UvA-VU Master Course: Advanced Solid State Physics

Transcription

1 UvA-VU Maste Couse: Advanced Solid State Pysics Contents in 005: Diffaction fom peiodic stuctues (wee 6, AdV Electonic band stuctue of solids (wee 7, AdV Motion of electons and tanspot penomena (wee 8, AdV Supeconductivity (wee 9&0, RW Magnetism (wee &,JB Anne de Visse Rine Wijngaaden Jügen Buscow

2 Liteatue, softwae and omewo Te couse is based on te boo: H. Ibac and H. Lüt: Solid State Pysics 3 d edition (Spinge-Velag, Belin, 003 ISBN X See also: N.W. Ascoft and N.D. Memin: Solid State Pysics (Saundes College Publ. ISBN Compute simulations fom an essential pat of te couse: R.H. Silsbee and J. Däge: Simulations fo Solid State Pysics (Cambidge Univesity Pess, Cambidge 997 ISBN Softwae (feewae: Homewo execises will be distibuted tougout te couse Completing te couse gives 6 ECTS ~ 6 x 8 ous

3 Couse 3: Motions of electons and tanspot penomena ne τ E( σ m m* ij j i j Pictues ae taen fom te Solid State Couse by Ma Jael (Cincinnati Univesity, fom Ibac and Lüt, fom Ascoft and Memin and fom seveal souces on te web.

4 Couse 3: Motions of electons and tanspot penomena Equation of motion of electons Dude and Sommefeld models fo conductivity Cystal momentum is not momentum! Motion of electons in bands and te effective mass tenso Cuents in bands and oles Scatteing of electons in bands Electical conductivity of metals Quantum oscillations and te topology of Femi sufaces Quantum Hall effect Pictues ae taen fom te Solid State Couse by Ma Jael (Cincinnati Univesity, fom te boo of Ibac and Lüt, fom te boo of Ascoft and Memin and fom seveal souces on te web.

5 Equation of motion of electons Classical equation of motion in E and B field: dv F m e( E + v B dt dv v F m + e E + v B dt τ ( witout collisions wit collisions v ~ e t /τ v decays exponentially elaxation time τ steady-state aveage velocity eeτ v vav m cuent density j 0 eeτ nev av ne m v av

6 Dude model fo conductivity Classical model: dilute gas of electons neglect inteactions wit ote electons and ions between collisions independent electon appoximation fee electon appoximation collisions pobability /τ (time between collisions τ temal equilibium toug collisions j nev j av ne σ E ρ E eeτ m ne τ E m ρ esistivity σ conductivity τ ~ s, v T ~ 0 5 m/s mean fee pat l v T τ -0 Å σ ne τ m Paul Dude ( Maxwell-Boltzmann velocity distibution - equipatition of enegy / mv T 3/ B T electon tanspot wit v av v D dift velocity Impotant failue Dude: mean fee pat l can be >> inteatomic distance

7 Sommefeld model fo conductivity Quantum mecanical desciption ψ V e i Femi velocity Femi enegy v E F Femi-Diac velocity distibution Semi classical enegy gain electons Femi spee δe ee vδt v E( displaced δe E( δ v δ m in space d fee electons only! dift ee eτ dt δ stationay state EOM d dv F m e( E + v B mean fee: l v dt dt F τ (use Femi velocity! example coppe: v F.6x0 6 m/s ne τ τ ~ x0-9 at 4K l 4K 3x0-3 m ne τ j nevav E σ τ ~ x0-4 at 300K l 300K 3x0-8 m m m Sommefeld wos also at low T! Lie Dude! F m F F m x E x

8 Intemezzo: Cystal momentum is not momentum! Fee electon wit enegy ε in state ψ Hψ ε ψ ; H m d dx ψ ix e ; ε L m Momentum expectation value fee electons d ix d ix p i ψ ψ e i e dx dx L dx L Bloc electons d H + V ( x m dx ψ ( x u ( x e ix p u * ( x e i eal momentum ix u * * u ( x i du ( x dx dx d dx u ( x e ix dx cystal momentum

9 Motion of electons in bands and effective mass tenso te eal wold: electon state is wave pacet ψ ( x, t i( x ω( t U ( e d π goup velocity and dispesion ω v ; ω c( velocity of cystal electon depends on dispesion E( v ω( E( fee electons: E /m v /m p/m U ( const. δ ( 0 delocalized ψ ( x, t U ( const. ψ ( x, t δ ( x i e ( 0 x ωt localized

10 Velocity of cystal electon Example: tigt binding dispesion elation ε E at + A + B cos( a velocity v E( v Basin( a velocity is constant at fixed vey diffeent fom classical pictue v 0

11 ate of cange of goup velocity component wit effective mass tenso (invese eq. of motion j E j e & ( j j j i i i E E dt d v & & j j i ij E m ( * ( j j j i i ee E v & ( j ij i ee m v * & Semi classical eq. of motion in electic field

12 Effective mass nd deivative * m d E d flatte band ige m* dv / d > 0 m * > 0 dv / d < 0 m * < 0 Negative effective mass! effective mass appoximation m* constant wen E ( E + ( * x + y + m 0 z

13 Cystal (Bloc electon in electic field Foce due to electic field is equal to time deivative of cystal momentum d ee dt Bloc state evolves, afte time t: ψ ψ + wen state eaces BZ π/a -π/a Bloc oscillations NB Scatteing pevents obsevation Bloc osc.

14 Cuent fo Bloc states in a alf filled band Scatteing poduces steady state ψ ψ + ee τ τ elaxation time netto velocity E 0 cuent I e v occup. states eeτ/ ~5 m - Wit τ ~ 0-4 s and E ~ V/m << BZ ~0 0 m - in eality small cange

15 Cuents in bands and oles paticle cuent density of d at dj n v electical cuent density integate ove fist Billouin zone j full band d 8π 8π ( 3 3 e π 3 E( 8 st B. z. E( d cuent 0 insulato d v( E( E( v( density states in d /(π 3 diffeent occupied states mae diffeent contibutions to te cuent density lattice wit invesion symmety E( E(

16 patially filled band: E field edistibutes states symmety aound 0 lost cuent of positive cage, paticles in unoccupied states (oles + empty empty z B st occupied d v e d v e d v e d v e j ( 8 ( 8 ( 8 ( π π π π

17 nea top of te band ( taen fom top E( E 0 m * oles oles at te top of te band ave positive effective mass! v& d dt E( m * & e m * E insulatos conduct at T 0 n ~ exp(-e g / B T

18 Scatteing of electons in bands Wat did we lean: equation of motion electons/oles acceleate Bloc waves in pefect lattice no esistivity Tis cannot be tue: scatteing! deviations fom peiodicity (defects, lattice vibations electon-electon collisions - q scatteing at a defect o ponon momentum and enegy consevation E + E E3 + E4 ; scatteing esticted to naow -sell nea F 4 τ e e ~ ρ e e BT ~ EF nea 300 K τ e-e ~0-0 s >> τ e-p o τ e-d, scattes into 3, 4

19 Boltzmann eq. descibes non-equilibium steady state diving foce due to E and B field dissipation due to scatteing temal equilibium distibution EB0 cange of f in time (t-dt t + effect of scatteing expanding up to tems linea in dt Boltzmann equation Boltzmann equation and elaxation time appoximation s t f f E e f v t f +,, ( ( / ( ( T E E E B F e t f f dt t f dt t dt ee vdt f t f s + +,, (,, ( ( ( ( 0 f f t f s τ Relaxation time appoximation: ate at wic f etuns to equilibium deviation of f fom f 0

20 Paticle cuent density j n 8π Electical conductivity of metals v( f ( 3 st BZ d - linea effects in electic field (Oms law - isotopic medium, cubic lattice - lineaized Boltzmann eq. e f 0 σ jx / Ex vx ( τ ( d 3 8π E e vx ( σ τ ( df 3 8π E EF v( inset distibution function E only states at Femi suface impotant Femi suface Conductivity expessed as integal ove te Femi suface, depends on v(e F and τ(e F Fo paabolic band tis educes to: σ e τ ( E * m F n

21 Electical conductivity of metals Mattiesen s ule τ τ τ τ τ def e e p mag + τ CEF +... ρ constant 5 5 x dx ρ p a( T / θ ρ ρ0 + ρe e + ρ p + ρmag + ρcef +... x ( e ( e ρ 0 e e AT θ / T 0 x esistance of sodium 3 diff. defect concentations esistivity of coppenicel alloys ponon (Debye esistance

22 Electical conductivity of metals: examples esistivity of eavyfemion compounds esistivity of supeconducting cupates: La -x S x CuO 4

23 Quantum oscillations and te topology of Femi sufaces Motion of electons and oles in magnetic field Loentz foce dv m e( v B dt fo wave pacet mv d e [ E( B] dt Electons move: in plane B tangential to suface of constant E( open obits closed obits

24 Peiod of obit in magnetic field T dt eb [ d E( ] d d de ds de Fee electons Sπ and E /m Subniov-de Haas effect ds π m* T T.3 K eb de eb ω π T c eb m* cycloton fequency wy oscillations?

25 de Haas-van Alpen effect Landau quantization E ( n + ω ; ω n c c eb m* de Haas van Alpen ( ( Landau tubes Enegy splitting : Femi suface aea' s : Peiod of E oscillations : n+ E n ω π eb Sn+ Sn SF, ext B π e c eb m* π T S F,ext (λ+n S Landau tubes coss E F wit peiod (/B

26 Some numbes: peiod T π/ω c 3.6x0 - s in T quantum numbe: (n+/ω c ~ E F fo silve E F 5.5 ev n ~ 4.6x0 4 fo B T absence of temal smeaing: B T/ω c < B T/ω c B m/e(t/b.34(t/b low T & ig B B dhva signal (magnetization in silve B [,,] T.3 K two peiods nec and belly obits S (belly/s (nec 5

27 Quantum Hall effect in D systems D electon gas fomed at inteface of lattice matced eteostuctues o quantum wells GaAs/Al 0.3 Ga 0.7 As In 0.53 Ga 0.47 As/InP

28 Heteostuctues and quantum wells potential well E-gap AlGaAs G GaAs Heteostuctue E-gap Quantum well AlGaAs GaAs AlGaAs

29 Magnetotanspot in Hall ba geomety Classically R xx V I xx ρ xx R xx b L m* ne τ Vxx I neµ b L b L R xy V xy I ρ xy mobility µ v D /E Vxy R xy I B ne Hall esistance linea in field gives caie concentation and type (electons o oles

30 Te quantum Hall effect ρ xx (Ω InGaAs/AlGaAs T 30 mk n.8x0 5 m - µ3.4 m /Vs ρ xy /ie i ρ xy (Ω Klaus von Klitzing Nobel Pize Pysics 985 fo te discovey of te quantum Hall effect B (T QHE: ρ xy quantized (esistance standad allows pecise detemination fine stuctue constant ie ρxy α 5.8 (Ω i e µ c 0 37

31 Enegy levels in band i split into discete Landau levels in magnetic field Ei, n ( n + ω c + gµ BB n intege Landau quantization Zeeman tem ωc eb m* cycloton fequency Lev Landau Nobel Pize Pysics 96 fo is pioneeing teoies fo condensed matte DOS D m* π N L states pe unit aea pe Landau level N L eb Filling facto ν ν n N D L nd eb

32 Simple explanation QHE Landau level Extended states Localised Localized states disode/impuities extended and localised states in Landau levels i+ Slope i+ i R xy B wit inceasing field B Landau levels pused to above E F plateau-plateau tansitions B c Widt B R xx B B conduction wen Landau level in extended states widt extended states 0 wen T 0

33 Typical numbes Enegy distance between levels B T~ 5 mev nea 300 K ω c ~.6 mev/t (m* 0.067m e B T << ω c T< 4 K ω c eb / m* Filling faction typical n D x0 5 m - quantum limit ν B~ 8.5 T ν n N D L nd eb ig B/T needed R xy (Ω R xx (Ω K.07 K K 0.89 K K Field (T 4.50 K.07 K K 0.89 K K InGaAs/GaAs quantum well n x0 5 m - T K

34 Factional quantum Hall effect νe ρxy 5.8 (Ω ν ν /3, /3, /5, /5 etc. factional quantum Hall effect in ig mobility GaAs/AlGaAs eteostuctue T 0.5 K