1. Brillouin zones of rectangular lattice. Make a plot of the first two Brillouin zones of a primitive rectangular two-dimensional lattice with axes

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1 Chap9 練 Brilloui oe of rectagular lattice Mae a plot of the firt two Brilloui oe of a priitive rectagular two-dieioal lattice with axe a b=3a Brilloui oe rectagular lattice A two-dieioal etal ha oe ato of valecy oe i a iple rectagular priitive cell a=å; b=4 Å (a) Draw the firt Brilloui oe ive it dieio i c - (b) Calculate the radiu of the free electro Feri phere i c - (c) Draw thi phere to cale o a drawig of the firt Brilloui oe Mae aother etch to how the firt few period of the free electro bad i the periodic oe chee for both the firt ad ecod eergy bad Aue there i a all eergy gap at the oe boudary 3 Hexagoal cloe-paced tructure Coider the firt Brilloui oe of a crytal with a iple hexagoal lattice i three dieio with lattice cotat a ad c Let c deote the hortet reciprocal lattice vector parallel to the c axi of the crytal lattice (a) Show that for a hexagoal cloe-paced crytal tructure the Fourier copoet U( c ) of the crytal potetial U(r) i ero (b) I U( c ) alo ero? (c) Why i it poible i priciple to obtai a iulator ade up of divalet ato at the lattice poit of a iple hexagoal lattice? (d) Why i it ot poible to obtai a iulator ade up oovalet ato i a hexagoal-cloe-paced tructure? 4 Brilloui oe of two-dieio divalet etal A two-dieioal etal i the for of a quare lattice ha two coductio electro per ato I the alot free electro approxiatio etch carefully the electro ad hole eergy urface For the electro chooe a oe chee uch that the Feri urface i how a cloed 5 Ope orbit A ope orbit i a oovalet tetragoal etal coect oppoite face of the boudary of a Brilloui oe The face are eparated by = 0 8 c - A agetic field B=0 3 gau= 0 - tela i oral to the plae of the ope orbit (a) What i the order of agitude of the period of the otio i pace? Tae v~0 8 c/ec (b) Decribe i real pace the otio of a electro o thi orbit i the preece of the agetic field 6 De Haa-va Alphe period of potaiu (a) Calculate the period (/B) expected for potaiu o the free electro odel (b) What i the area i real pace of the extreal orbit for B=0 = T? The ae period applie to ocillatio i the electrical reitivity ow a the Shubiov-de Haa effect 7 Bad edge tructure o p perturbatio theory Coider a odegeerate orbital ψ at =0 i the bad of a cubic crytal Ue ecod-order perturbatio theory to fid the reult

2 h h ' ( ) = (0) 0 p 0 (0) (0) where the u i over all other orbital ψ at =0 The effective a at thi poit i ' 0p 0 = (0) (0) The a at the coductio bad edge i a arrow gap eicoductor if ofte doiated by the effect of the valece bad edge whece c pυ E g υ where the u i over the valece bad; E g i the eergy gap For give atrix eleet all gap lead to all ae 8 Waier fuctio The Waier fuctio of a bad are defied i ter of the Bloch fuctio of the ae bad by / w ( r r ) = N exp( i r ) ψ where r i a lattice poit (a) Prove that Waier fuctio about differet lattice poit are orthogoal: dvw ( r r ) w( r r ) = 0 Thi orthogoality property ae the fuctio ofte of greater ue tha atoic orbital cetered o differet lattice ite becaue the latter are ot geerally orthogoal (b) The Waier fuctio are peaed aroud the lattice ite Show that for / ix ψ = N e u x) the Waier fuctio i 0 ( iπ ( x x ) / a w( x x ) = u0 ( x) π ( x x ) / a for N ato o a lie of lattice cotat a 9 Ladau level The vector potetial of a uifor agetic field B ẑ i A = Byxˆ i the Ladau gauge The Hailtoia of a free electro without pi i H h = ih eyb y x (i SI uit) We will loo for a eigefuctio of the wave equatio ψ = χ( y)exp[ i( x )] x Hψ = ψ i the for

3 (a) Show that χ(y) atifie the equatio h d χ h ( 0 ) = 0 ωc y y χ dy where ω c = eb / ad y0 = h x / eb (b) Show that thi i the wave equatio of a haroic ocillator with frequecy ω c where h = ( ) h ωc (c) Calculate the degeeracy per uit area of each Ladau level 0 3 金 利 量 來 粒 Nearly Free Electro Feri Surface Near a Sigle Bragg Plae To ivetigate the early free electro bad tructure ear a Bragg plae it i coveiet to eaure the wave vector with repect to the poit o the Bragg plae (Brilloui oe boudary) If we write = q ad reolve q ito it copoet parallel (q // ) ad perpedicular ( q ) to the the electro eergy ca be writte a / h q h = λ / ± 4λ / // q U where h λ / = ad U i the Fourier copoet of U(r) at It i alo coveiet to eaure the Feri eergy F with repect to the lowet value aued by the equatio above i the Bragg plae writig: F = λ / U o that whe <0 o Feri urface iterect the Bragg plae (a) Show that whe 0< < U the Feri urface lie etirey i the lower bad ad iterect the Bragg plae i a circle of radiu ρ = h (b) Show that if > U the Feri urface lie i both bad cuttig the Bragg plae i two circle of radii ρ ad ρ ad that the differece i the area of the two circle i 4π ( ρ ρ ) U π = h Verify that i a crytal with a fcc oatoic Bravai lattice the free electro

4 Feri phere for valece reache (6/3π ) /6 =0903 of the way fro the origi to the oe face i the [] directio 3 Alali etal lie Li Na etc have bcc lattice tructure What i the ratio of the Feri wavevector ( F ) to the hortet ditace betwee the Brilloui oe ceter Γ to the oe boudary (N)? Aue the Feri urface i ot ditorted by the lattice potetial Explai why the traport propertie of alali etal ca be explaied by Soerfeld free Feri ga theory? 4 Doe the Feri urface of a alalie earth etal eg Be Mg exted to the d Brilloui oe? Why? 5 Decribe the hape of the Feri urface of the oble etal (Cu Ag ad Au)? I it poible to fid a hole-lie orbit i the De Haa-va Alphe effect? 6 Uig the tight bidig ethod coiderig oly the orbital at each atoic ite how that for a iple cubic tructure with a lattice cotat a the electro eergy ca be writte a: ( ) γ (co a co a co a) x y where dv ϕ dvϕ ( r ρ ) ad ρ i the poitio of the earet eighbor What i the width of the bad? Calculate the ivere effective a teor ear the oe ceter 7 Uig the tight bidig ethod coiderig oly the orbital at each atoic ite how that for a bcc tructure with a lattice cotat a the electro eergy ca be writte a: ( ) 8γ co aco aco a x y where dv ϕ dvϕ ( r ρ ) ad ρ i the poitio of the earet eighbor What i the width of the bad? Calculate the ivere effective a teor ear the oe ceter 8 Uig the tight bidig ethod coiderig oly the orbital at each atoic ite how that for a fcc tructure with a lattice cotat a the electro eergy ca be writte a: ( ) 4γ ( aco co co co co y a a xa a xaco a y ) where dv ϕ dvϕ ( r ρ ) ad ρ i the

5 poitio of the earet eighbor What i the width of the bad? Calculate the ivere effective a teor ear the oe ceter 9 Explai the de Haa-va Alphe effect 0 Explai the Shubiov-de Haa effect Explai why i the de Haa-va Alphe ocillatio of gold there are two ditict period for agetic field i the <> directio? The right figure how the de Haa-va Alphe ocillatio i Ag The agetic field i alog a <> directio a how i the lower figure How ca you deterie the area ratio of the two extreal orbit o the Feri urface? What i the ratio? Why? 3 (a) I the de Haa-va Alphe eaureet of Au oe fid that the (/B) period of the agetic uceptibility i 0-5 T - What i the area ecloed by the correpodig extreal orbit o the Feri urface? (b) If the agetic field i directed i a pecific directio oe would fid a dog boe haped hole-lie orbit ad the area ecloed by thi orbit i roughly 04 of the area foud i (a) What i the directio of the agetic field? What i the (/B) period for thi orbit? 4 Explai agetic breadow? 5 A two-dieio electro ga (DE) ha a areal deity of 0 c - How large i the agetic field perpedicular to the DE eeded to have the Ladau fillig factor to be Aue that the pi are already plit by the Zeea effect 6 The agetic field tregth o the urface of the Earth i about 05 gau Etiate the degeeracy of each Ladau level if uch a field i applied to the oral directio of a two-dieioal electro yte 7 Draw qualitatively the deity of tate veru eergy for a three-dieioal electro yte if there i a fixed agetic field alog the directio