PH672 WINTER Problem Set #1. Hint: The tight-binding band function for an fcc crystal is [ ] (a) The tight-binding Hamiltonian (8.
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1 PH67 WINTER 5 Poblm St # Mad, hapt, poblm # 6 Hint: Th tight-binding band function fo an fcc cstal is ( U t cos( a / cos( a / cos( a / cos( a / cos( a / cos( a / ε [ ] (a Th tight-binding Hamiltonian (85 is H TB R t R R,δ ng band function in th nast-nighbo appoimation is ε In on-dimnsion, ε δ ± a so that U t (W can st U without loss of gnalit R R U R U t ia ia ( U t cos a ε t cos a and th Th dnsit of stats in on-dimnsion will b givn b D( ε d δ( ε ε Sinc ε t cosa cos w hav ε a t Taing th absolut valu (D(ε must b positiv: and L π dε d ta ε δ i δ t D L π dε ta ( ε δ( ε ε ε t L πta ε t (b Fo th fcc lattic with t <, th minimum ng occus at (G, ε(g - t Th maimum ng occus at th X point ( π/a,, ε(x t Thus, th bandwidth W 6 t
2 (c Th dnsit of stats givn b th Fidl modl is D ( ε fo W ε W and D( ε othwis W Fo Z lctons p atom, th stats will b occupid in th ang Z Z W ε ε ma wh εma W W W 5 Th cohsiv ng is ε εma εma W εd W ε W ( ε dε Z( Z t Z( Z W Not that th maimum stabilit (minimum cohsiv ng occus fo Z 5, i whn th d-band is half-filld 5
3 Show that th bul modulus of a cubic cstal pssd in tms of th lastic constants is givn b B [ ], i div Eq (9 fom Eq (7 In th notation intoducd b Eq (5, th f ng fo a cubic cstal, Eq ( can b wittn as in Eq (7: F d 6 αα α, In th cas of unifom stain, Eq (7 bcoms 6 F α α d In mati fom, this can b wittn α α, α, 6 α F ( wh us is mad of th fact that, and δ aing out th mati multiplication and intoducing, w gt F [ 6 ] [ ] [ ] 6 δ onsid a Talo pansion of th f ng about th quilibium volum: F F F( ( ( F δ δ δ Th lina tm vanishs and w can idntif th scond divativ F 6 [ ] F [ ] Finall, B [ ] F
4 Mad, hapt, poblm # (a Div Eqs (, ( and ( Th suggstd appoach is to appl a 5 o otation to th mati α and qui that th f ng b invaiant und this tansfomation * Th tansfomation quation (a is α ( RαγRδ In mati fom, ( ( ( ( ( ( Now, substitut th componnts of th otatd mati in Eq ( and insist that th f ng main unchangd onsid fist th tm in : [ ] ( ( [ ] [ ( 8 ] Subtacting th ight-hand sid fom th lft-hand sid, ( [ ]
5 5 Similal, [ ] [( ( ( ( ] [( ] ( And, finall, [ ] [ ] [( ( ( ] ( ( ( [ ] Putting it all togth, w hav [ ] Raanging, this can b put in th fom of Eq (: ( ( ( Stting th fist facto qual to o, w hav immdiatl, Eq (:
6 6 To gt Eq (, bgin b substituting into Eq (: F d ( ( ( ( ( ( d Intoducing th Lamé constants λ and µ w hav Eq (, F d λαα µ α, α, α d λ α αα µ α, α (b Div Eq (8 Th gnal pssion fo th stss tnso is Eq (: σ α α Suppos α Thn, σ λ γ γγ µ ( λ µ µ Not, all oth lastic constants,,, tc µ Gnaliing, fo th diagonal componnts, w can wit σ λ αα γγ µ γ Similal, σ µ and gnaliing again, σα µ α fo α Thn, combining ths sults, w hav α λδα γγ µ γ σ α αα
7 7 Mad, hapt, poblm #5 & (a Stat with Eq (5: ρu& α σα( wh σα α Loo at th componnt α and wit out componnts σ : σ ( (Oth lastic constants such as in cubic smmt Similal, σ and σ Thn, ρ & [ ( ] [ ] [ ] u σ σ σ Ta divativs using α u α u α and substitut: ρu&& u u u u u u u ( u u u u u ( ( ( u u ( ( u a and gnaliing to gt th vcto fom of th quation of motion, u ρ t u u u ( iˆ jˆ ˆ ( ( u u u (b Assum u(, t w p( i iωt Thn, ρ ω w p( i iωt Again, loo at th -componnt: t u w p ( i iωt
8 8 u ( iw iw iw p( i iωt ( u ( w w w p( i iωt u ρω w u u u ( w w w p( i iωt w p( i iωt ( w ( ( w w w [( ] w ( w ( w Th complt mati quation is w ρω w w ( ( ( w ( ( ( ( ( ( w w w (c To find th spd of sound, w nd to solv th ignvalu quation to gt th dispsion lation: ω c wh c is th spd of sound associatd with a paticula ignvalu Fo [ ],, and th scula dtminant is ρω ρω ρω Th is a singl oot, ω giving th spd of longitudinal wavs ρ ρ c L Fo undopd Silicon, Tabl givs 65 Gpa; th dnsit of ρ silicon is g/m 65 N/m Thus, c L 8 m/s g/m Similal, fo th two dgnat tansvs wavs, c T ρ
9 9 T and with 79 Gpa, c 58 m/s (d [, ], and th scula dtminant is ( ρω ( ( ( ( ρω ( ( ( ( ρω A B B Th dtminant is of th fom B A B A AB B ( A B( A B B B A Th fist oot is ρω ( ( ( ( so th spd of longitudinal wavs is c L 9 m/s ρ Th dgnat oots a ρω ( ( ( ( Th spd of th tansvs wavs is c T 58 m/s ρ hc: If th cstal w isotopic w would hav and w would pct th spd of sound to b th sam in th [ ] and [ ] dictions Fo th longitudinal wavs: ( c L c ρ ρ ρ L Similal, fo th tansvs wavs: c ρ ρ ρ T c T
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