An alternative way to control the spin relaxation rate in 2DEG. - PowerPoint PPT Presentation

An alternative way to control the spin relaxation rate in 2DEG. Phys. Rev. Lett. 104, 226601 (2010) Oleg Chalaev Giovanni Vignale Department of Physics, University of Missouri, Columbia, MO 65211 USA Work supported by ARO Grant No. W911NF-08-1-0317 October 13, 2010

The usual way to control the spin relaxation rate E c ( z ) E v ( z ) – 2D electron gas with Rashba SOI: H = ˆ p 2 ˆ 2 m + α ( σ x ˆ p y − σ y ˆ p x ) , α = 5 P 2 � � � 1 1 � � � n � ∂ z U v ( z ) � n , − � � E 2 ( E g + ∆) 2 9 � g where U v ( z ) is the effective potential acting on electrons in the valence band.

Rashba SOI in GaAs n -doped quantum wells E c ( z ) The one-subband case: | 1 � H = ˆ p 2 ˆ 2 m + α ( σ x ˆ p y − σ y ˆ p x ) E v ( z ) α = P 2 � � 1 1 3 � � 1 | ∂ z U v | 1 � − E 2 ( E g + ∆) 2 g � ˆ � � p x � � � P = � , U v ( z ) = ✘✘✘ U ext ( z ) + E v ( z ) + U H ( z ) S � X ❳❳❳ ✘ � � � � ❳ m �

Rashba SOI in GaAs n -doped quantum wells E c ( z ) | 2 � The two-subband case: | 1 � H 1 , 2 = ˆ p 2 ˆ 2 m + α 1 , 2 ( σ x ˆ p y − σ y ˆ p x ) E v ( z ) α n = P 2 � � 1 1 3 � � n | ∂ z U v | n � − E 2 ( E g + ∆) 2 g � ˆ � � p x � � � P = � , U v ( z ) = ✘✘✘ U ext ( z ) + E v ( z ) + U H ( z ) S � X ❳❳❳ ✘ � � � � ❳ m �

The idea: controlling the total spin relaxation rate E c ( z ) | 2 � | 1 � α n ∝ � n | ∂ z U v | n � on trolling E v ( z ) on trolling the total p opulation SOI strength of subbands α 1 ≪ α 2

Task#1: E c ( z ) | 2 � | 1 � α n ∝ � n | ∂ z U v | n � E v ( z ) Design such a quantum well shape that α 1 ≪ α 2

The double Darboux Transformation (DDT) [Zakhariev & Chabanov] I ϕ ( 0 ) � r 2 − 1 � � z rA E c = E ( 0 ) − 2 [ ϕ 2 A ] ′ , ϕ 1 = ϕ ( 0 ) ϕ ( 0 ) n ( y ) ϕ ( 0 ) 2 1 − AI 21 , ϕ 2 = , A = , I nm ( z ) = m ( y ) d y , c r 2 − 1 � r 2 − 1 � 1 + I 22 −∞ where r is the transformation parameter. w w w w �� �� doping doping �� �� �� �� � � layers � � layers E c ( z ) E c ( z ) �� �� �� �� � � � � ǫ 2 ϕ 2 ( z ) � � �� �� �� �� � � � � �� �� �� �� � � ǫ 1 ϕ 1 ( z ) � � �� �� �� �� � � DDT ( r ) � � � � �� �� �� �� = ⇒ δ L δ L δ R δ R δ L � = δ R spectrum conserved log r = 0 . 82 E v ( z ) a E v ( z ) a Using DDT one can manipulate energy spectrum and/or wave functions.

The double Darboux Transformation (DDT) [Zakhariev & Chabanov] II H 1 , 2 = ˆ p 2 ˆ 2 m + α 1 , 2 ( σ x ˆ p y − σ y ˆ p x ) � � α n = P 2 1 1 3 � � n | ∂ z U v | n � − E 2 ( E g + ∆) 2 g 1 h ) ( 4 µ eV · nm / ¯ 0.18 | α 2 ( r ) | 0.1 0.01 | α 1 ( r ) | At log r = 0 . 82 choosing 0.001 amplitude α 1 = 0, optimal 0.0001 value of r but α 2 � 0! 1e−05 SOI 1e−06 −1 −0.5 0 0.5 0 . 82 1 log r

Wave function engineering α 2 = 7 m/s = 4 × 10 − 3 nm · me V / ¯ �� �� doping �� �� �� �� layers E c ( z ) �� �� �� �� ϕ 2 ( z ) h �� �� �� �� �� �� �� �� ϕ 1 ( z ) α 1 = 0 �� �� �� �� 400 � �� �� �� �� E v ( z ) ◮ The shape of the well is obtained using the inverse scattering theory. [Zakhariev & Chabanov] ◮ Spin-orbit amplitude strongly depends on the level population. How can we change the population of the subbands?

Wave function engineering α 2 = 7 m/s = 4 × 10 − 3 nm · me V / ¯ �� �� doping �� �� �� �� layers E c ( z ) �� �� �� �� ϕ 2 ( z ) h �� �� �� �� �� �� �� �� ϕ 1 ( z ) α 1 = 0 �� �� �� �� 400 � �� �� �� �� E v ( z ) ◮ The shape of the well is obtained using the inverse scattering theory. [Zakhariev & Chabanov] ◮ Spin-orbit amplitude strongly depends on the level population. How can we change the population of the subbands?

Task#2: E c ( z ) | 2 � α n ∝ � n | ∂ z U v | n � | 1 � We already achieved α 1 ≪ α 2 , now we must E v ( z ) learn to manipulate subbands population with an applied voltage.

Energy distribution inside the current-carrying sample I D / L 2 ≫ τ − 1 Requirements: in = ⇒ ◮ short wires ◮ low temperatures ◮ high mobilities

ele trons Energy distribution inside the current-carrying sample II b orro w ed f E from the lo w est subband ǫ 1 ǫ 2 E

Inhomogeneous population and spin-orbit amplitudes 1.2 n 1 1 0.8 and n 2 V = 1 . 5( ǫ 2 − ǫ 1 ) 0.6 n 1 The size of the zone where 0.4 is on trolled b y V 0.2 n 2 0 0 0.2 0.6 0.8 1 x/L n 2 � = 0

Inhomogeneous population and spin-orbit amplitudes 1.2 n 1 1 0.8 and n 2 V = 1 . 5( ǫ 2 − ǫ 1 ) 0.6 n 1 0.4 0.2 n 2 0 0 0.2 0.6 0.8 1 x/L 0.2 α 2 0.15 amplitudes 0.1 V = 1 . 5( ǫ 2 − ǫ 1 ) 0.05 SOI 0 −0.05 α 1 −0.1 x/L 0 0.2 0.6 0.8 1

Conclusions The width of the “red zone”, where the spin relaxation occurs is controlled by the applied voltage = ⇒ an alternative method to control the spin polarization. Thank you!

Conclusions The width of the “red zone”, where the spin relaxation occurs is controlled by the applied voltage = ⇒ an alternative method to control the spin polarization. Thank you!

References Ronald Winkler, “Spin-Orbit Effects in Two-Dimensional Electron and Hole Systems.” Springer, 2003. B. N. Zakhariev and V. M. Chabanov, “Submissive quantum mechanics. New status of the theory in inverse problem approach.” Nova Science Publishers, Inc. New York, 2008. Pothier, H. et al, Phys. Rev. Lett., 79 , 3490 (1997).

The energy distribution in the middle of the hot ele trons with energies > µ ( x ∗ ) sample 4.5 ǫ 2 (1) Energy µ R 3.5 3 ǫ 1 (1) µ ( x ∗ ) and f E (˜ ǫ 2 (0) µ L 1.5 1 V = 1 . 5( ǫ 2 − ǫ 1 ) /e ǫ 1 (0) 0 x ∗ ˜ 0 0.2 0.6 0.8 1 x ∗ ) x ˜