Nonlinear Quantum Mechanics

Transcription

1 The Mechanics Björn Center for Complex and Science and Department of Mathematics, UC Santa Barbara and University of Iceland, Reykjavík UCSB, January 2018 Co-authors: Miguel Ruiz-Garcia, Jonathan Essen, Manuel Carretero, Luis Bonilla, Mark Sherwin Department of Physics, University of California, Santa Barbara Gregorio Millán Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics, Universidad Carlos III de Madrid, Spain Supported by U. S. Army Research Office under contract/grant number

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3 Question: Can nonlinear phenomena, i.e. bifurcations, be observed in and Superlattices? The Extend prior work of Galdrikian, Batista and [6, 5, 3, 2] Develop theory of strongly driven, laterally inhomogeneous quantum wells Approach: Theory of homogeneous quantum wells Local density approximation Intersubband absorption Time-dependent local density approximation phenomena Extension to DC driven superlattices Weakly coupled superlattices, Gigahertz range Strongly coupled superlattices, Terahertz range

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5 The two-step Well The Figure: The stationary self-consistent potential of the assymetric GaAs-AlGaAs quantum well from Batista et al. [3]. The eigenstates energy levels are indicated by the horizontal lines.

6 The Heisenberg Equations The The Heisenberg equation for the electron operator ψ are i h ψ = [ψ,h]. t Let the x- and z-coordinates parameterize the lateral and growth directions of the heterostructure, respectively. The mean field Hamiltonian is H(t) = ψ (x,z,t)[ h2 2m 2 + v(x,z) + w(x,z,t) ezf(x,t)]ψ(x,z,t) d 2 xdz, where v and w are the time-independent and time-dependent parts of the electric potential, respectively, e is the electron charge and m is the effective mass.

7 The Electron Density creates the ity The The electric potential is coupled to the electron density n by Poisson s equation 2 [v(x,z) + w(x,z,t)] = e ε n(x,z,t), n(x,z,t) = ψ (x,z,t)ψ(x,z,t), is the electron density. The electron operator is expressed as ψ(x,z,t) = α e ik x ξ α (z)a kα (t) d 2 k 2π The envelope wavefunctions ξ α (z) form a complete orthonormal basis. (1)

8 Homogeneous quantum wells The Envelope wavefunctions ψ(x,y,z) = A 1/2 Parabolic subbands k x,k y,α a kx k y α e ik x ξ α (z) E kx k y α = h2 2m (k 2 x + k 2 y ) + E α Schrödinger equation { } h2 2 2m z 2 + v(z) ξ α (z) = E α ξ α (z) Poisson equation 2 z 2 v(z) = e2 ε n(z)

9 Local density approximation The Partition function yields relationship between chemical potential µ, sheet density N s and subband energies E α, which gives the thermal weights w α N s = m { } π h 2 log 1 + e β(e α µ) m β α π h 2 w α β α Electron density Hartree iteration: n(z) = w α ξ α (z) 2 α 1 Solve Schrödinger equation for {ξ α (z),e α } 2 Determine {µ,w α } from {N s,e α } and update n(z) 3 Solve Poisson s equation and update v(z), Repeat until converged

10 Local density approximation The

11 Intersubband absorption The Depolarization shift Theory by Zaluzny (PRB 1993) Experiment by Craig et al. (PRL 1996) Collective oscillations Relaxation time approximation: Γ 1 = Depopulation rate Γ 2 = Depolarization rate

12 Time-dependent local density approximation The Electric field ẑf(t) induces self-consistent fluctuations in potential δv(z,t) and electron density δn(z,t) H(t) = H ezf(t)+δv(z,t); Liouville-von Neumann equation ρ(z,z,t) t Hartree iteration: 2 δv(z,t) = e2 z2 ε δn(z,t) = ī h [ H(t),ρ(z,z,t)] R[ρ(z,z,t)] 1 Evolve ρ(z,z,t) until ic response 2 Get electron density n(z,t) = ρ(z,z,t) z =z 3 Solve Poisson s equation and update δv(z, t) Repeat until converged

13 phenomena Period-doubling: Galdrikian and (PRL 1996) Supercritical Hopf: Batista and (PRB 2003) The

14 phenomena Experimental search by Morris and Sherwin (Thesis 2011) The Observed superharmonics, but not subharmonics Possible issues Excessive heating? Domain formation? Addressed by Superlattices Design of SL cools active region Domain formation is suppressed with number of quantum wells Weak signals are enhanced

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16 layered together in a Superlattice The Figure: A mesa-shaped SL of 1.2 mm (square) width and 1.5 µm thickness. (a) shows a dc voltage-biased SL consisting of two contact regions of about 0.5 µm thickness each and 50 s, formed by the two semiconductor layers of different bandgaps (b).

17 The We study the usual sequential tunneling model of electron transport in a weakly coupled n-doped SL, under a (DC) voltage bias and search for nonlinear bifurcations of coherent electron states, parametrized by DC voltage V [10, 4]. The model is adapted from M. Alvaro, M. Carretero, and L. Bonilla, [1]. Let the electric field, in well i, be F i, where i = 1,...,N(N 50) is the number of SL s, and J i i+1 and J(t) is the tunneling current density from well i to well i + 1, and the total current density, respectively.

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19 The Bifurcations The 1.8 V 2.1 V V V J(t) F6 vs F5 PF6 vs PF6 P [J] V t (ns) F5 (a.u.) F6 (a.u.) f (GHz)

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21 Bifurcations and Devices The We simulate SLs that can be used as halvers and, random and mixers, at room temperature Hopf bifurcation: The first bifurcation is the Hopf bifurcation from stationary state to ic orbit. Period doubling bifurcation: The second bifurcation that we find in the SL is the -doubling bifurcation. Period doubling cascade: The doubling of the ic orbit continues into a -doubling cascade to a strange attractor. A practical application of the dynamics in this regime is ultrafast generation of random number s [9].

22 The Design of the Superlattice The design (b) that effectively quenches the background noise is taken from Huang et al. [8, 7]. The Figure: Two superlattices, respectively GaAs/AlAs (a) and (b) GaAs/Al 0.45 Ga 0.55 As.

23 Conclusions The This work demonstrates that semiconductor heterostructures, consisting of a superlattice of weakly-coupled quantum wells under a DC voltage bias, are nonlinear quantum systems The coherent electronic dynamics in the superlattice bifurcate with increased voltage. In our simulations, we find a Hopf bifurcation, a -doubling of the resulting ic orbit and a -doubling cascade to a strange attractor These bifurcations are observable at room temperature. These enable the design of a number of devices operating in the GHz range.

24 The M Alvaro, M Carretero, and LL Bonilla. Noise-enhanced spontaneous chaos in semiconductor superlattices at room temperature. EPL (Europhysics Letters), 107(3):37002, Adriano A. Batista, Bjorn, P. I. Tamborenea, and D. S. Citrin. Period-doubling and hopf bifurcations in far-infrared driven quantum well intersubband transitions. Phys. Rev. B, 68:035307, Jul Adriano A Batista, PI Tamborenea, Bjorn, Mark S Sherwin, and DS Citrin. dynamics in far-infrared driven quantum-well intersubband transitions. Physical Review B, 66(19):195325, 2002.

25 The J. Essen, M. Ruiz-Garcia, I. Jenkins, M. Carretero, L. Bonilla, and B.. Parameter dependence of high- nonlinear oscillations and intrinsic chaos in short GaAs/(Al,Ga)As superlattices. (in submission), B. Galdrikian and B.. Phys. Rev. Lett., 76:3308, Bryan Galdrikian, Mark Sherwin, and Björn. Self-consistent floquet states for ically driven quantum wells. Phys. Rev. B, 49: , May Yuyang Huang, Wen Li, Wenquan Ma, Hua Qin, Holger T Grahn, and Yaohui Zhang.

26 The Spontaneous quasi-ic current self-oscillations in a weakly coupled gaas/(al, ga) as superlattice at room temperature. Applied Physics Letters, 102(24):242107, YuYang Huang, Wen Li, WenQuan Ma, Hua Qin, and YaoHui Zhang. Experimental observation of spontaneous chaotic current oscillations in gaas/al0. 45ga0. 55as superlattices at room temperature. Chinese Science Bulletin, 57(17): , Wen Li, Igor Reidler, Yaara Aviad, Yuyang Huang, Helong Song, Yaohui Zhang, Michael Rosenbluh, and Ido Kanter.

27 The Fast physical random-number generation based on room-temperature chaotic oscillations in weakly coupled superlattices. Phys. Rev. Lett., 111:044102, Jul M Ruiz-Garcia, J Essen, M Carretero, LL Bonilla, and B. Enhancing chaotic behavior at room temperature in GaAs/(Al, Ga) As superlattices. Physical Review B, 95(8):085204, 2017.