Quantum description of spin Spin current and spin accumulation Magnetotransport Spin-torque Spintronics devices Hall effect Materials for spintronics Semiconductors Nanosensors and spintronics Jaroslav Hamrle [email protected] April 14, 2014 Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Outline One FM layer 1 Quantum description of spin Two FM layer Non-relativistic description Magnetization dynamics of spin (LLG equations) Schr¨odingerequation Experimental examples Addition of angular Spin-pumping momentum Spin-torque oscillators Zeeman effect: angular Domain wall moment in magnetic field 5 Spintronics devices Magnetism and relativity: 6 Hall effect classical picture Hall effect Direc equation Spin-Hall effect 2 Spin current and spin Spin caloritronics accumulation 7 Materials for spintronics 3 Magnetotransport 8 Semiconductors Giant magnetoresistance Jaroslav Hamrle NanosensorsSpins and spintronics in semiconductors Tunnel magnetoresistance Spin injection to 4 Spin-torque semiconductors Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Quantum description of electron and its spin Spin of particles (spin of electron): Consequence of relativity, but can be postulated as particle property. 1 Electron (and also proton, neutron) has quantized spin s = 2 . In following, we first postulate existence of spin. Then, we show how spin originates from relativistic quantum theory. Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Angular momentum I Total angular momentum = orbital angular momentum + spin angular momentum Non-relativistic Schr¨odingerequation does not have spin (only angular momentum) ) spin can be included ad-hoc. Definition of angular momentum L^ = ^r × ^p = ~^r × r (1) i Commutation relations of angular momentum operator: ^ ^ ^ ^ ^ ^ [Lx; Ly] = i~Lz [Ly; Lz] = i~Lx (2) ^ ^ ^ ^2 ^ [Lz; Lx] = i~Ly [L ; Li] = 0 (3) and ^ ^ ^ ^ [Lx; y^] = −[Ly; x^] = i~z^ [Lx; p^y] = −[Ly; p^x] = i~p^z (4) [L^x; x^] = [L^x; p^x] = 0 (5) Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Example of determining commutators between angular momenta E.g. [Lx;Ly] = [(YPz−ZPy); (ZPx−XPZ )] = Y [Pz;Z]Px+XPY [Z; PZ ] = i~(−YPx + XPy) = i~Lz Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Angular and spin momentum II Spin in non-relativistic description: intrinsic property of the electron ) can not be defined similar to Eq. (1) ) spin is defined as quantity obeying the same commutation equations as L^. Total angular momentum J^ = L^ + S^ (6) where J^ obeys equal commutation relation. Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Angular momentum III: eigenvalues Total angular momentum eigenvalues: ^2 mj 2 mj J j = j(j + 1)~ j (7) ^ mj mj Jz j = mj~ j (8) where −j ≤ mj ≤ j. http://hyperphysics.phy-astr.gsu.edu/ hbase/quantum/vecmod.html Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Angular momentum IV: length of momentum Maximum value of J in z-direction is jmjj = j However, length of J is pj(j + 1) ) angular momentum can never points exactly in z (or in any other) direction classical limit: j ! 1 http://hyperphysics.phy-astr.gsu.edu/ hbase/quantum/vecmod.html Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Angular momentum V: raising/lowering operators Lowering/raising operators: J^± = J^x ± iJ^y 2 2 2 2 J+J− = Jx + Jy + ~Jz = J − Jz + ~Jz 2 2 2 2 J−J+ = Jx + Jy − ~Jz = J − Jz − ~Jz [Jz;J+] = +~J+ [Jz;J−] = −~J− [J+;J−] = 2~Jz 2 2 2 [J ;J+] = [J ;J−] = [J ;Jz] = 0 Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Angular momentum V: raising/lowering operators value of mj can be increased/decreased by raising/lowering operator J^± = J^x ± iJ^y, working as q ^ jJ+jj; mji = ~ (j − mj)(j + mj + 1) jj; mj + 1i q ^ jJ−jj; mji = ~ (j + mj)(j − mj + 1) jj; mj − 1i Can be derived by two steps: Apply J^± to eigenstate jj; mji (by using commutator relations; note: jJzjj; mji = ~mj jj; mji): ^ ^ ^ ^ ^ ^ ^ ^ jJzJ±jj; mji = jJ±Jz + [Jz; J±]jj; mji = jJ±Jz + ~J±jj; mji = ^ ^ (mj±1)~ jJ±jj; mji = (mj±1)~ jj; mj ± 1i = jJzjj; mj ± 1i 2 j jJ+jj; mji j = hj; mjjJ−J+jj; mji = 2 2 2 hj; mjjJ − Jz − ~Jzjj; mji = ~ [j(j + 1) − mj(mj + 1)] p ) jJ+jj; mji = ~ j(j + 1) − mj(mj + 1) jj; mji = p ~ (j − mj)(j + mj + 1) jj; mji Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Angular momentum VI: particles with moment s = 1=2 1 The same valid for spin j ! s = 2 , −s ≤ ms ≤ s ) ms = 1=2: spin-up spin ("); ms = −1=2: spin-down spin (#) http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/ Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Non-relativistic Schr¨odingerequation " # @ (~r; t) 1 2 i r = ~r − eA~(~r; t) + eΦ(~r; t) (~r; t) (9) ~ @t 2m i r @ (~r; t) i r = H^ (~r; t) (10) ~ @t r where A(~r; t) is the vector potential (B~ = r × A~) ~ @A~ eΦ(~r; t) is the scalar potential (E = −∇Φ − @t ) Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic description Spin-torque of spin Spintronics Schr¨odingerequation devices Hall effectAddition Materials of angular for spintronics momentum SemiconductorsZeeman effect: angular moment in magnetic field Magnetism and relativity: classical picture Direc equation Schr¨odingerequation with spin Spin can be superimposed into Sch¨odingerequation by product of ms time-space dependent part r(~r; t) and spin-dependent part χs ms ms s = r(~r; t)χs (11) However, this is only valid when spin-freedom is strictly independent on its time-space part. This is not valid for e.g. spin-orbit coupling. Then, one can express spin-time-space wavefunction as c" r;"(~r; t) c" (~r; t) = c" r;"(~r; t)+c# r;#(~r; t) ≡ ≈ r(~r; t) c# r;#(~r; t) c# (12) where following eigenvectors were used for definition 1 0 χ = χ = (13) " 0 # 1 Jaroslav Hamrle Nanosensors and spintronics Quantum description of spin Spin current and spin accumulation MagnetotransportNon-relativistic