SPIN AND CHARGE EFFECTS IN ONE AND TWO DIMENSIONAL SYSTEMS WITH SPIN-ORBIT INTERACTION. A Numerical Perspective. Catalina Marinescu

Transcription

1 SPIN AND CHARG FFCTS IN ON AND TWO DIMNSIONAL SYSTMS WITH SPIN-ORBIT INTRACTION A Numral Prspv Caala Marsu Clmso Uvrsy Collaboraor: C. Pasu Moa (Uvrsy of Orada) Work suppord by DO D-FG-R45897

2 OUTLIN Msosop Quaum Traspor Pur Logudal ad Sp-Hall Coduas D ad D sysms Tm-Dpd Algorhms dg Sp Aumulao Colusos

3 TH SPIN HALL FFCT AND TH SPIN ORBIT INTRACTION (SOI) H p m ( σ p σ p ) β ( σ p σ p ) α x y y x Rashba x x x x Drsslhaus H p α m ( σ p σ p ) β ( σ p σ p ) x y y x x x y y Ivrso asymmry of h ofg poal Bulk asymmry Wh oly Rashba s osdrd uvrsal valu of h sp Hall oduvy [Sova al. PRL 9 4] σ sh 8π

4 QUSTIONS. Robusss udr dsordr *Callao of h ff f D sysms h prs of fsmal amous of dsordr Prlmary alulaos wh h Kubo formalsm. Propr dfo of h sp urr 3. Th uvrsaly of h rasvrs oduvy 4. Salg wh h dmsoaly of h sampl xprms ar do f sz sampls

5 or MSOSCOPIC QUANTUM FORMALISM Lads fr of SOI ad dsordr * ( m a ) V 3 M. Bukr PRL (986) T.P. Park PRL (4) B.K.Nkol PRB (5) dor I SOI I RGION V 4 T pq s h rasmsso probably of a lro bw lads p ad q (rgardlss of sp) I I p sp p μ Boudary odos h q p T pq 4π q p ν V I I I 3 4 ( V V ) T I p μν pq α α ( V V ) I I p q sp 3 α sp 4 α q

6 TH TIGHT-BINDING APPROXIMATION Sgl bad gh-bdg modl T. P. Park ad P. Bruo PRB () P. Bruo PRL (997) N x N la of osa a o Nars ghbor hoppg paramr * m a Dsordr s rodud as loal s rgy W W ε V R α V D β a a

7 TH TIGHT BINDING HAMILTONIAN H R H D H H ( ) ( ) [ ] ( )( ) ( ) [ ] D D R R y y x x y y x x V H V H H δ δ δ δ δ δ δ δ α α α α α α ε Th Hamloa orporas h sp-dpd bad sruur ad h spdpd dsordr. Th SOI auss hoppg alog h dagoal ad s h sour of sp-flp sarg

8 TH RCURSIV GRN S FUNCTION MTHOD T T T T h T h G [ ] A q R p pq G G Tr T ν μ μν Γ Γ ( ) Σ Σ Γ μ μ μ p p p Th slf rgy marx Σ Σ Σ p p p 4 Σ p p F R H G [S. Daa lro Traspor Msosop Sysms 995] Th rardd slf-rgy fuo h solad lad p for sp hal μ μ Σ p Σ Σ p p For a prf mall lad

9 LONGITUDINAL AND SPIN HALL CONDUCTANCS T T T pq pq ou pq T T pq pq T pq T T T pq pq pq T T T pq pq pq T T T pq pq pq S. Souma ad B. K. Nkol PRL (5) G G sh L I V I sp 3 V h I sp 3 ou ou ou ( T T T T T ) ( T.5T.5T ) 8π

10 G sh Dpd o F V R V D Cla x sysm F - For V R V D ; G sh a ay F For V R /V D > ad F < (hol-lk) G sh > whl for V R /V D < G sh < PRB (5) Th sp Hall urr s grad h dro of h maor drvg fld. N. A. Ssy al. PRB 7 83 (4) B. K. Nkol al PRB (5) L. Shg al. PRL (5) Typal a 5m m *.7m 9mV V R -.6 mv V R /.5~.8

11 G L AS A FUNCTION OF F FOR DIFFRNT DRSSLHAUS COUPLINGS

12 Logudal odua as a fuo of V R ad V D F - G L (V R V D ) G L (V D V R ) V R V D s a symmry l

13 Sp-Hall odua as a fuo of V R ad V D F - G sh s asymmr alog h V R V D l

14 Sp-Hall odua salg G sh s almos osa up o 5x5 bu h boudars may b mpora G sh dpds o F ad sysm sz S also B. K. Nkol al. PRB (5); L. Shg al. PRL (5)

15 G sh AND TH FFCT OF DISORDR F -.

16 COMPARISON WITH TH KUBO FORMALISM ( ) ( ) ( ) ( ) ( ) ( ) σ σ r v r f f r r r v r f f r r y z x sh x x L ( ) α α α ψ b [ ] { } x z z x v H r v 4 σ Th gral Kubo formulas Sgl parl sas ar osrud from h loal orbal bass ( )( ) [ ] ( ) ( )( ) [ ] ( ) { } H H H r r H r r v z z ~ ~ 4 * * σ β ψ α ψ β ψ α ψ αβ β α αβ β α

17 Comparso of h umral rsuls wh h Kubo formula Ts h ff of h lads Th G SH s o a osa bu a fuo of F ad h SOI ouplg

18 D AHARONOV-BOHM RING ( ) ( ) [ ] ( ) ( ) [ ] x y y x r r m H σ ϕ β ϕ α σ ϕ β ϕ α φ φ σ ϕ β ϕ α σ ϕ β ϕ α φ φ ϕ ϕ os s s os os s s os * h B r / φ π φ Frusagla ad Rhr PRB (4) modfd o orpora h Drsslhaus rm (CPM&DCM) J. Phys.:Cods. Ma. 8 7 (6)

19 TH TIGHT BINDING HAMILTONIAN ( )( ) y x D D y x R R N σ ϕ ϕ σ ϕ ϕ σ ϕ ϕ σ ϕ ϕ φ φ π os s s os / / ( ) N N rg h H.. ε

20 LONGITUDINAL AND SPIN HALL CONDUCTANC r.μm.6mv R Q R.6mV D R D / ( N / π ) W. W. W. Also S. Souma ad B. K. Nkol PRL (5) oly for Rashba sysms

21 LONGITUDINAL AND SPIN HALL CONDUCTANC IN A DISORDRD SYSTM N Avrags ovr sampls Q R.954; Q D. Q R 6.366; Q D 3.83

22 G L ad G sh AS FUNCTIONS OF TH RASHBA SOI STRNGTH W. (shfd up by.) W. (shfd up by.7) W.3 (shfd up by.3) F. W.5 Sold ls : Q D. Dashd ls: Q D 3.83 W.3.5 Wh rasd dsordr h G L amplud s rdud bu h prody s prsrvd G sh xhbs dampd osllaos Q R

23 G L IN TH MAGNTIC FILD W. Q R. Q D.; Q R.954 Q D.; Q R Q D.; Q R Q D 3.83

24 SPIN HALL CONDUCTIVITY IN TH PRSNC OF TH MAGNTIC FILD W. Q R. Q D.; Q R.954 Q D.; Q R Q D.; Q R Q D 3.83 F.

25 SPIN ACCUMULATION FFCTS Nd for m-dpd hory Ladaur-Kldysh formalsm S G V d π [ ] F < ( r ) Tr ˆ σg ( r r; V ) < F V / / sp r < a ( ) G ( ) Σ ( ) G ( ) Nkol al PRL (5) Th o-qulbrum lssr Gr s fuo G < () h sasfs h sady sa Kldysh quao for quaum raspor a o-rag sysm G Σ r < [ ] r r ( ) H ΣL( V / ) ΣR( V / ) ( ) [ ImΣ ( V / ) ImΣ ( V / ) ] H s h gh bdg Hamloa L R

26 Th ou-of-pla ompo <S z (r)> of h o-qulbrum sp aumulao

27 Nkol al PRL (5) x s h dro of propagao

28 DG SPIN POLARIZATION FFCTS Dr umral grao of h Shrodgr quao sod ordr dffrg shm (MSD) (Th lap frog mhod) d ψ d ψ Δ ψ ψ T. Iaka PRB (994) Askar J. Chm. Phys. 68(6) 794 (978) H ψ xp( HΔ) ψ MSD(sod ordr dffrg shm) Δ ψ Δ [ xp( HΔ) xp( HΔ) ] ( τ Δτ ) ψ ( τ Δτ ) HΔτψ ( τ ) ψ Shm s aura up o (HΔ) Δ.

29 ψ α k σ WAV PACKT PROPAGATION Cosdr a pak propagag alog y ( x y τ ) C s[ k ( x x) ] x a π ( L ) Δk y -umbr of op hals x k y y y ( y y ) ( ) σ χα k a V mv m*.4m y R F.35a 3.3m mv α 5 8mVA.5.4mV

30 LOCAL SPIN DNSITIS * Sx( τ ) R{ ψ ( τ ) ψ ( τ )} * S Coour plos for S y ( τ ) Imψ ( τ ) ψ ( τ ) z * (prpdular o h dro of S ( τ ) ψ ( τ ) ψ ( τ ) z propagao) { } V R N.4 8 τ 4 τ 4 τ S y alog h dro of propagao

31 AVRAG SPIN DNSITIS y s h dro of propagao! Avrag sp aumulao <S x > <S y > <S z > Trasvrs oorda (m) Avrag sp aumulao <S x > <S y > <S z > Trasvrs oorda (m) 4 ( / ) V R. N 8 Avrags ar do alog h logudal dro

32 AVRAG SPIN ACCUMULATIONS FOR DIFFRNT V R Avrag sp aumulao <S x > <S y > <S z > V R. V R.4 4 ( / ) Trasvrs oorda (m) N 8

33 AVRAG SPIN POLARIZATION AS A FUNCTION OF DISORDR <S x > W W.5 W. Avrag sp aumulao <S y > <S z > Trasvrs oorda (m) 4 ( / ) Avrag ovr al dsordr ofguraos V R 4. N 8 Sp aumulao prsss up o W.

34 () ( ) ( ) ( ) ( ) [ ] H d H d d H H θ ; ~ ; ~ ; ~ ; ~ TIM DPNDNT FORMALISM Th parl sour mhod [T. Iaka Taaka PRB 57 (998)] * valua h marx lms of h Gr s fuo ad hos of hr produs wh ohr quaum opraors for larg dsordrd sysms *Avods dr dagoalzao of h Hamloa wh h gh-bdg Hamloa s usd. (CPU m ad mmory sz lar dpd of h sysm sz) For >> oly h frs rm s rlva Sgl frquy sour rm

35 ( ) ( ) ( ) ( ) ( )T T T T T G G H T ; ~ ; ~ lm lδ T Ay sa a b hos as h al sa! ( ) AG valua h marx lms of a produ ludg svral Gr s fuos ad ohr opraors Th marx lms of h Gr s fuo a b alulad a may dffr rgy valus by solvg h quao oly o. ( ) () ( ) ( ) ( ) ( ) ( )T l l l l H l H l l l l l G H d H d d θ ; ~ ; ~ ; ~ ; ~

36 > ar h gh bdg orbals θ ar radom umbrs [ π] ( ) ( ) G d T l T l ; ~ T Δ δ Th umral soluo o h Shrodgr quao (lap-frog mhod) ( ) ( ) / ; ~ ; ~ ; ~ max < Δ Δ Δ Δ Δ α α θ H Th al sa s a radom k N θ Φ s δ θ θ Φ Φ A A

37 SPIN ACCUMULATION IN TH BALISTIC RGIM Sz V dy f f σ z vy L ε ε ε ε K. Nomura al. PRB (5) <S z > Kubo formalsm wav pak propagao L 5a L / < L s W. W Trasvrs oorda (m). V R.3 Δε.4 / V R.4 N F. τ 9

38 σ z x sh v y Tr Ω 4 [ y H ] { σ v } z TH KUBO FORMULA Dpd o F z ( H ) G v { δ } x F x R y Avrag ovr al sa vors ad ovr dsordrd sampls /8π Nkol al. PRB (5)

39 CHARACTRISTIC LNGTH SCALS Sp prosso lgh L s vf αk F Ma fr pah (smlassal xprsso) kfa l πd( μ) W RGIMS L S < L smlassal rgm L S > L msosop rgm l < L dffusv rgm l > L balls rgm L s h sysm sz

40 SCALING OF TH SPIN HALL CONDUCTIVITY Th bhavor of GsH s srogly orrlad wh L s

41 SCALING OF TH SPIN HALL CONDUCTIVITY (Logarhm sal) G sh L /ξ F.

42 CONCLUSIONS Numral valuao of h raspor paramrs D ad D sysms wh SOI rao suppor h xs of h sp Hall ff as a robus propry of h sysm.