We now moveonto thenextorderofthe tight-bindingapproximation (1.9.20),namely termsfor nearest neighbors. For simplicity, we will use the example of asimple cubic lattice, inwhich each atom has sixnearest neighbors, as illustrated inFigure 1.42.For nearestneighborsinthe x-direction,wemustcomputetermsofthefollowingtype:
Uxx= −i±2 m2c2
à
d3r ´∗x(´r)[(∇xU)∇y− (∇yU)∇x]´x(´r − aˆx), (1.13.12) Uxy= −i±2
m2c2 à
d3r ´∗x(´r)[(∇xU)∇y− (∇yU)∇x]´y(´r − aˆx). (1.13.13)
±Fig.1.42 Adjacentp-orbitalsinoneplaneofasimplecubiclattice.
83 1.13 SpininElectronBands
Becausethetwoorbitals ´n(´r)and ´(´r − aˆx)arenotcenteredonthesamelocation,par- ity in the x-directionis no longera concern. However,parity in the y- and z-directions stillmatters,and Uxx vanishesdueto negativeparity inthe y-direction.However, Uxy is nonzero.
Ifwechangethevariable x in(1.13.13)to −x,thenthe x-integralbecomes I(a) =
à ∞
−∞dx ´∗x(x, y, z)[(∇xU)∇y− (∇yU)∇x]´y(x − a, y, z).
=à ∞
−∞dx ´∗x(−x, y, z)[(−∇xU)∇y+ (∇yU)∇x]´y(−x − a, y, z). (1.13.14) Since ´∗(−x, y, z) = −´(x, y, z)and ´y(−x, y, z) = ´y(x, y, z),wethereforehave I(a) = I(−a).Therefore,thesumofthetwonearest-neighbortermsinthe x-directioninthetight- bindingformula(1.9.20)is
iσzUxy(eikxa+ e−ikxa) = 2iσzUxycos kxa. (1.13.15) Wetakeintoaccountonlynearest-neighbortermswithorthogonal p-orbitals,assuming that termswith parallelorbitals have negligible overlap in space. Accounting for these nearest-neighborterms,thespin–orbitinteraction(1.13.7)isthen
HSO =
⎛
⎝ 0 −iσzU(kx, ky) iσyU(kx, kz) iσzU(kx, ky) 0 −iσxU(ky, kz)
−iσyU(kx, kz) iσxU(ky, kz) 0
⎞
⎠ , (1.13.16)
where U(ki, kj) = U0+ 2Uxy(cos kia + cos kja).
If ´k issmall,wecanassumethattheeigenstatesarestillwellapproximatedbythestates givenin(1.13.9)and(1.13.10).For ´k inthe x-direction,theenergiesofthesestates(when properlynormalized)are
³²3
2,±32|HSO|²3
2,±32²= U0+ 2Uxy(1 + cos kxa)
³²3
2,±12|HSO|²3
2,±12²= U0+23Uxy(5 + cos kxa)
³²1
2,±12|HSO|²1
2,±12²=−2U0− 83Uxy(2 + cos kxa). (1.13.17) Note that states that have the same magnitude of angular momentum but opposite k-directionhave the sameenergy. Thisis aconsequence ofthe fact that the cubic sys- temwehavechosenis centrosymmetric;thatis, U(−´r) = U(´r).(Thisisnotthesameas havingacentralpotential,whichrequires U(´r) = U(|´r|).)Centrosymmetryledtotheform ofthecouplingterms(1.13.15),whichgaveonlytermswithcos kxa.
If U(´r)hasanantisymmetricterm,thediagonaltermsforthe p-statesin(1.13.16)will nolongernecessarilyvanish. Then ∇xU willnotchangesign onthechange ofvariable x →−x,inwhichcasewehaveatermwith I(−a) =−I(a)in(1.13.14),whichgivesacon- tributionproportionaltosin kxa.Wetaketheleading-ordercontributionasonlythoseterms thatcouplenearest-neighbororbitalsalignedalongthesamedirectionasthevectorsepa- ratingthem,becausethesecorrespondtothemaximalspatialoverlapoftheorbitalwave functions.Letusassumethat U(´r)hasantisymmetrictermsalongthe x-and y-axes.Then, accounting again for parity in alldirections, the contributiondue to the antisymmetric termsis
HSO(a)=
⎛
⎝ 2Uxxσzsin kxa 0 0 0 2Uxxσzsin kya 0
0 0 0
⎞
⎠ . (1.13.18)
Whenthisisappliedtothestates(1.13.9)and(1.13.10)for ´k inthe x-direction,weobtain
³²3
2,±32|HSO|²3
2,±32²=±Uxx(sin kxa + sin kya)
³²3
2,±12|HSO|²3
2,±12²=±13Uxx(sin kxa + sin kya)
³²1
2,±12|HSO|²1
2,±12²=±23Uxx(sin kxa + sin kya). (1.13.19) Notethatinthiscase,stateswithoppositeangularmomentumhaveoppositeenergyshift.
Inotherwords,thebandsactasthoughthereisaneffectivemagneticfieldproportionalto themagnitudeof ´k.Thisis generallyapossibilityinnon-centrosymmetriccrystals.Fig- ure1.43 illustrates theeffectofthe terms(1.13.17)and (1.13.19)onthe p-states inour example.
If welookatonlyoneofthestatesinFigure1.43,itviolates theversionofKramers’
rule (1.6.16)wededuced inSection1.6, becausethesin kxa termgivesoppositeenergy shiftfor ´k →−´k.ThatversionofKramers’ruledidnottakeintoaccountspin.Themore generalversionofKramers’rule,whichwillbeproveninSection6.9,is
En,−mJ(−´k) = En,mJ(´k), (1.13.20) where mJistheprojectionofthetotalangularmomentum ´J = ´L + ´S.Itiseasytoseethat thisruleissatisfiedforthebandsinFigure1.43.
This general form of Kramers’ theorem can be viewed as a simple consequence of time-reversalsymmetry.Timereversalofanelectroninabandstate correspondstoflip- pingitspropagationdirectionfrom ´k to −´k andalsoflippingthesignofitstotalangular
±Fig.1.43 Energyofthesixstates(1.13.9)and(1.13.10)forthespin–orbitterms(1.13.16)and(1.13.18),forU0= 0.5,
Uxy= 0.1,and Uxx= 0.5.
85 1.13 SpininElectronBands
momentum.IntheabsenceofanytermsintheHamiltonianthatbreaktime-reversalsym- metry(suchasarealmagneticfield),theeigenstatesoftheHamiltonianmustalsosatisfy time-reversalsymmetry.
Exercise1.13.2 (a)Showexplicitlythatparityrulesgivethespin–orbitterm(1.13.18)for U(´r)antisymmetricinthe x-and y-directionsandsymmetricinthe z-directions.
(b) What would this term be if U(´r) had an antisymmetric term only in the x-direction?
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J.R. Chelokowski, D.J. Chadi, and M.L. Cohen, “Calculated valence-band densities of statesandphotoemissionspectraofdiamondandzinc-blendesemiconductors,” Physical Review B 8,2786(1973).
C.Cohen-Tannoudji,B.Diu,andF.Laloở, QuantumMechanics (Wiley,1977).
M.Z.HasanandC.L.Kane,“TopologicalInsulators,”ReviewsofModernPhysics 82,3045 (2010).
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Inthischapter,wewilleventuallydiscussafairamountofsemiconductorand transistor technology. Unfortunately,many physicist students don’t studythese in detail, thinking that they are “applied” science rather than “fundamental” science. But as we will see, to properly understand that technology, one must engage with many very fundamental concepts.Onesuchconceptistheideaofthe“renormalizedvacuum,”inwhichtheentire crystalinitsgroundstateisviewedasanewvacuum,andonlyexcitationsoutoftheground statecountasparticlesofinterest.
Thephysicsofsemiconductorsledtosomeofthemostfascinatingfundamentalphysics results of the twentiethcentury, including the fractionalquantum Hall effect,in which electronsactasthoughtheyhavechargeofsomefractionof e.Wewillreviewthisatthe endofthischapter,butquiteabitoffoundationmustbelaidtogetthere.