Lecture 4. Sommerfeld s theory does not explain all

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1 Lectue 4 Nely Fee Electon Model 4. Nely Fee Electon Model 4.. Billoiun Zone 4.. Enegy Gps 4. Tnsltionl Symmety Bloch s Theoem 4. Konig-Penney Model 4.4 Tight-Binding Appoximtion 4.5 Exmples Refeences:. Mde, Chptes 7-8. Kittel, Chpte 7. Ashcoft nd Memin, Chpte 9 4. Kxis, Chpte Lectue 4 5. Ibch, Chpte 7 Sommefeld s theoy does not explin ll Metl s conduction electons fom highly degenete Femi gs Fee electon model: woks only fo metls - het cpcity, theml nd electicl conductivity, mgnetic susceptibility, etc Dwbcks: pedicted electon men pth is too long inceses with tempetue positive vlues fo the Hll coefficient, mgnetotnspot diffeence between good conducto (0-0 Ohm-cm) nd good insulto (0 - Ohm-cm) 0!!! Lectue 4

Electon Occupncy of Allowed Enegy Bnds Adpted fom Kittel No electons cn move in n electic field (enegy bnd is completely filled o empty) insulto ; One o moe bnds e ptly filled conducto Bsic

2 Electon Occupncy of Allowed Enegy Bnds Adpted fom Kittel No electons cn move in n electic field (enegy bnd is completely filled o empty) insulto ; One o moe bnds e ptly filled conducto Bsic Assumptions: - cystl stuctue is peiodic - peiodicity leds to fomtion of enegy bnds (llowed enegy levels) - enegy bnds e septed by enegy Lectue gps 4 o bnd gps (egion in enegy fo which no wvelike electon obitl exist) 4. Nely Fee Electon Model In fee electon model: ll enegy vlues fom 0 to infinity e llowed h h ε = k = ( kx + k y + kz ) k m m Wvefunctions e in the fom: ψ ( ) = exp( ik ), whee the components of the wvevecto k k π 4π e: k x = 0; ± ; ± ;... L L Nely fee electon model: wek petubtion of electons by peiodic potentil of ions Lectue 4 4

3 Nely Fee Electons Conside the effects due to peiodic cystl stuctue n Unde condition k = ± G = ± electon wve will undego Bgg eflection π, the Enegy gps develop t these k due to these eflections At k = n π/ the wvefunctions e not the tveling wve of fee electons The egion between - π/ nd π/ : fist Billouin zone of this D lttice Lectue Billoiun Zone in D: extended, educed nd epeted Repeted Extended Reduced Lectue 4 6 BZ boundies

4 Lectue 4 7 Reduction to the fist Billouin zone This genel demnd of peiodicity implies tht the possible electon sttes e not esticted to single pbol in k-spce, but cn be found on ny pbol shifted by

4 4 Lectue 4 7 Reduction to the fist Billouin zone This genel demnd of peiodicity implies tht the possible electon sttes e not esticted to single pbol in k-spce, but cn be found on ny pbol shifted by ny G-vecto: Fo D cse: ) ( ) ( G k m G k k h + = + = ε ε G G π = Lectue 4 8 Billoiun Zone in D: Wigne-Seitz cell of the ecipocl lttice ; ; ; such tht e bsic vectos,, whee, b b b b b b b n b n n b G = = = + + = π π π Billoiun Zone in D Recll: ecipocl lttice vecto Some popeties of ecipocl lttice: The diect lttice is the ecipocl of its own ecipocl lttice The unit cell of the ecipocl lttice need not be plellopiped, e.g., Wigne-Seitz cell fist Billoin Zone (BZ) of the fcc lttice

5 4.. Oigin of the Enegy Gp Cystl Potentil - U The pobbility density of the pticle is ψ*ψ = ψ Fo pue tveling wve: ρ = exp( ikx)exp( ikx) = Fo plne wves the chge density is not constnt: fo the wveψ ( + ) : ρ( + ) = ψ ( + ) cos fo the wveψ ( ) : ρ( ) = ψ ( ) sin πx πx Lectue 4 9 Mgnitude of the Enegy Gp The potentil enegy due to the cystl cn be ppoximted s: πx U ( x) = U cos This potentil hs the peiodicity of the lttice, U(x) = U (x + ) The wvefunctions t the Billouin zone boundy k = π/ (nomlized ove unit length of line, ) e x cos πx nd sin π The diffeence between the two stnding wve sttes is πx πx πx E g = U ( x)[ ψ ( + ) ψ ( ) ] dx = U cos (cos sin ) dx = U 0 The gp is equl to the Fouie component of the cystl potentil Lectue 4 0 5

6 4. Tnsltionl Symmety Bloch s Theoem Bloch s theoem: the wve functions of the electons in cystl must be of specil fom (the Bloch fom) ψ ( ) = exp( ik ) uk ( ) k u ( ) = u ( + T ) k k u k () the peiodicity of the lttice (depends on the wve vecto!) Note: the Bloch function cn be decomposed into sum of tveling wves In D: Conside cystl of length L = N (N pimitive u. c. of length on ing) The peiodic boundy condition demns tht Cψ ( x) = ψ ( x + ); hee C - const. Addition of the tnsltionl symmety gives : N ψ(x + N) = ψ ( x) = C ψ ( x) iπs ψ( x + n) = ψ ( x) nd C = exp ; s = 0,,,..., N - N iπsx Theefoe ψ ( x) = uk ( x)exp( ) N Lectue 4 Kittel, pp Bloch s Theoem Fo non-intecting electons moving in peiodic potentil, U () U ( + R) = U ( ) Bloch wve functions e peiodic functions u () modulted by plne wve of longe peiod Peiodic function u () Lectue 4 6

7 Bloch s solution Fo non-intecting electons moving in peiodic potentil, U () ˆ p H = + U ( R) m ψ ( + R) =ψ ( ) - tempting, but WRONG! Lectue 4 Tnsltion Opetos Let Tˆ tnslte wve function by R i : h Tˆ R = e R PR ˆ Theoem: if one hs collection of Hemitin opetos tht commute with one nothe, they cn be digonlized simultneously Any eigenvecto of the Hmiltonin cn be tken s n eigenfunction of ll the tnsltionl opetos s well: Use theoem: PR ˆ i Tˆ h ψ = e ψ = C R ψ ( + R) = Cψ ( ) R R ψ Lectue 4 4 7

8 Tnsltion Opetos Opeting with eigenfunction of momentum: eithe ikr e k ψ = C R ikr C R = e o k ψ = 0 k ψ k : hk : n : Bloch wve vecto Cystl momentum Bnd index Fo given vlue of Bloch wvevecto, thee is still the possibility of mny enegy eigenvlues (cn be lbeled by the bnd index n) The eigenfunctions mde possible by peiodicity is: Hˆ ψ ikr ψ ( + R) = e ψ ( ) nk nk ik u ( ) = e ψ ( ) nk nk o Tˆ ψ R nk nk = E = e nk ikr nk nk Lectue 4 5 ψ ψ Enegy Bnds Lectue 4 6 8

9 Allowed vlues of k If cystl is peiodic with (mcoscopic) dimensions M, M, M then equiing exp[ ik ] to be peiodic constins k to m l k = bl, 0 ml M l, whee b... b e such tht bl l ' = πδ ll' M l= l Peiodic boundy condition plce condition on how smll k cn be Demnding tht C R = e ikr be unique plces conditions on how big k cn be Numbe of points in cystl equls numbe of unique Bloch wve vectos Lectue 4 7 Enegy Bnds nd Goup Velocities Velocity of electons in the nth bnd with wve numbe k is: v = E nk k nk h Note: this is simil to the solution of wve equtions fo goup velocity: v = ω k ik ' ie t / h Wve pcket: k ' ik ' W (, k, t) = w( k ' k ) e ψ e dk ' k ' ik ' ie t / h ik ' ie t / h k ' k ' e w( k '') dk ' e Lectue 4 8 9

10 4. Konig-Penney Model The wve eqution is h d ψ + U ( x) ψ = εψ m dx In the egion 0<x< (U = 0), the eigenfunction is line combintion of plne h K wves tveling to the ight nd to the left with enegyε = m ψ = ikx In the egion b < x < 0 within the bie the solution is Ae ψ = Ce Qx ikx + Be + De Qx h Q wheeu 0 ε = m, Lectue 4 9 Konig-Penney Model Solution must be in the Bloch fom: ψ ( < x < + b) = ψ ( b < x < 0) e ik ( + b) The constnts A, B, C, D e chosen so tht wvefunction nd its deivtive e continuous t x = 0 nd x = At x = 0 At x = Solution: A + B = C + D i K (A-B) = Q (C - D) Ae ik ik( Ae P K + Be ik ik Be = ( Ce ik Qb + De ) = Q( Ce Qb Qb sin K + cos K = cos k ) e ik ( + b) De Qb ) e ik ( + b) [( Q K )QK ]sinh Qbsin K + cosh Qb cos K = cos k( + b) In the limit Q >> K nd Qb << Lectue 4 0 0

11 Functions nd Enegy fo the K-P potentil Lectue 4 Fist Billouin Zone fo fcc lttice Lectue 4

12 Fist Billouin Zone fo bcc lttice 4π Lectue 4 Fist Billouin Zone fo hcp lttice Lectue 4 4

13 Exmple: Nely Fee Electon in D Electons of mss m e confined to one dimension. A wek peiodic potentil is pplied: πx 4πx V ( x) = Vo + V cos + V cos () Unde wht conditions will the nely fee-electon ppoximtion wok? (b) Sketch the thee lowest enegy bnds in the fist Billouin zone. Numbe the enegy bnds (stting fom one t the lowest bnd) (c) Clculte (to fist ode) the enegy gp t k = π / (between the fist nd second bnd) nd k = 0 (between the second nd thid bnd) Lectue 4 5 Lectue 4, continued N th Billioun zone: geometicl view Pocedue: pependicul bisectos e dwn between the oigin nd ll neby ecipocl lttice points zone boundies the st, nd, nd d BZ e shded in diffeent colo (sme volume) electon esponse to the extenl electic field sme s fo fee electon till it ppoches zone boundy plne n electon once in the n th BZ emins in the n th BZ Lectue 4 6

14 Exmple in two dimensions Suppose D lttice hs two conduction el. pe lttice sites The # of k-sttes in BZ = the # of lttice points Fo wek potentil, shpe of the enegy sufce ~ sphee Fo lttice spcing The ecipocl lttice π/ The volume of st BZ 4π / The Femi sphee must hve sme volume the Femi sufce slightly out of the st BZ Lectue 4 7 Exmple in two dimensions Femi sufce completely enclosing the st BZ shpe of sufce is modified ne the zone boundy potion of the Femi sufce in nd BZ is mpped bck into the st zone potion of the Femi sufce in d BZ is mde continuous by tnsltion though ecipocl lttice vectos Hison constuction: n th BZ mpped into st BZ Lectue 4 8 4

Nely Fee Electon Femi Sufce Glley http://www.phys.ufl.edu/femisufce/peiodic_tble.

15 Nely Fee Electon Femi Sufce Glley Lectue 4 9 Symmety Popeties Bsic ide: fee molecule with N-toms N degees of feedom n 6 noml modes of vibtion (o line molecule hs n 5 noml modes of vibtion becuse ottion bout its molecul xis cnnot be obseved) Wht e symmety opetions of the molecule? Opetions (eflection, ottion, etc.) which leve molecule invint Lectue 4 0 5

16 Symmety opetions:. Symmety plne: σ. n-fold xis of symmety C n ottion by π/n Lectue 4 Symmety opetions:. Invesion symmety, i 4. Identity, E 5. n-fold ottion + eflection S n ottion by π/n + eflection though plne to ottionl xis Lectue 4 6

Point Goups nd Thei Repesenttions Demonstte concept by exmple with H O molecule Conside H O molecule lying in xz-plne: z Which symmety opetions leve molecule unchnged?

17 Point Goups nd Thei Repesenttions Demonstte concept by exmple with H O molecule Conside H O molecule lying in xz-plne: z Which symmety opetions leve molecule unchnged? y As we shll see, these fou opetions fom C ν point goup To estblish goup must conside how opetions (i.e., elements) multiply Ide: fom poduct tble If R nd R -symmety opetion, define poduct R R s consecutive ppliction of R nd R x Lectue 4 R element Goup of Elements Goup of elements is set of elements tht stisfy : ) E is membe of ) Obey ssocitive ule of multipliction : R(R' R'') ) Poduct R R' is membe of 4) R - exists set set = (R R') R'' Point goup t lest one point is left unchnged unde symmety opetion Clsses of conjugted elements: R, R e conjugte if R = S - R S, whee S is nothe element in the goup Typiclly ssocites opetions such s ottions o eflections, whee S tkes it bout nothe plne o xis Lectue 4 4 7

18 Mtix Repesenttions Mtix epesenttion of element R of goup [ D ij (R) ] Chcte of epesenttive mtix X (R) = tce [ D ij (R) ] C Fo the bove epesenttion of C ν the chctes e: X (E)= X (C )= X (σ xz )= X (σ yz )= Note: mtix el-ts fo the epesenttions of ll 4 opetos of C ν e ± Lectue 4 5 Goups nd Vibtions Genel poblem: how to connect epesenttions of highe dimensions to those of lowe dimensions Define foml sum: The dimensionl epesenttion of C ν tht we hve used cn be expessed s Lectue 4 6 8

19 Goups nd Vibtions Moe common poblem is to tke high dimensionl epesenttion nd educe to sum of low dimensionl epesenttions Poceed s guided by theoem: ) Given ny n-dimensionl epesenttion Γ n, with mtices D n (R ), D n (R ), nd coesponding chctes X(D n (R )), X(D n (R )), whee R i is n opetion of the point goup. If D j0 (R ), D jn (R ),, e the mtices nd X j (D jn (R )), X j (D jn (R )), the chctes of the ieducible epesenttions Γ oj of the point goup, then ) The numbe of ieducible epesenttions = numbe of clsses Lectue 4 7 Exmple Let s etun bck to H O Lectue 4 8 9

20 Digession: finding the j s Lectue 4 9 Extended chcte tble Some of the modes chcteize tnsltionl nd ottionl degees of feedom, eminde e vibtionl Lectue

Electric Potential. and Equipotentials

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