Phys463.nb Conductivity. Another equivalent definition of the Fermi velocity is

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Griffin Harper
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1 39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h nx chapr), w us h firs dfiniion vf â âk kkf Th boo lin: h firs dfiniion is or gric. Th scond on H pf L only works for k Conduciviy Undr a saic lcric fild (), lcrons fl a consan forc. (6.4) F Th scond Law of won ll us ha âp (6.4) F â If w hav a consan forc, P incrass linarly as a funcion of PHL PH L F (6.43) In quanu chanics, P k, so w hav k HL k H L F k H L (6.44) Hr, if w urn on h fild a, a, h Fri sa is a sphr crd a k. A >, h Fri sphr is crd a k HL, bcaus h wavvcor of vry lcron is shifd by his aoun a i. For fr lcrons in a prfc crysal, khl as. Howvr, in a rals solid, khl will NOT divrg, bcaus hr ar collisions bw lcrons, bw lcrons and phonons, and bw lcrons and ipuriis. Ths scarings will rduc h ou owards k. Thy will inroduc viscosiy o h Fri liquid, which rducs h vlociy and ou of h liquid. vually, h scarings and h fild will balanc ach ohr and k will urn o a fixd valu Th oal forc can b wri as F p (6.45) v Hr h firs r is h lcric forc and h scond r is a viscosiy r, which is proporional o v, rducing h spd. Th cofficis and ar h lcron ass and h collision i (avrag i bw wo collisions). Using h scond law of won: âp p p (6.46) â whr p is h ou bfor w urn on h lcric fild and w assu h iniial condiion ph L p. I is asy o find ha h soluion o his quaion is p p I ã M (6.47) This ans ha h chang of ou p p p I ã M In h saic lii H L, (6.48)

2 4 (6.49) p Th chang in wavvcor k (6.5) This approxiaion is known as h Drud approxiaion (h Drud odl). Th classical hory (ignor h fac ha lcrons ar frions) Th oal lcric curr is n k j n v n (6.5) This is h Oh s Law (6.5) j Σ whr h conduciviy n (6.53) Σ Th quanu hory (ak ino accoun h Pauli xclusiv principl) Assuing ha h fild is along h z dircion. Th oal vlociy along z is j X n v\ [ N v_ kf vf k Π H ΠL3 à â Θ cos Θ Π à Π â Φ à â Θ kf k cos Θ vf cos Θ Π kf vf k kf vf 4 Π Bcaus 4 Π kf kf vf 4 Π vf 4 Π (6.54) kf 3 4 Π 3 N kf 3 Π (6.55) 3Π kf 3 j 4 Π N 4 Π 3 4 (6.56) 3 (6.57) Σ 4 Typically, w absorb h xra 3/4 facor ino h dfiniion of, so h conduciviy urns back o is classical for (6.58) Σ which is idical o h classical forula shown abov. is, w us h an fr pah l vf in h forula, insad of. Th physical aning of h an fr pah is h avrag disanc ha an lcron ravls bw wo collisions. Using l, w hav l Σ (6.59) vf Th rsisiviy Ρ is

3 Ρ (6.6) Σ 4 Collision i Th collision i cos fro collisions bw lcrons collisions bw lcrons and phonons: p collisions bw lcrons and ipuriis: i If w assu ha diffr scarings ar indpd, h oal saisfis p (6.6) i Noic ha Ρ (6.6) n Ρ p n i n p n i (6.63) If w dfin Ρ n and Ρp n p and Ρi n i (6.64) so w found Ρ Ρ Ρp Ρi (6.65) Rsisiviy in a ral al a fini praur In a ral al a fini praur, h approxiaion usd abov ar sill valid, bu h collision i will show praurdpdc. As a rsul, Ρ is a funcion of T. Typically, Ρ dcrass as T gos down. A T, Ρ gos o a fini valu ΡHT L, which is known as h rsidu rsisiviy. A low T, Ρ and Ρp dcrass o as T is rducd down o, whil Ρi is T indpd. ry ypically, Ρ µ T, Ρp µ T 5 and Ρi µ consan Ρ Ρi A T Ap T 5 (6.66) A low praur, Ρi >> Ρ >> Ρp, so ipuriy scaring is h doina conribuion for Ρ. A high praur, Ρ is ypically a linar funcion of T Ρ AT (6.67) This is bcaus h nubr of phonons is proporional o T a high praur. n xpj N» J OHT LN (6.68) h Hall ffc (par I) Considr a hin sapl (lcrons oving in D). W apply a agnic fild prpdicular o h plan (in h z dircion) F K v BO (6.69) Th scond r is h Lorz forc, which is prpdicular o h vlociy. If w pass a curr along h x dircion. Th lcrons will fl a Lorz forc along h y dircion and hus posiiv (ngaiv) charg will accuula along h op and boo dg of h sapl. This charg accuulaion will induc a lcric fild along y, prpdicular o h curr. If w hav a saic curr, his lcric fild will balanc h :Lorz forc.

4 4 Th scond r is h Lorz forc, which is prpdicular o h vlociy. If w pass a curr along h x dircion. Th lcrons will fl a Lorz forc along h y dircion and hus posiiv (ngaiv) charg will accuula along h op and boo dg of h sapl. This charg accuulaion will induc a lcric fild along y, prpdicular o h curr. If w hav a saic curr, his lcric fild will balanc h :Lorz forc. ` ` y y v B vx B y (6.7) y vx B (6.7) Th curr along x is jx n vx (6.7) h raio bw y and j X, which is calld h Hall rsisanc, is y Rxy vx B jx n vx B (6.73) Th raio bw Rxy and B is known as h Hall coffici Rxy RH (6.74) B RH is proporional o n (invrs of h lcron dsiy), his asur is h sandard chniqu o drin h dsiy of lcrons in a arial. n (6.75) RH Noic ha h dsiy n hr is D lcron dsiy (nubr of lcron pr ara) (6.76) n n D A For a 3D arial, h 3D lcron dsiy is n n3 D Ad n D d (6.77) whr d is h hicknss of h sapl h Hall ffc (par II wih scarings) In h calculaions abov, w didn considr h lcron collisions. If w ra h collisions using h Drud approxiaion (assuing ha h collisions ar dscribd by a singl parar, h collision i), h forc on an lcron is F K v BO v (6.78) Using h scond law of won, w find ha âv â K v BO (6.79) v w hav wo quaions of oion â vx vx â â vy (6.8) vx B y (6.8) vy â v y B x Using h iniial condiion vx v y a, h soluion of hs diffrial quaions is vx Ix B y M B ã AIx B y M cos B I y B x M sin B B (6.8)

5 vy I y B x M ã AI y B x M cos B x Ωc y Ix Ωc y M cos HΩc L I y Ωc x M sinhωc L ã Ωc ã Ωc B (6.83) (6.84) Ωc y Ωc x vy Ix B y M sin B W can dfin h cycloron frqucy Ωc B, so ha vx B 43 I y Ωc x M coshωc L Ix Ωc y M sinhωc L (6.85) Ωc Th saic soluion a is x Ωc y vx (6.86) Ωc y Ωc x vy (6.87) Ωc If h curr is along x Iv y M, w find ha y Ωc x (6.88) x Ωc x x Ωc y vx Ωc x (6.89) Ωc Thrfor, y Rxy y jx Ωc x n vx Ωc x B B (6.9) And hus Rxy RH (6.9) B Fr lcrons in 3D a fini praur Nubr of lcrons on a quanu sa wih rgy (h FriDirac disribuion funcion) f HL xpj Μ N (6.9) Noic ha f HL, which agrs wih h Pauli xclusiv principl. Dsiy of sas (DOS) How any sas do w hav in h window of â? DHL â? (6.93) To answr his qusion, l s considr h oal nubr of lcrons Nà F DHL â Toal nubr of lcrons can also b wri as (6.94)

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors

Boyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar