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

Ec(z)

Ev(z)

– 2D electron gas with Rashba SOI: ˆ H = ˆ p2 2m + α(σx ˆ py − σy ˆ px), α =5P2 9

• 1

E2

g

− 1 (Eg + ∆)2 n

• ∂zUv(z)
• n
• ,

where Uv(z) is the effective potential acting on electrons in the valence band.

Rashba SOI in GaAs n-doped quantum wells

Ev(z) |1 Ec(z)

The one-subband case: ˆ H = ˆ p2 2m + α(σx ˆ py − σy ˆ px) α = P2 3 1 |∂zUv| 1

• 1

E2

g

− 1 (Eg + ∆)2

• P =
• S
• ˆ

px m

• X
• ,

Uv(z) = ✘✘✘

✘ ❳❳❳ ❳

Uext(z) + Ev(z) + UH(z)

Rashba SOI in GaAs n-doped quantum wells

Ev(z) Ec(z) |1 |2

The two-subband case: ˆ H1,2 = ˆ p2 2m + α1,2(σx ˆ py − σy ˆ px) αn = P2 3 n |∂zUv| n

• 1

E2

g

− 1 (Eg + ∆)2

• P =
• S
• ˆ

px m

• X
• ,

Uv(z) = ✘✘✘

✘ ❳❳❳ ❳

Uext(z) + Ev(z) + UH(z)

The idea: controlling the total spin relaxation rate

Ev(z) Ec(z) |1 |2

αn ∝ n |∂zUv| n

• pulation
• f

• n

• n

α1 ≪ α2

Task#1:

Ev(z) Ec(z) |1 |2

αn ∝ n |∂zUv| n

Design such a quantum well shape that α1 ≪ α2

The double Darboux Transformation (DDT) [Zakhariev & Chabanov] I

Ec = E(0)

c

−2 [ϕ2A]′ , ϕ1 = ϕ(0)

1 −AI21,

ϕ2 = rA r2 − 1 , A =

• r2 − 1
• ϕ(0)

2

1 +

• r2 − 1
• I22

, Inm(z) = z

−∞

ϕ(0)

n (y)ϕ(0) m (y)dy,

where r is the transformation parameter.

• layers

doping

ǫ2 ǫ1

w w a δL

Ec(z) Ev(z)

δL = δR δR

DDT(r)

=⇒

spectrum conserved

• layers

doping

log r = 0.82 w w

ϕ1(z) Ec(z) Ev(z) ϕ2(z)

δL a δR

Using DDT one can manipulate energy spectrum and/or wave functions.

The double Darboux Transformation (DDT) [Zakhariev & Chabanov] II

ˆ H1,2 = ˆ p2 2m + α1,2(σx ˆ py − σy ˆ px) αn = P2 3 n |∂zUv| n

• 1

E2

g

− 1 (Eg + ∆)2

• choosing
• ptimal

value of r

1e−06 1e−05 0.0001 0.001 0.01 0.1 1 −1 −0.5 0.5 1 0.18

|α1(r)| |α2(r)|

log r

0.82

h)

At log r = 0.82 α1 = 0, but α2 0!

Wave function engineering

• layers

doping ϕ1(z) Ec(z) Ev(z) ϕ2(z)

400

α1 = 0 α2 = 7m/s= 4 × 10−3

h

◮ 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

• layers

doping ϕ1(z) Ec(z) Ev(z) ϕ2(z)

400

α1 = 0 α2 = 7m/s= 4 × 10−3

h

◮ 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:

Ev(z) Ec(z) |1 |2

αn ∝ n |∂zUv| n We already achieved α1 ≪ α2, now we must learn to

manipulate subbands population with an applied voltage.

Energy distribution inside the current-carrying sample I

Requirements: D/L 2 ≫ τ−1

in =⇒ ◮ short wires ◮ low temperatures ◮ high mobilities

Energy distribution inside the current-carrying sample II

E fE

• rro

ǫ1 ǫ2

Inhomogeneous population and spin-orbit amplitudes

0.2 0.4 0.6 0.8 1 1.2 0.2 0.6 0.8 1

V = 1.5(ǫ2 − ǫ1) n2 n1

• f

n2 = 0

• n

n1

x/L

Inhomogeneous population and spin-orbit amplitudes

0.2 0.4 0.6 0.8 1 1.2 0.2 0.6 0.8 1

V = 1.5(ǫ2 − ǫ1) n2 n1 n1

x/L

−0.1 −0.05 0.05 0.1 0.15 0.2 0.2 0.6 0.8 1

V = 1.5(ǫ2 − ǫ1) α2 α1

x/L

Conclusions

The width of the “red zone”, where the spin relaxation

• ccurs 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

• ccurs 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 sample

1 1.5 3 3.5 4.5 0.2 0.6 0.8 1

ǫ2(0) ǫ1(0) ǫ2(1) ǫ1(1) µL

V = 1.5(ǫ2 − ǫ1)/e

µ(x∗)

µR

˜ x∗

˜ x

x∗)