The spin susceptibility enhancement in wide AlAs quantum wells

Transcription

1 The spin susceptibility enhancement in wide AlAs quantum wells Mariapia Marchi, S. De Palo, S. Moroni, and G. Senatore SISSA and University of Trieste, Trieste (Italy) QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 1

2 Outline Motivation Devices and 2D electron systems Spin susceptibility: experiments and previous QMC results Technique in brief and details of QMC simulations Predictions and comparison with experiments: AlAs Quantum Wells (QWs) Conclusions QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 2

3 Motivation 2DEG model system for 2D systems of electrons in solid-state devices. Improvements in fabrication techniques: cleaner and more dilute devices. Previously unexpected metal insulator transition. Measurements of the spin susceptibility enhancement: χ s /χ 0 increases as the density decreases, but depends on device details. QMC capable of accounting for experimental evidence. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 3

4 Definition of the system Two-dimensional electron gas (2DEG): system of electrons in two dimensions interacting via 1/r potential in a uniform neutralizing background of positive charges. 2DEG is a good approximation of 2D systems of electrons in solid state devices (e.g. in Si-MOSFETs, AlAs quantum wells, GaAs HIGFET, etc.). QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 4

5 Heterojunctions and quantum wells QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 5

6 2D confinement in the HIGFET QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 6

7 Important features of the devices: Valley degeneracy: 1-valley systems: k, σ =, ; 2-valley systems: k, σ =,, v = 1,2. Finite transverse thickness. Disorder: scattering of carriers from different impurity sources. Mass anisotropy. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 7

8 Valley degeneracy and mass anisotropy Schematic drawing of the Brillouin zone and constant energy surfaces of the lowest energy bands for bulk GaAs (a) and bulk AlAs (b) QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 8

9 Measurements ofχ s Spin susceptibility enhancementχ s /χ 0 increases with the couplingr s (r s n 1/2 ), but depends on device details (number of occupied valleys, thickness, disorder, mass anisotropy). 7 AlAs: Vakili et al,prl 04 6 GaAs: Zhu et al, PRL 03 Si-MOSFET: Pudalov et al, PRL 02 χ s /χ AlAs GaAs Shashkin et al, PRL 01 2 Si-MOSFET r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 9

10 Theory and experiments: one valley De Palo et al, PRL. 2005: Effect of thickness 7 6 2DEG χ s /χ AlAs GaAs r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 10

11 Effects onχ s /χ 0 Z: Vakili et al, PRL 04 (1v isotropic masses) X,XY: Gunawan et al., PRL 06 (1v,2v anisotropic masses) χ s /χ Z X 2 XY 1 0 AlAs QWs r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 11

12 Experiments in 15nm 1v AlAs QW g*m*/g b m b n (10 11 cm -2 ) g*m*/g b m b ζ Spin Polarization (ζ) n (10 11 cm -2 ) T. Gokmen et al., PRB 07 QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 12

13 Description of the theoretical model I 2DEG att = 0 andb = 0 with valley degeneracy and mass anisotropy: [ ] [ ] H = 2 N/2 2 i,x + 2 N/2 2 i,y i,x i,y 2 m x m y + 1 2A i=1 m y i=1 m x v(q)[ρ q ρ q N], ρ q = e iq r iν. ν,i ν q 0 m x = m t = 0.205m 0, m y = m l = 1.05m 0 (T. Gokmen et al., PRB 2007); v(q) = (2πe 2 /ǫq)f(q). F(q) = 1 for a strictly 2D egas. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 13

14 Description of the theoretical model II Important parameters of the system: r s parameter: 1/n = A/N = π(r s a B )2,a B = 2 ǫ/(m b e 2 ). Spin polarizationζ = (N N )/N (symmetric valleys). Effective massm b = m x m y 0.46m 0 or anisotropy factor α = (m y /m x ) 1/4 5 1/4 H = 1 r 2 s + 1 r s 1 A N/2 i=1 q 0 [ α 2 2 i,x + 2 i,y α 2 ] + 2π q F(q)[ρ qρ q N] N/2 i=1 (energy in Rydberg and lengths in units ofr s a B ). [ 2 i,x α 2 +α2 2 i,y ] QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 14

15 The thickness for AlAs QWs ( ) 1 F(q) = 3aq + 8π2 32π4 1 e aq 4π 2 +a 2 q 2 aq a 2 q 2 4π 2 +a 2 q 2 ǫ = 10; (Gold, PRB 1987). a the QW width; in the considered experiments: a = 11nm, 15nm. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 15

16 Before going on... By considering suitable coordinate transformations it is possible to transfer the anisotropy into the potential energy term. In the 1v case, e.g. x αx, y y/α and H = 1 r 2 s N i=1 i r s 1 A q 0 2π q F(q)[ ρ q ρ q N], where ρ = i expi[αq xx+q y y/α]. In the non-interacting limit this transformation exactly maps the anisotropic system onto the isotropic one. For non-zero coupling, we have an effective Hamiltonian with anisotropic interaction. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 16

17 The Spin Susceptibility A 2D electron gas withn andn up and down spin electrons, ζ = (N N )/N, has an energy per particleε(ζ) (... from QMC). In an in-plane magnetic fieldb = Bẑ, ε(ζ,b) = ε(ζ)+γζb, Equilibrium polarization ζ(b) γ = gµ B 2 ε(ζ, B) ζ = 0 ε (ζ) = γb ζ(b) The spin susceptibility is defined as χ s = dm db = nγ dζ B=0 db = nγ 2 1 B=0 ε (ζ) Knowledge ofε(ζ) ( QMC) allows forχ s estimates!!! ζ=0 QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 17

18 Technique QMC (and DMC) simulations provide very accurate results: ground state energies; pair distribution functions; static responses. Fermion wave functions have nodes! Fixed-node (or fixed-phase) approximation: we assume the nodal (phase) structure ofφ 0 is the same as forψ T. FN-DMC (or FP-DMC) satisfies a variational principle. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 18

19 Simulation details Choice of trial wave function: Ψ T = J(R) σ,ν D σν (R σν ), D σν (R σν ) = det[exp(ik iσν s jσν )]. We have considered: Plane-wave (PW) nodes: s jσν = r jσν J(R):u( r) = a r+b r2 1+c r+d r 2, with r 2 αβ = p(x) αβ x2 αβ +p(y) αβ y2 αβ. Size extrapolation to get results in the thermodynamic limit. Use of twist-averaged boundary conditions (TABC) to reduce size effects on the kinetic energy at finiten and to getǫ(ζ) at fixed N. We have performed DMC simulations at 8 different values ofζ for variousr s, for 1v and 2v, strictly-2d or system with finite thickness. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 19

20 Energy per particle: Example I TABC-DMC energy for 1v strictly-2d anisotropic system N=80 rs=5 1v ani ε N (r s, ζ) ζ QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 20

21 Energy per particle: Example II TABC-DMC energy for 1v quasi-2d anisotropic system N=80 rs=5 1v ani L=11nm ε t (r s,ζ) ζ QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 21

22 Rough ζ-fit of the energies (givenr s!) At givenr s we fit ε(r s,ζ) = 2 i=0 a i ζ 2i ε N (r s,ζ) k(r s ) T N (r s,ζ), T N (r s,ζ) = ε 0N (r s,ζ) ε 0 (r s,ζ), ε 0 (r s,ζ) = 1+ζ2 g v r 2 s. Enhancement ratio: χ s = ε 0(0) χ 0 ε (0) = 1 g v a 1 rs 2. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 22

23 Approximate treatment of mass anisotropy T. Gokmen et al., PRB 2007 Neglected intrinsic dependence on anisotropy Thickness within perturbation theory ζ=1 ζ=0 g*m*/g b m b w=0 ζ w=15nm r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 23

24 QMC full simulation results I Strictly-2DEG iso and aniso 11nm 2DEG iso and aniso 1v χ s /χ v iso 1v ani 1v ani-th r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 24

25 QMC full simulation results II Strictly-2DEG iso and aniso 11nm 2DEG iso and aniso 1v and 2v χ s /χ v iso 1v ani 1v ani-th 2v iso 2v ani 2v ani-th r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 25

26 DMC and anisotropic 11nm-AlAs Qws 1v and 2v Experiment,1v QMC, 1v Experiment,2v QMC, 2v χ s /χ r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 26

27 DMC vs 11nm and 15nm AlAs Qw I 1v 2DEG 15nm and 11nm Experiment 15nm, 1v QMC 15nm,1v Experiment 11nm, 1v QMC 11nm,1v χ s /χ r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 27

28 DMC vs 11nm and 15nm AlAs Qw II 1v 2DEG 15nm and 11nm χ s /χ Experiment 15nm, 1v QMC 15nm,1v QMC 15nm, 1v, B p Experiment 11nm, 1v QMC 11nm,1v QMC 11nm, 1v, B p r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 28

29 L=11nm, finiteζ χ s /χ v,ζ=0.75 1v,ζ=0.67 1v,ζ=0.40 2v,ζ=0.67 1v 2v 1v,ζ 2v,ζ r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 29

30 Conclusions Valley degeneracy and thickness reduces spin-susceptibility enhancement. Mass anisotropy suppresses the spin-susceptibility enhancement in 1v systems. For 2v mass anisotropy is counterbalanced by the different orientation of the Fermi surfaces. Finiteζ enhancesχ s /χ 0 (2v?). QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 30

31 Measuring the spin susceptibility Transport measurements. Tilted field technique (Fang & Stiles, 1968): exploits Shubnikov-de Haas oscillations and gives access to quasiparticle parameters (g, m ). Magnetoresistance saturation (Okamoto, 1989; Shashkin, 2001). Changes in the behavior of magnetoresistance with the parallel component ofb allows the determination of the polarization fieldb p, from whichg m can be extracted (with ad-hoc assumptions). Thermodynamic measurements (Prus, 2003; Shashkin, 2006). Measurement of µ/ B (µ= chemical potential here) allows for the determination of M(B). QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 31

32 Boundary conditions N particles in a box of sidel. PBC:Ψ(r 1,..., r i +Lˆx,...) = Ψ(r 1,..., r i,...). TABC: ([H,T Ln ] = 0) Ψ(r 1,..., r i +Lˆx α,...) = e iθ xα Ψ(r1,..., r i,...), (α = 1,..., d). How to do the twist average O θ = 1 (2π) d dθ Ψ(R, θ) O Ψ(R, θ)? Calculation of i ω io i using a grid defined by pointsθ i with weightsω i (such that i ω i = 1) in the region in which one averages. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 32

33 Fixed Phase (FP) approximation Consider e.g. real H and complex Ψ(R) = Ψ(R) exp[iϕ(r)]. Real and imaginary parts of Schrödinger equation yield [ 2 2m N i i=1 N 2 i ] Ψ(R) +Ṽ(R) = E Ψ(R), i=1 [ ] Ψ 2 (R) i ϕ(r) = 0, with effective potential Ṽ(R) = V + 2 2m N i=1 [ i ϕ(r)] 2. FP approximation: make a choice for the phaseϕand solve bosonic problem for Ψ. QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 33

34 2DEG energies: iso vs aniso, 1v-2v r s 3/2 (E(rs )-c 1 /r s ) (Ry) v iso z=0 2v iso z=1 1v iso z=1 2v aniso z=0 2v aniso z=1 1v aniso z=0 1v aniso z= r s QMC in the Apuan Alps, Vallico di Sotto, July 26th, 2010 Mariapia Marchi Spin susceptibility enhancement... p. 34