g-factors in quantum dots

Transcription

1 g-factors in quantum dots Craig Pryor Dept. of Physics and Astronomy University of Iowa With: Michael Flatté, Joseph Pingenot, Amrit De supported by DARPA/ARO DAAD

2 g-factors in quantum dots bulk g-factors g-factors in self-assembled quantum dots g-factors nanowhisker dots modulation by E-fields, g-tmr With: Michael Flatté, Joseph Pingenot, Amrit De Pryor and Flatté PRL 96,

3 free electron µ s = g 2 µ B σ g = 2 + O(1/137) H = 1 2m ( P + e ) 2 A c + g 2 µ B σ B

4 band electron µ s = g 2 µ B σ g = 2 something H = 1 2m ( other bands P + e ) 2 A c + g 2 µ B σ B

5 atomic Landé g-factor orbital l + spin s! total j g = 1 + j(j + 1) + s(s + 1) l(l + 1) 2j(j + 1)

6 Bloch state g-factors (zone center) g = 2 : l = 0, s = 1 2, j = 1 2 E v E g! g = 4/3 : l = 1, s = 1 2, j = 3 2 g = 2/3 : l = 1, s = 1 2, j = 1 2 g = 1 + j(j + 1) + s(s + 1) l(l + 1) 2j(j + 1)

7 bulk semiconductor g-factors (k.p) H = 1 2m ( P + e ) 2 A c + g 2 µ B σ B ψ(r) = i,s c i,s e ik r u i,s (r) g = 2 2i m k n n p x k k p y n n p y k k p x n E 0 n E0 k Roth et al. Phys. Rev 114, 90 (1959)

8 bulk semiconductor g-factors (k.p) electron magnetic moment B µ s = g 2 µ B σ bulk semiconductor g = 2 2 E p 3E g (E g + ) E v E g! Roth et al. Phys. Rev 114, 90 (1959)

9 ornia, Los Angeles, Electrical Engineering Dept., Angeles, California 90024, USA trical and Computer Engineering, North Carolina sity, Raleigh, North Carolina 27695, USA (a) 5 AlAs GaP electron g-factor InP 0 GaAs Ga0.47In0.53As -5 GaSb -10 experimental theory direct gap theory indirect gap GaInSb InSb InAs lattice constant (!) 2.5 AlP A l As et al.gap Electronics Letters 37, 464 gap Eg (ev) -V semiconduconic devices for ventions for imband structure ures [1]. Pracns [2], [3] is exphoto-detectors tted by photon (b) hen transferred To maintain the Kosaka absorb equally hus the electron ortunately, the ctive mass can AlSb GaAs AlSb Al0.48In0.52 As InP th (!m) photo-detector n. Up and down arized photons, energies, ge 0 r versus lattice e constant, that alloys and quan- bulk semiconductor g-factors (k.p)

10 strained InAs/GaAs quantum dots 1.0 E (ev) GaAs no strain E g InAs GaAs with strain E g GaAs InAs [001] distance along [001] g = 2 2 E p 3E g (E g + )

11 strained InAs/GaAs quantum dots 1.0 E (ev) GaAs no strain E g InAs GaAs with strain E g GaAs InAs [001] distance along [001] unstrained: ginas = strained: ginas -6.1 confinement: ginas g = 2 2 E p 3E g (E g + )

12 quantum dot k.p calculations! calculate strain using finite elements! Schrödinger equation: Hψ(r) = Eψ(r)! ψ(r) is an 8-component vector, and H is an 8x8 matrix with elements like A = E c (r)! put system on a grid: 2 ( 2 x 2m + 2 y + 2 z 0 ) + a c (r) i e ii (r) 2 x ψ i(r) ψ i(r + ɛˆx) + ψ i (r ɛˆx) 2ψ i (r) ɛ 2! solve for eigenvalues and eigenvectors as a sparse matrix problem using the Lanczos algorithm

13 nanostructures in B-fields coupling to envelope (c.f. lattice gauge theory) ψ( r + ɛˆx) ψ( r ɛˆx) 2ɛ ψ( r + ɛˆx)u x( r) ψ( r ɛˆx)u x ( r ɛˆx) 2ɛ U x ( r)u y ( r + ɛˆx)u x ( r + ɛŷ)u y ( r) = exp(iɛ2 B e/ ) Pauli term for Bloch function spin H s = µ B 2 B 2 σ J σ

14 g-factors in spherical dots g nanocrystal -6 2E P g = 2-8 3E g (E g + ) nm nm 0.45 ev ev InAs -14 bulk InAs E g (ev)

15 angular momentum quenching bulk B=0 bulk B>0 r atom H = 1 ( P e 2m + c ) 2 g A + 2 µ B 1 dv + 4m 2 c 2 dr σ B

16 angular momentum quenching bulk B=0 bulk B>0 atom

17 InAs/GaAs dots [001] h GaAs InAs d [110] e = d[1 10] 10 nm [010] [010] [100] [001] [001] Electron [100] Hole d [110] [100] d [1 10] [100]

18 g-factors in InAs/GaAs dots experiment [001] B in [001] direction h GaAs InAs d [110] e = d[1 10] g d [110] 0.5 d [1 10] e = 1.0 e = 1.4 h=1.7 nm h=2.8 nm electron E g (ev) experiment: G. Medeiros-Ribeiro et al. Appl. Phys. A (2003)

19 in-plane g-factors h d [110] GaAs B [001] InAs d [1 10] d [110] e = d[1 10] g electron e = d [110] e = 1.0 d [110] + e = 1.4 B in [110] direction E g (ev) Eg (ev) h=2.8 nm h=1.7 nm hole

20 whisker dots Amrit De quantum well inside a nanowhisker N. Panev et al, APL, 83, 2238 (2003) resonant-magneto-tunneling spectroscopy M. T. Bjork, et al, PRB 72,201307,(R) ( 2005) Hexagonal cross-section, <111> orientation, wurtzite

21 whisker orientation binp = 2.6 nm WInAs = 6 nm D = 40 nm

22 theory-vs-experiment

23 theory-vs-experiment }

24 coupling to leads x = 0 x = g factor raised barrier normal barrier x = 1

25 whisker dots clean geometry and composition allows better comparison than for self-assembled dots much closer to experiment than Roth formula coupling to leads alters g wurtzite structure still a complication

26 electrical control of g-factors, g-tmr Joseph Pingenot [110] g nm Eg=1.2 ev [001] GaAs InAs 13 nm 0.2 [001] E (kv/cm)

27 g-tmr Hs= g µ B B S Hs= µ B B α g αβ S β = Ω S controllable Beff

28 g-tmr experiments: quantum wells Kato et al., Science 299, (2003) electric field to modify the electron wavefunction in a parabolic quantum well

29 g-tmr in self-assembled InAs/GaAs dots E: growth direction B: different directions different dot heights, elongations

30 g-tmr in self-assembled InAs/GaAs dots -100 kv/cm [110] 100 kv/cm e=1 e=1.25 e=1.67 fixed height, vary base size large dot small dot

31 g-tmr in self-assembled InAs/GaAs dots [001]