (aJ the ",","I point of the ronduetion band OC'OJrs at ttrsame ,~olk as the highrsl point of the . - . - band. A dlrecl: optialtranlillOn is drawn \"ertical1y with no signi5cant dIange of k. because doe aboorbed pholon ha5 a ''''ry llnall "~·e\·t'Ctor. 1be thrcshokf frequency .... for ablOrptiOll by the direct lnn.itioo determines the energy gap £ ... !'i&l., TIle indirt'd tnnsitioo in



ooth a photon and I phonon beuuse the band ed~ of the oonductioll and \'$knce bands arc widely Icparat...d in k IpllCC. l'he threshold ent'.!,')' for thc indi'l' be COIlse.,..ed.



(h) j'lVol\'es



,~



,, ,



..



(j



S.,micond"cfor Crlld""



-------------



,~



J!



i . w'