SEVENTH EDITION
`Introduction to
`Solid State Physics
`
`CHARLES KITTEL
`
`Greenthread Ex. 2084, p. 1 of 27
`Semiconductor v. Greenthread
`
`

`

`CI-IARLES KITTEL
`
`Introduction
`to
`Solid State
`Physics
`
`SEVENTH EDITION
`
`Jolin \Viley & Sons, foe ., New York, Chicl,ester,
`
`Brisbone, Toronto, Singapore
`
`Greenthread Ex. 2084, p. 2 of 27
`Semiconductor v. Greenthread
`
`

`

`-------- ---
`
`About the Author
`
`Charles Kittel t2ught soHd state physics al Berkeley from 1951 to 1978;
`earlier he was a member of the solid state group at the ~ll Laboratories. His
`undergraduate work was at M.I.T. and at Cambridge Unjv~nity, follow~ by
`graduate work at the University of Wisconsin. He .is a member of tho National
`Ac.ademy of Science and of the Ammcan Academy of Arts and Sciences.
`His research in solids began with studies of forromngnetic. antifuromag(cid:173)
`ncUc, and paramagnetic resonance, along with work on ~etic domains, spin
`waves, and domain boundaries in ferronugncts and ferrocl~rics , His work on
`the single dorrnain structure of fine particles Ii.is had br03d application in mag(cid:173)
`netic recording, gcomagneHsm, and bioma,gnetism. Along witl1 colLiborntors ut
`Berkeley he did the first work on cyclotron resonance in s.en1iconducton, which
`led to the understanding of the band structure of silicon, germanium, and in(cid:173)
`dium antimonide, together witJa the theory of their impurity states. He also
`worked on tlw intcrprct:ntion of m~gnetoplasm~ resonance in semiconductors
`and of Alh•~n re~onancc in electron-bole drops in germanium.
`The first edition of JSSP int~grated the elementary aspects of soUd smtc
`physics for study hy seniors and beginning gr.uluate students. Now Jn Its sev(cid:173)
`enth edition, ISSP pb)'S tJae same p.art for the current generation of students.
`
`Copyright O l~, 1956, 1066, 1971, 1Q76, 1Q66, 1906 by John WJ.tylit Soru, lnc.
`All right1 reserved. PuhluhcJ ,imultaneousl)' ln C&nMla.
`Reproduction or translat.ion ol any put of this work bt-)ood tlw pt"rmitted by S«tions 107 or IOR
`of the 1976 United States Cop)Tigbt Act without the pt-rmillloo ol the copyricht owner h
`unlawful. RNJUt'ltJ (or permiuion o.- further infonnaUon ,bouJd be ~ to the P«mlnlons
`Department. John Wiley 6c Sons, loc.
`
`Ubrary of Cor14:n"..u Cat.ologfni 111 Publlatio11 Oat.a:
`
`Charles Kittel
`Introductioa lo 10lid state ph)'1ia / Charles Kittel - 7th ed.
`P· cm.
`Includes indea.
`ISBN 0-.71-11181-3 (doth : alk. paper)
`J. Solid atate ph)'lics. L 1\tle.
`QC176.K5 1Q96
`53(U' l--<ic20
`
`95-18445
`ClP
`
`Printed in the United States of America
`
`1098 7
`
`Greenthread Ex. 2084, p. 3 of 27
`Semiconductor v. Greenthread
`
`

`

`Preface
`
`111is book is the seventh edition of an elementary text on solid state physics
`fi,r ~cnior and beginning ~raduatc students of physical science and engineering.
`'!'lie book is on update of the sixth edition of 1986 and includes ndditions,
`1111provc111ents, an<l corrections made in that edition in 13 su<.-cessive prinUngs(cid:173)
`wliid1 it was time to pull together-and a nurnbcr of new topics besides. Signif(cid:173)
`i<-ant advances in the field hnvc been o<l<lc<l or discussed more fully: thus high
`h·mpcraturc superconductors urc treated , and results of scanning tunneling
`111i<:roscopy ore <lisplayc<l; the treatment of fiber optics is cxpam.Jcd. n,crc arc
`discussions, amon~ other topics, of nanostructures, supcrlatticcs, Bloch/
`Wannicr levels, Zener tunneling, light-emitting diodes, and 11cw magnetic
`materials. Tho addition have been made within a boundary <.•<m<lUion intended
`to keep the text within one volume nnd nt n rensonable price.
`11ie thcoreti<.-al level of the text itself hM not been changed. n,crc is more
`discussion of useful materials. The trentment of elastic ronstnnls und elastic
`waves which was dropped alter the fourth edition has now bcon returned be(cid:173)
`cause, as many have 1x,intc<l out, the matter is useful W1d not easily accessible
`elsewhere. The treatment of superconductors is much more extensive than is
`usual in n text nt tJ1is level: either you do it or you don't.
`Solid ~tale phy ics is conceme<l with the properties, often astonhhing an<l
`often of ~real utility, that result from the distribution of clcctro11s in metals,
`semiconductors, and insulators. The book also tells how the exciUtions and
`imperfections of real solids can be understood with simple model~ whose power
`and scope are now firmly established. TI1e subjec-t matter supports a profitable
`interplay of experiment, application, and theory. Tilc book, in English and in
`many translations, has helped give several generations of students a picture of
`the process. Students also find the field uttn1ctivc because of the frequent possi(cid:173)
`bility of working in smaJI groups.
`Instructors will use the book as the foundation of a course in their own
`way, yet there are two gener.il patterns to the introduction, sdcction and order
`of the basic materiaJ. If students have a significant preparation in elementary
`quantum mechanics, they will like to begin with the quantum theory of elec-
`
`iii
`
`Greenthread Ex. 2084, p. 4 of 27
`Semiconductor v. Greenthread
`
`

`

`iv
`
`trons in one-di men ionaJ solids. startin~ with the free ck'(.iron ~as i11 Chapter 6
`and energy bands in Chapter 7. One will nce<l to tre.it tJ,c rt'(.·iprocaJ lattice in
`three dimensions (Cluptcr 2) before plunging into ~crni<."(m<lu,·tors (Chaptc-r 8)
`and Fermi surface (Clwptcr 9). Cry ta1 strnctur • ·, cry-'tal hia,din~. and pho(cid:173)
`nons could be considered as rc,Teational reading. Inn more ~r.1d11;1I nppmad,.
`the 0rst eight chapters tJ1rou~h the physics of scmiconduc:tors :m· n·a<I oms<."{' 11-
`tively as a one-semester introduction to the flcl<l.
`What about the necc sary statistical mechanics? A v-.1~11•· <liS<.'l1111fort .,t tlic
`thought of the chemical potential is still dw-.actcristi<- of a phy ·ics education.
`Tiais intellectual W&P i due to the obscurity of the writ in~ of J. \Vilt,nl Cil,l,s,
`who discovered ond understood the matter 100 years aJ:o. 1 lcrl>t·rt Kn>t·mt•r
`and J have dariflcd tJie physi~ of the clacmic-.u potential in tli • t•;1rly d,aptt•N of
`our book on thermal physics.
`Review series give excellent extended trc.itmcnts of all tli • ~111,jeds
`treated in this book and many more besides; thus with ~00(1 ,·onsc..:i,·11<..'\' I ,-tiv ~
`few references to origjnaJ papers. In these omissions nc> l.,l·k of l11>11or i-i in(cid:173)
`tended to those who flrst set sail on these seas.
`The crystaJlogrJphic notation ("(>nforms with curn•nt 11~~" i11 physics.
`Important equations arc rep<•:ttt•<.l in SI an<.1 CCS-Cau,,;i,111 unit,. wll<'n: tl1t'\t'
`differ. Exceptiom arc figure cnptio11 •. cliJptcr ~11111111.arics. ,rn,w prnl1lt•111'i, :uul
`any Ion~ section or tcAt where a lli11~lc i11Jic::1tecl ~1111\t it11ti1111 will tr.111\l.,tc· from
`CCS to SI. ChaptN Co11tc·11h p.,~l•,; di,e11s'i crnl\'L'lllinn, mloplt·d to 111.1kc· 1uml(cid:173)
`lcl usage llimpk•. Tlw dual magc· i11 this hook ha'i ,.._.,,,, fo1111cl 11,;c•f11I 1111cl :t('\'<'pt·
`ahlc.
`Tuhlcs nre 111 t'1HlV<.'ntionJI units. Tlw 11)11111>01 ,. d,·111>ll's ti.,• d1.1~c• on tlic
`proton and is po~itivc. Tho 11olutlo11 (Hi) rcfor • to Equation (IX) of tlll' nirn·nt
`chapter, hut (3.18) refers to Equation 18 of Ch:.aptcr 3. A c.,rd • m•c•r a V<•t·tor
`refers to a unit vector. Few of the problems arc exactly c·asy; IIHhl wc•rc· de·\ ised
`to carry forward tho ubjcct of the chapter. \Vith a few •xccptio11"i, tlw prnl,(cid:173)
`lems nrc thos<: of the original sixth edition.
`111is edition owes much to tlac advice of Profc sor Stevt•11 G. IJ111it·. For
`collected corrections, data, and illu trot ions I am ~ratcful to I'. Allc•n, M. Uc•as(cid:173)
`loy, D . Chcmla, T.-C. Chiang. M. L. Cohen, M. C. CrJforJ, A. E. C11rm11.
`D . Eigler, L. M. Falicov, R. B. Frankel, J. Friedel, T. JI. C •l,all •, D. M.
`Ginsberg, C. Herrin~. H. F. Hess, N. Holonyak, Jr., M. ja(-ob, J. Mami11.
`P. McEuen, J. C . Mullen, J. C. Phillips, D. E. Prober, Marta Puebla, D. S.
`Hokhsar, L. Takacs, Tingye Li, M. A. Van Hove, E . R. \Vcbcr, n. M. \Vhitl',
`J. P. \1/olfc, and A. Zcttl. Of the \Vilcy staff J have particularly great debts to
`Clifford Mills for publication supervision, to Cathy Donovan for her ingenuity
`in processing the additions between the thirteen suoc"Cssivc printings, and to
`Su:zannc Ingrao of lngrao A sociates for her sldll and undcrstarading during the
`editorial process.
`
`Greenthread Ex. 2084, p. 5 of 27
`Semiconductor v. Greenthread
`
`

`

`Prefece
`
`v
`
`Corrections and suggestions will be gratefully received and may be ad-
`tressed to the author at the Department of Physics, University of California,
`Kerkeley, CA 94720-7300, by email to kittel@uclink4. Berkeley.edu; and by fax
`tu (510) 643-9473.
`
`C. Kittel
`
`An Instructor's Manual is available for this revision; several problems have
`heen added (to Chapter 3 and Chapter 6); one dropped (from Chapter 4), and
`several corrections made. Instructors who have adopted the text for classroom
`use should direct a request on departmental letterhead to John Wiley & Sons,
`Inc., 605 Third Avenue, New York, NY 10158-0012. Limited requests for per-
`mission to copy figures or other material should be addressed to the Permis-
`sions Editor at this address.
`
`Greenthread Ex. 2084, p. 6 of 27
`Semiconductor v. Greenthread
`
`
`
`Greenthread Ex. 2084, p. 6 of 27
`Semiconductor v. Greenthread
`
`

`

`8
`Semiconductor Crystals
`
`BAND GAP
`
`199
`
`EQUATIONS OF MOTION
`2-03
`Physical derivation of hf, = F
`!05
`206
`Holes
`Eff c<.1ivc mass
`209
`Physical interpretation of the effective mau 110
`Effective masses in semioonducton
`tit
`Silicon ond gennonium
`21-4
`
`INTRINSIC CARRIER CONCENTRATION
`Intrinsic mobility
`
`IMPURITY CONDUCTIVITT
`Donor states
`Acceptor states
`TI1cnnal iooiution of donors and ac-«pton
`
`THERMOELECTRIC EFFECTS
`
`SEMIMETALS
`
`SUPERLATTICES
`
`SUMMARY
`
`PROBLEMS
`
`l. Impurity orbits
`2. Ionization of donon
`3. Hall effect with two c.arricr t}-pes
`~. Cyclotron resononce
`5. Magnetoresistance with two carrier types
`
`REFERENCES
`
`216
`HO
`
`tu
`IH
`lt-4
`tie
`
`UT
`
`ll8
`
`W
`
`!31
`
`!31
`
`131
`131
`131
`131
`!31
`
`131
`
`NOTE: The discussion of carrier orbits in applied fields is continued in Chapter 9.
`Amorphous semiconductors are treated in Chapter 17. Junctions and banien arc
`treated f n Chapter 19.
`
`Greenthread Ex. 2084, p. 7 of 27
`Semiconductor v. Greenthread
`
`

`

`\ •111fr1,("(.;IJ
`
`-
`
`to-' '
`
`- 1<>1''
`
`:
`
`" '
`10 19 e
`-~ :s
`10" i
`
`-
`
`~
`IO'i l
`~
`~
`-IO'n ~
`
`Semlronduc1or1 (at roo111 lt·rnpcrJluri.')
`
`------- - - - -- - - - - -- -- - - - -- ----101
`Figure I Carrier concentrations for mctaJs, semi metals, llnd semiconduc-ton. TI.e semiconductor
`range may be e,d ended upward b)' increasing the impurity l'Oocenlr.ltion, and the nng.e can be
`extended downward lo merge eventually with the insulator ni.ngo.
`
`)
`
`198
`
`Greenthread Ex. 2084, p. 8 of 27
`Semiconductor v. Greenthread
`
`

`

`CHAPTER 8: SEMICONDUCTOR CRYSTALS
`
`Carrier concentrations representative of meta1s, semimetals, and semicon(cid:173)
`ductors arc shown in Fig. 1. Semiconductors are generally classified by their
`electrical resistivity at room tempcrnture, with values in the range of 10-2 to
`10° ohm-cm, and strongly dependent on temperature. At absolute zero a pure,
`perfect crystal of most semicondu<:tors will be an insulator, if we arbitrarily
`define an insulator as having a resistivity above 101 ◄ ohm-<:m.
`Devices based on semiconductors include transistors, switches, diodes,
`photovoltaic cells, deteC'tors, and thermistors. TI1ese may be used as single
`circuit clements or as components of integrated circuits. \Ve discuss in this
`chapter the central physic-.il features of the classical semiconductor crystals,
`particularly silicon , germanium, and gallium arsenide.
`Some useful nomenclature: the semiconductor compounds of chemical for(cid:173)
`mula AB, where A is a trivalent clement and B is a pentavalcnt element, arc
`called 111-V (three-five) compounds. Examples are indium nntimonide nnd gaJ(cid:173)
`liuin arsenide. Where A is divalent and B is hexavalent, the c.-ompound is called
`a II-VJ c.-ompound; examples arc zinc sulfide and cadmium sulfide. Silicon and
`germanium arc sometimes called dion10nd-typc semiconductors, because they
`have the crystal structure of diamond. Diamond itself Is more an insulator
`rather than a semiconductor. Silicon carbide SiC is a IV-JV compound.
`A highly purified scmiconducto; exhibits intrinsic conductivity, as distin(cid:173)
`~uishcd from the impurity conductivity of less pure sp<.-cimcns. In the intrinsic
`temperature range the electric-al properties of n semiconductor arc not essen(cid:173)
`tially modifled by impurities in the crystal. An elt.>ctronic band scheme leadin~
`to intrinsic oon<luctivity is indicated in Fig. 2. TI1e conduction band is vacant at
`absolute zero un<l is scparnkd by an energy gap E,: from the filled valence band.
`The band gnp is the difference in energy between the lowest point of the
`conduction band und the highest point of tJ1e valence band. The lowest point' in
`the conduction band is called the conduction band edge; the highest point in
`the valence band is called the valence band edge.
`As the ternpernturc is increased, electrons arc thennally excited from the
`\'alencc band to the conduction band (Fig. 3). Both the electrons in the conduc(cid:173)
`t ion band and the vacant orbitals or holes left behind in the valence band
`<:ontributc to the electrical c-onductivity.
`
`BAND GAP
`
`TI--ie intrinsic c.-onductivity and intrinsic carrier concentrations are largely
`<.·on trolled by EJk8 T, the ratio of the band gap to the temperature. When this
`, atio is large, the concentration of intrinsic carriers will be low and the conduc-
`1 ivity will be low. Band gaps of representative semiconductors are given in
`'fol>lc 1. The best values of the band gap are obtained by optical absorption.
`
`199
`
`. ..
`
`Greenthread Ex. 2084, p. 9 of 27
`Semiconductor v. Greenthread
`
`

`

`Figure 2 Band scheme for intrinsic conducthity in a semiconductor. At O K the conducti'1,ity is
`-zero because All states in the valence b:and are Oiled and :all sbtcs in the conduction band arc
`Vllelnt. As the temperature is increased, electrons are thermally excited from the l--alencc band to
`the conduction band, where they become mobile.
`
`101•r---...--..---------....------
`
`10•:1
`
`.. e ..
`~ c., .i
`
`~
`
`10•2
`
`.5
`.§
`~
`
`t C e
`
`C e
`11
`~
`
`10••
`
`10•0
`
`,,_
`
`!; ..
`
`~
`C
`:,
`
`1: a
`
`.E
`C:
`.2
`~
`C: t
`
`;
`
`1011
`
`C: e e
`t
`~
`
`Temperature, K
`
`(a)
`
`lo" 275
`
`300
`
`325
`
`350 375
`
`-'00
`
`-&25
`
`450
`
`Tt."fll per.Lt un:. K
`
`(b)
`
`Intrinsic electron conccntr.ation as a function of temperature for (a) gerIJl3nium and
`Figure 3
`(h) silicon. Under intrinsic conditions the hole concentration is equal to the electron concentration.
`The intrinsic concentration ot a given temperature is higher in Ge th2D in Si bcc:iusc the energy gnp
`is narrower in Ce (0.66 eV) tho.n in Si (1.11 eV). (After W. C. Dunbp.)
`
`Greenthread Ex. 2084, p. 10 of 27
`Semiconductor v. Greenthread
`
`

`

`8 S~IIC'forC~
`
`!01
`
`Table I Enc~ gap be™·cen t.he valence and conduction band.s
`(I = lndirec-t gap; d = direct gap)
`
`£~ cV
`
`£_. cV
`
`Cry,la.l
`
`C ap
`
`OK
`
`300 K
`
`Cryrtal
`
`C ap
`
`OK
`
`300 K
`
`Diamond
`SI
`Cc
`aSn
`lnSh
`lnAs
`lnP
`c ~,P
`CaAs
`CaSh
`AISh
`SiCO, ·x)
`Tc
`ZnSb
`
`i
`i
`d
`d
`,I
`ti
`i
`d
`d
`
`i
`ti
`
`5.-4
`1.17
`0.74-4
`0.00
`0.23
`0.43
`1.42
`2.32
`1.52
`0.81
`1.6.5
`3.0
`0.33
`0.56
`
`1.11
`0.66
`0.00
`0.17
`0.36
`1.27
`2.25
`1.43
`0.68
`J.6
`
`0.56
`
`Hg'fc'"
`Pl,S
`PbSe
`PbTe
`CdS
`CdSe
`CdTc
`ZnO
`ZnS
`SnTe
`AgCI
`Agl
`CuiO
`TiO1
`
`d
`d
`
`'
`
`d
`,I
`d
`
`d
`
`d
`
`-0.30
`0.286
`0.165
`0.190
`2.582
`1.8-40
`1.607
`3.436
`3.91
`0.3
`
`2.172
`3.03
`
`0.34-0.37
`0.27
`0.29
`2.42
`1.74
`1.44
`3.2
`3.6
`0.18
`3.2
`2.8
`
`•11,crc Ii u t.t•mimc•lul. Ilic h.111Js ovcrl.ap.
`
`111c threshold of ctmtinuous opti<.·aJ absorption at frequency W,:, determines
`the band gap E,:, = l,w,,, in Figs. 4a and Sa. In the direct absorption process a
`photon is uhsorhcd hy the crystal with the creation of an electron and a hole.
`Jn the indirect absorption process in Figs. 4b and Sb the minimum energy
`gap of the band structure involves electrons and holes separated by a substan(cid:173)
`tial wavcvcctor le<. Herc a direct photon transition at the energy of the mini(cid:173)
`mum ~ap c.annot satisfy the requirement of conservation of wavevector, be(cid:173)
`(.'auso photon wavevectors arc negligible at the energy range of interest. But if
`a phonon of wavevector K and frequency n is created in the process, then we
`can have
`
`le(photon) = le,. + K :! 0 ;
`lrw = E,:, + 1,n ,
`as required by the conservation laws. TI1e phonon energy Ml will generally be
`111uch less than E~: a phonon even of high wavevector is an easily accessible
`~ourcc of crystal momentum because the phonon energies are characteristically
`~mall (~0.01 to 0.03 cV) in comparison with the energy gap. If the temperature
`b high enough that the necessary phonon is already thermally excited in the
`<:rystal, it is possible also to have a photon absorption process in which the
`plionon is absorbed.
`
`Greenthread Ex. 2084, p. 11 of 27
`Semiconductor v. Greenthread
`
`

`

`CK\' .TA I. \\ ITII UIIHX T G·\11
`
`AJ..,,.,,_ • .,
`
`Tr.anfPU"'I
`mdoo
`
`011...,-1 u( J_iu, I
`plou(uo
`lr.tn,/IIOfl
`
`(a)
`
`C. + AO 1:... ..
`....,._...,.~&..(cid:173)
`
`(1,)
`
`Flgun! -4 Opt m l abwrpllon In pure lnsulalon at abtOlutc n-ro. In (a) tht- th~hold tkh:n•lm•~ tlK·
`energy p p u
`111 (h ) tlic optical ahsorpclon ls •~«.-r ~.a.r the.• thrnl,olcl:
`t
`1-:11 •
`lii.111•
`""' - e. + Ml • photoo ,~ a h~(heJ with the <Tr.Ilion o( thrtt p;ar1~: • frtt t'k-ctruu, II '""''
`hok, and • phonon of cncrJ,,,Y AO. In (b) lM n><'l'JO' £ _, n....rb tlk" t i~ fur the <'t'Ntlo11 o( •
`fn:.-c clcdron 111d I free.• hole;•, witli no phonon IM·ohc.-d. Sue.ft a lr•mil lon b oJk.J ,crtk.,1. 11 b
`,Im/I.tr 10 the d lri :-ct Ir.ans II Ion in (a). 1l1CjC plot, do nol how alnorpeion tint'~ 1h11t wmc-lim<-s ar('
`sc;,-cn lylngJtnl lo the low co rgy side o( the thrnhold. Such llM"I am du to tlie m•.ilion of a l >01111ll
`ck•dron-hole paJr, ClltcJ an cxcitun.
`
`f
`
`Figure 5 In (11) the lowt·st point of the conduction band occvn at the !W1lC v.Juc of k IIS the hl~11
`point of the valcnre band. A direct optk-'.al traiuition Ls dr.t•'ll ,-c,tic;JJy v.itl, 09 llignif1C:U1t dw,gc ul
`k. bco&usc the absorhc<l photon 113.S a \'Cry sowl V.'3\'~rt"tOr. 11K" tlucshos.d (rc,qutncy "'• ~
`absorption b)• the d irect lnan Ilion dete nnines the t'~f'g)' pp E« .._ A.111• 11K' iodlroet tnruiUon in
`(h) invoh'tls both a photon and a phonon becaulC the b.ind od~n d the roaductioo aod , ~
`l>.lnds are wide!)' S<'par•lod In le Spac<:', The threshold~• f« the indlrm prottu in (b) i_s great_n(cid:173)
`than the true band ftllp. The absorption threshold for tbc loc:lirect tnnsitioo bdwtt11 the ba.nd
`edgt"S Is al "°' - £. + An. where n
`is tht' ftt,qU<'OC')' of an nnitted phonon of w:avn-«tor
`K a -k.,. At higher tempc.r.a_tures phonons arc alrc~)' prt'K'Ot; if a phonon is absorbed along with a
`photon, the threshold t'n<'rgy Is l,a, • r.. - An. Note: 11w- figure ,hows onJy the thrcJ)old tnnd(cid:173)
`tions. Tr,ansilions ()('(.'Ur gent'rally bctwttn almost all points of tM two ~d.s Sor wbk-h the
`wavevectors and t'ncrgy c-.an be c-onserYed.
`
`Greenthread Ex. 2084, p. 12 of 27
`Semiconductor v. Greenthread
`
`

`

`J o ' , - - - - - - - - - - - - - - - - - - - ,
`
`-. e
`.s
`~ j
`~
`j JO' .
`
`<
`
`l, L.I ----..JU..--10_3 ___
`
`......_ __ O~l---- -0.:-":7: - -~
`
`n,turt.• 6 Oplk.11 ril1,01111ion in pu1t• i11Ji11111 1111U 111u 11tJl', l11SL. TI.e lr:amlUOfl "dm-c1 ht'C"llUlC
`Lotli n11ul11dion 1111J \'.&ll'm,· l,,111d t-d~t·~ un.· :ii tltt" <'i.'nl<>r of Iii<• Unllauln 1~. Ir • 0 . Not~ tlic
`~11.irp 11,rc·~hold. (Mt<•r c.. W. Cot.di nd II . Y. Fan.)
`
`111e bnn<l gap mny ulso he deduced from the temperature depcndcn('C of
`tho conductivity or of the carrier ronccntntion in the intrinsic range. The
`carrier ronc<'ntrntion is obtainro from measurements of the Hall voltage (Chap(cid:173)
`ter 6), sometimes supplemented b)1 c-011t.luctivity measurements. Optical meas•
`uremcnts determine whether the g-ap i dirt-ct or indirect. The ~nd <.:.dgcs in
`Ce and in Si are ronne<:tcd by indirect transitions; the rond edges in InSb arc
`connected by u <lirect tn1nsition (Fig. 6). n1c gap in aSn is direct and is euc.1ly
`zero; HgTc and H~Sc are semimetaJs and have negative gaps-the bands
`OVt•rlap.
`
`EQUATIONS OF MOTION
`
`\Ve derive the equation of motion of an electron in an energy band. We
`look at the motion of a wave packet in an applied electric field. Suppose that the
`wave packet is made up of wavefunctions near a particular wavevector k. The
`
`Greenthread Ex. 2084, p. 13 of 27
`Semiconductor v. Greenthread
`
`

`

`group velocity is Vi = dwldk. The frequency as.sociatN:I with 11 wavefunction of
`energy ~ is w -= tlla, and so
`
`or
`
`(1)
`
`TI1e effects of the crystal on the electron motion arc contained in the dispcnion
`relation f(I<).
`111c work & done on the electron by the electric field E in the interval
`6t is
`
`& = -eEv" 6t .
`
`(2)
`
`\Ve observe that
`
`& = (dedk)Sk = lav" 6k .
`using (1). On comparing (2) with (3) we have
`6k = -(eFJli)l,t .
`whence lidkldt = -eE, the same relation as for free electrons.
`\Ve may write (4) in terms of the cxtema.J force f' as
`
`lh~=F.J
`
`(3)
`
`(4)
`
`(5)
`
`TI1is is an important relation: in a crystal fu/kldt is equal to the external force on
`tho electron . Jn free space d(mv)ldt is equal to the forre. \Ve have not over(cid:173)
`thrown Newton's second law of motion: t]w electron in the cry. tnl is subject to
`forces from the crystal lattice o.s well as from external sources.
`lnc force term in (5) also includes the Lorentz force on an electron in a
`ma~nelic flcld, under ordinary conditions where the magnetic fle)d is not so
`strong that it breaks down the band structure. Thus the equation of motion of
`an cluctron of group velocity v in n constant magnetic flcld B is
`
`dk
`(CCS) f1- = --v ,c B :
`,It
`
`C
`
`C
`
`die
`(SI) fidt = -,"' >< B ,
`
`(6)
`
`where the right-hand side is the Lorentz force on the electron. \Virh the group
`velocity f,v = gradi.-t, the rate of change of the wavcvector is
`
`(CCS)
`
`(SI)
`
`die
`-
`dt
`
`C
`= - -V1ct >< B
`f, 2
`
`(7)
`
`where now both sides of the equation refer to the coordinates in k space.
`We see from the vector cross-product in (7) that in a magnetic field an
`electron moves in I< space in a direction normal to the direction of the gradient
`of the energy E, so that the electron moves on a surface of constant energy.
`
`Greenthread Ex. 2084, p. 14 of 27
`Semiconductor v. Greenthread
`
`

`

`8 ~ Cl')IIIWa
`
`TI1e ,r.Juc of lhe project.ion k,, of k on 8 is corul21lt durin_g the motion. The
`moUoo in le sp.c-e [ on plane oonnal to Ute diroctioo of B, and the orbit is
`defuled by the I nte~lion of lhiJ pl.lne "ith 2 urbc.-e of roru:~nt energy.
`
`Phv ical Derloalion of !1~ = F
`We oonsidcr the Bloch cigrnfuoc:tion ~ hdoogin.g to tl,e en<'rgy f'igeu(cid:173)
`vaJuc f 1, and w ,·evec--tor le
`
`tJ,,,, = L C(lc + C) ap{i(1c + C) • r]
`
`G
`11,c ex pee-tat ion value of the mom<-ntum of an electron ln the st~tc It h
`
`(8)
`
`r
`I
`
`'
`
`...
`
`'I
`
`u ~l11~ :tlCO< + C )l2 = 1.
`We cumi11e the transfer of momentum bctwttu the electron llnd the lit•
`tkc when the state k of the cleL1run I cl,anged to k + ~ by the pplicatlon of
`a 11 cx lcmal force. We imagine an in uwllng Cf) W clttirostidically n<-utr.il
`cxc-c·pt for 11 single d c<:tron in the
`lr1tc k of an othcrwu.e e mpty hand.
`Wo sup1>osc that a weak cite m aJ fort'(' Is ar>pfic.'<! for
`time l11kr.".a.l 11d1
`)' te m ls J • fF Jt. If the
`that the total impul~c ~ivcn to the en tire CJ)'S'tal
`conduction dcctron w('rc fr(·c (m• • m), the total momentum lmr>3rtcd to tho
`crystal syste m hy the lmpul~• would appear In the change of momentum of the
`<'011duction clcctro11:
`
`'Ille ne ut ral crystal suffers no net interacHon with the eltt<tric fidd, either
`dir<.'<:tly or indirectly through llie free electron.
`If the conduction elC"<·tron interJcts \\ith t.he (lt'riodic potcnti.11 of the cryi·
`ta.I latti<'<', we must have
`
`J = 6p,.__ .. = 6Pt,., + 6p .. ,
`Frum the re.suit (9) for p,.1 we have
`
`.
`
`6p I = la~k + L /1Cl(V~IC(k + C)l7) . M]
`
`C
`
`(11)
`
`(12)
`
`TI10 change 6p1.., in the lattice momentum resulting from the du111ge or
`~tate of tJ1e electron may be derived by an elementary ph)·sical consideration.
`An electron reflected by the lattice lransfcn momentum to the l~ttice. If an
`incident electron with plane wave component of momentum hk is reflected
`
`..
`
`-
`
`Greenthread Ex. 2084, p. 15 of 27
`Semiconductor v. Greenthread
`
`

`

`with momentum li(lc + C). the lattice acquires the momentum -1,G, as re(cid:173)
`quired by momentum conservation. n,e momentum transfer to the lattkc
`when the state cf,,., goes over to c/,,,,.+i1k is
`
`6Pt., = -1,L G[(Vi.lC{I( + G)l2 - 6k] .
`
`c;
`
`ns the portion
`
`(13)
`
`(14 )
`
`of each individual component of the initial state is r •Oected during the state
`change 6k.
`The total momentum change is therefore
`
`(15)
`euctly as for free electrons, Eq. (10). Thus from the definition of J, we hav'
`l,dk/dt = F ,
`(16)
`derived in (5) by a different method . A rigorous derivation of (16) by an cntircl)'
`different method is given in Appendix E .
`
`llok,
`The properties of va<.'ant orbitals in an otherwise filled band arc important
`in semiconductor physics and in solid state electronics. Vacant orbital · in a
`hand arc c'Omn,only called holes. A hole acts in applied electric and magnetic
`fields ns if it has a positive charge + c. TI1e reason Is given in five step · in thl'
`boxes that follow.
`
`(17)
`1.
`The totol w-.ivevector of the electrons in a filled b.ind is zero: ~le = 0. This
`result follows from the ~comctrical symmetry of the Brillouin zone: every
`fundamental lattice type has symmetry under the inve~ion opcnltion
`that the Brillouin zone of tho
`r ~ - r about any lattice point; it follow
`lattice also has inversion symmetry. If the band is filled all pairs of orbitals
`le and -le arc filled, and the total w·.ivevector is zero.
`If an electron is missing from an orbital of wavevector le~, the total
`wavevcctor of the system is -le,. and is attributed to the hole. This result
`is surp1 ;sing: the electron is missing from k,. and the position of the hole is
`usually indicated graphically as situated at k,., as in Fig. 7. But the true
`wavevector le,, of the hole is -k,., which is the wavevector of the point G
`if the hole is at E. The wavevector - k,. enters into selection rules for
`photon absorption.
`
`Greenthread Ex. 2084, p. 16 of 27
`Semiconductor v. Greenthread
`
`

`

`a S--'c 0Nluc1or Cryai.u
`
`I le 11, r 7 Abwq,tion of a photon of crlerJtY 1,., and ~
`jg_,bk \\"l\ T \'N:'for t.ucn an ckc:tron fr0tn £
`111 d lf' fill,-J Vlllt'oce ha_nd to Qin the- coodurtioo b.ind. Jf k,,. .,.-.u the \\"3\ CY«tor ohhe cltttroo a t
`,t f.,,, .omc-s the wavevector oftl~ dcrtron at Q. 11.e total "' ,·~ o r ol'the vakn«• Land al\rr
`,
`ti 11 ,ii ""' pllon h, - k,,, and thu IJ the- ~V<"Vttior w e mwt .scribe to the hole I( we ck-scribe the
`l ►.mJ as OC'<.-upkd l,y one hole. Thus k,. • -k,,; the Wa\'e"'fflor o(~ ~ Is the lame u tbe
`, ,1, 111 ~ -
`, , . , 1111 o(the electron whk:h ~m.afos al C. For the entin, syalern the toul "',.,-e,·mor after the
`" 111 ,
`I,., ''I" 11111 11( I he plioton 11 k,, + lc4 • 0, to th~, tlx! total wa\~\'fflot ls u~ by tht, ab,tQrp-
`11111, .. 1 llw photon 11nd tJ.c CTc~ tion of • free clf'ctroo and frtt ~
`-
`
`(
`
`'11,<.· hole is an alternate description of a band with one missing elec-
`1, Ptt . a11cl we either say that tl1c hole has w-avcvcctor -~ or th.at the band
`"111, 1111<· mi s ing electron has total wa\'evector -~.
`
`(18)
`
`J •·I t l,c • zero of energy of the valence b:md be at the top of the band. The
`l11,,T 1 iu the band the missing eltttron lies, the higher the energy of the
`~p 1, · 111 The cnf'rgy of the ho]e is opposite in sign to the energy of the
`t11l1i\l 11 ~ t'l(·c:tron, because it tales more work to remove an electron from
`,1 l11w 111 l,ila] than from a high orbital. Thus if the band is syrnmetrlc. 1
`cc.( - ~) = -E,,(-~) = -E,-{](11). \Ve construct in Fig. 8 a band
`(~. )
`tu•1111· lo represent the properties of a hole. This hole band is a helpful
`H•J111 •w11tation because it appears right "ide up.
`
`IJ 11.f, ""
`.,hrn~11 ~ymmetric under the fovenK>n 1(- -1( if the qxn-orhit intt'rac-Uon ls ne•
`-l 11 t., , . ., ",ti, ,pin-orbit interaction. bands ar~ aJ,.-ays symmdric if the- U)'Sbl stlucture
`.. llh 1111, ''" 1· , ,11,11 opc·r.1tlon. Without a ct"nter of sym~try, but with spin-orbit intttadion, the
`It• ,, 11111wl1t(· ,r we romp.uc subbands for which the ,pen dirmion iJ rn-cncd:
`I
`I j
`, f A. J l Sc·•· ()TS. ChaplC'r 9.
`
`Greenthread Ex. 2084, p. 17 of 27
`Semiconductor v. Greenthread
`
`

`

`llok b.-cl ~
`-li.. ad
`Wltit .. -
`•A>•--.
`~ d y
`o1 ......
`
`Fi,fu~ 8 1hc uppn ha.If o( the fl ure '~"' the hok band th.it • u~t
`lM d~u.unk-1 of a bok.
`ronstructcd by inHrslon of the \ 1fOC'C' b.ind In the- ori~n. 1hc ""~""T"Ct rand <"n<"~' of the- hok
`a.re ("QU.t1, but op~ile in siA:11, to th~ '111-.n·<"'·t tf or and c-~JltY oltht- ~mpt)' kct.roo orbits) ln the.(cid:173)
`J('fl('(.' b.lnd. We do not how th,• di sp<ttillon of Ill<' t·kdrou n.-nlO\nl ( m tile:' ,~ · lwxl at "..,.
`'
`
`3.
`
`(19)
`
`The velocity of the hole is equal to the velocity of the mis ing electron.
`From Fig. 8 we ec that V~11(k,.) = VE).lc#), so that v,.(k,_) = o,.(k.,).
`m,, = -m,r .
`We show below that the e ffective mass is inversely proportional to the
`curvature d 2u dk2 , and for the hole band thi has the: oppo itc ign to th t
`for an electron in the valence b:ind. Near th top of tl1c valence band na_. h
`negative, so that m,, is positive.
`elk,,
`l
`/1-- = c(E + -v,, • 8 )
`dJ
`C
`
`(20)
`
`(21)
`
`(22)
`
`5.
`
`This comes from the equation of motion
`la ~ = -e(E + _!_ v • 8)
`dt
`
`C
`
`f'
`
`(CCS)
`
`that opplies to the missing e lectron when we substitute -k,, for ~ and v,,
`for Ve. The equation of motion for a hole is that of a particle of positive
`charge e. The positive charge is consi tent with the electric curre nt car(cid:173)
`ried by the valence band of Fig. 9: the current is ronied by the unpaired
`electron in the orbital G:
`j = (-e)v(G) = (-e)(-,,(E)] = cv(E) ,
`which is just the current of a positive charge moving with the velocity
`ascribed to the missing electron at E . The current is shown in Fig. 10.
`
`(23)
`
`Greenthread Ex. 2084, p. 18 of 27
`Semiconductor v. Greenthread
`
`

`

`..
`
`- £ ,
`
`f
`
`f"lpre 9 (a) Al t • 0 all 1talr1 a.rt' llJJc..d ~ f ' aJ the top o( the l:»..,d; l~ , locity u u·ro at
`F bcnuk' <U.ldl, • 0. (b) An dectric Odd f. b applk-d :1
`~ x ~ The m-e or, the
`" lt"ctron1 b In the -k, dlm::tion and al.I ckruon1 m&b tnDJitou t~ ia tbc - l , .~.
`moving the hole lo the ,tale £ . (c) Aftt'r • further lnlt'n -.J t~ t'l«t,ons °'°"~ Cartb.rr ~ In l
`,p.ac,e and l1K" hok b now 11 D.
`
`Fl~ure 10 Motion o( drt-tron1 In th«' ronductkm l..;and aud t
`In tlic vakrK'<l ha111J In the t'1("(1r1c Odd E. TII<' IM>k i nd rn1ron
`dnfl \·rlodtlc, arc In opr>o,ltc dirl"(ilona, hut th<'lr rk-ctrit- t\lr•
`rc•nts arc In tlic ~ me d/r('('tlon, tl.c din.,c11oo o( the dc.'C1rw OdJ.
`
`liffectivc Mau
`When we look al the energy-wan .. vector rebtion t = (h1/2m)k2 for free
`electrons, Wl' sec thnt the c'Oeffi<:ient of k· detennlnes tbe <;'t.lrvaturo oft versu
`k. Turned uhout, we ca11 say that Jim, the rtX"iprocaJ m.us, determines the
`curvature. For c-lcdru11s in a band there can be r~ions of unu_nwJy high c·urva(cid:173)
`ture near the bund ~ap ut the zone boundary, a.~ we e • from the solutiom of the
`wave equation near tho zone boundary. If the energy g;ip b small in C'omparison
`with the free electron energy ,\ at the boundary, the curvature is enhanced by
`the factor Alf:,. and the r<-ciprocal mass is enhan('("d by the same factor.
`Jn semirondu<:tors tJ1e band width, whid1 is like the free elec-tron energy,
`is of the order of 20 cV, while the band gap is of the order of 0.2 to 2 eV. Thus
`the reciprocal mass is enhant'<'d hy a factor 10 to 100. and the effective mass ls
`reduced to 0.1-0.01 of the free electron mass. These values apply near the
`band gap; us wo go away from the gap tht, cur"atures are likely to approach
`those of free electrons.
`
`- - - - - - - - - - - - - - - - -
`
`Greenthread Ex. 2084, p. 19 of 27
`Semiconductor v. Greenthread
`
`

`

`tJO
`
`IJ .
`
`(24)
`
`To summMi.zc t.hc solutions of Chap«cr 7 for U po itive, an electron ne.u
`the lower