On a proper definition of spin current

Transcription

1 On a pope definition of pin cuent Qian Niu Univeity of Texa at Autin P. Zhang, Shi, Xiao, and Niu (cond-mat ) P. Zhang and Niu (cond-mat/ ) Culce, Sinova, Sintyn, Jungwith, MacDonald, and Niu (PRL,93,046602,2004)

2 Spin cuent in pin-obit coupled ytem Conventional definition Cannot be meaued diectly --- no conjugate foce exit. Can be finite even in Andeon inulato. Connection to pin accumulation S dx J But thi i Wong! J J & J & SS x

3 Pope pin cuent Focu on ytem with no bul pin geneation. Then, we can intoduce a new and meauable pin cuent, which hould be ued to infe pin accumulation Ha a conjugate foce and atifie Onage elation, and ha the deied popety to vanih in imple inulato.

4 Outline Intoduction A emiclaical pictue The pope pin cuent Linea epone and pin Hall effect Onage elation and meauement Spin Hall in inulato Concluion

5 A bighte futue with emiconducto pintonic Can do what metal do: GMR, pin tanfe,..., uing feomagnetic emiconducto Readily integated with emiconducto device: poible way aound impedance mimatch in pin injection. Tunable: tanpot, magnetic and optical popetie can be eadily contolled by doping, gating, and pumping. Spin-obit: tong in emiconducto, may lead to novel effect uch a electic geneation and tanpot of pin

6 Semiclaical pictue Spin-obit built into the band tuctue: not a petubation. Caie of chage and pin: epeented by wave pacet. Effect of electic field: Difting and band mixing. Impuity effect: catteing and elaxation.

7 + ' ' ' ' / n n n n n n n n ee u i u u u u ε ε Effect of electic field Effect of electic field Mixing Mixing Difting Difting n c c n c c ee Ω & h & & h ε u u i n n n Ω whee Bey cuvatue

8 Spin-chage caie Chage -e Spin S ( c, c ) ( c, c ) (, ) S (, t) d 3 c d 3 c f ( c, c, t) ˆ δ ( ˆ) c

9 Semiclaical pin continuity equation Culce et al (PRL,93,046602,2004) S t + J T + d 3 df dt ˆ Toque denity: T (, t) 3 3 d f (,, t) ˆ d f (,, t) ( ˆ ) ˆ Relaxation: S /

10 Spin geneation by electic field Inide a homogeneou ytem: S t ee d 3 f ( c,, t) ˆ S / Geneally noneo in inveion aymmetic cytal L. S. Levitov et al., Sov. Phy. JETP 61, 133 (1985) P. R. Hamma and M. Johnon, Phy. Rev. Lett. 88, (2002) Y. Kato et al, Cond-mat/ (2004). D.Culce,et al, Cond-mat/ (2004).

11 Spin cuent and accumulation Aume no pin-geneation in the bul Rahba (fo pin ), 4-band Luttinge, Sytem with inveion ymmety. Spin continuity equation: S t + ( J + P ) S J + P Spin accumulation. S dx ( J + P ) SS x

12 Geneal fomulation Continuity equation S t + J Τ Spin denity S Ψ + ( ) Ψ( ) Cuent denity J & Toque denity Τ whee, & [, H ] ih /

13 New pin cuent Aume eo pin geneation in the bul Toque dipole denity Τ 1 3 V d Τ ( ) 0 ( ) P ( ) Spin i coneved in the bul New pin cuent S t + ( J + ) J eff ( J P + P ) 0

14 Toque dipole denity A mateial popety P ( ) 0, outide ample Bounday toque n P

15 Diplacement pin cuent On aveage ove pace P & Maxwell diplacement cuent x Q & x Q& δ +Q -Q x Q& New pin cuent δx J J + P & + & eff ( )

16 On equilibium Linea epone Toque epone to electic field Τ ( q) χ ( q) E ( q) q χ E ( q) β Τ ( ) 0 P ( ) 0 β eveywhee eveywhee q β β Toque dipole denity New pin cuent J α eff P q ] Re[ i χ β Eβ ( σ + Re[ i χ ]) E α αβ q β β

17 Spin Hall effect: theoy Extinic: Dyaonov and Peel (71), J. E. Hich (99), S. Zhang (00) Intinic: Muaami et al Science (03) Sinova et al PRL (04) _ F SO _ F SO I

18 Spin Hall effect: expeiment Rahba 2D hole Wundelich et al PRL (05) n-type emiconducto Kato et al Science (04)

19 Spin Hall conductivity Uing conventional pin cuent 2d electon (Rahba( Rahba): e/8π 2d hole (cubic Rahba): -9e/8π 3d hole (Luttinge( Luttinge): e ( γ 1 + 2γ 2) ( ) 2 h l 6π 2γ 2 Uing ou new pin cuent 2d electon (Rahba( Rahba): -e/8π 2d hole (cubic Rahba): 9e/8π 3d hole (Luttinge( Luttinge): e( γ 1 γ 2) 2 6π γ 2 ( h l ) + e 6π 2 h

20 Spin-chage conductivity teno H H 0 E ( e) F ( ) Spin foce: Zeeman field gadient g facto gadient + Zeeman field Spin-dependent chemical potential gadient

21 Invee pin Hall effect Tanvee chage cuent induced by pin foce: J y σ yx F c c x 2d electon (Rahba( Rahba): σ yx c e 8π 2d hole (Cubic Rahba): σ yx c 9e 8π

22 Onage elation σ yx c σ xy c Violated if conventional pin cuent i ued Saved if ou new pin cuent i ued: J x d dt < x >< x & + x& > uual pin cuent toque dipole

23 Onage elation thee-line deivation If Then Antiymmetic in m n

24 Meauement Method Themodynamic method: dq dt J F J 1 F dq dt Electic method: J y σ xy c E y xy σ c E J σ xy c σ yx c σ yx c y c J / F x

25 Spin Hall in inulato Definition: Chage inulato with a pin Hall effect -- Muaami, Nagaoa,, Zhang Howeve, if we ue conventional pin cuent, then eentially all inulato with pin-obit coupling ae pin Hall inulato e.g., Yao and Fang found pin Hall conductivitie of 0.001,0.0015, and e/a fo undoped GaA, Si,, and Ge.

26 No pin Hall in imple inulato baed on ou pope pin cuent Kubo fomula: Fo localied eigentate, we can ue < α x& x& + β >< α x β > ( ε α ε β ) / < β y & α >< β y α > ( ε α ε ) / ih β ih Then σ c xy e α fα < α [ x, y] α > 0 h How about band inulato with localied Wannie obital?

27 Concluion Spin tanpot in ytem with no bul geneation Fo all component if thee i inveion ymmety Fo ome component (S( ) in many cae (2D electon,hole) Spin (S( ) i coneved in the bul atifie a oucele continuity equation new pin cuent conventional pin cuent + toque dipole denity -- Spin accumulation occu at ample bounday due to balance between the new cuent and elaxation Linea epone theoy fo the new cuent Yield damatically diffeent pin Hall conductivity (ign eveal) Conjugate foce exit Meauement fom heat geneation Onage elation i atified and can alo be ued fo meauement No pin Hall effect in imple inulato

28 Macocopic denitie Macocopic denitie Spin denity Spin denity Toque denity Toque denity Spin cuent denity Spin cuent denity p f d f d t S 3 3 ˆ ), ( c p ˆ ) ( ˆ p f d f d t T 3 3 ˆ ), ( c p ˆ ) ˆ ( f d f d t J ˆ ) ˆ ( ˆ ˆˆ ), ( 3 3 & & ] ˆ, ˆ [ ˆ H i h