nonlinear quantum bifurcations in semiconductor superlattices

NonlinearQuantum

Birnir

SemiconductorQuantumWells

Bifurcations inQuantumWells

Bifurcations inSuperlattices

The QuantumOscillator

The GHzsources,periodhalvers,frequencysqueezersand randomsequencegenerators

Nonlinear Quantum Bifurcations inSemiconductor Superlattices

Björn Birnir∗

Center for Complex and Nonlinear Scienceand

Department of Mathematics, UC Santa Barbaraand University of Iceland, Reykjavík

BIRS Banff August 2016

Co-authors Miguel Ruiz-Garcia†, Jonathan Essen?∗, Manuel Carretero†, LuisBonilla† ?Department of Physics, University of California, Santa Barbara

†Gregorio Millán Institute for Fluid Dynamics, Nanoscience and IndustrialMathematics, Universidad Carlos III de Madrid, Spain ∗Supported by U. S.

Army Research Office under contract/grant number 444045-22682

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Outline

1 Semiconductor Quantum Wells

2 Bifurcations in Quantum Wells

3 Bifurcations in Superlattices

4 The Quantum Oscillator

5 The GHz sources, period halvers, frequency squeezersand random sequence generators

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Nonlinear Coherent Electron States

In GaAs/AlGaAs quantum wells (QWs), the Coulombself-interaction of the two-dimensional electron gas(2DEG) leads to the emergence of experimentallyobservable nonlinear effects:

the depolarization shift of the intersubband absorptionpeakmodeled considering the interplay between theone-electron density matrix andmean-field (Hartree) potentialdipole-coupled optical driving fieldand thermal dissipation rates (depolarization anddecorrelation)

The nonlinear effects are most pronounced for highcarrier densities and driving field amplitudes

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Outline






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Nonperturbative theory of intersubbanddynamics in quantum wells

Galdrikian, Batista, and Birnir et al. (GBB) extended thetheory of Ando et al. [2]

By self-consistent numerical computation of thetime-dependent coherent electronic states [8, 7, 4, 3]Nonlinearities were enhanced in slightly assymetricquantum wells, containing one or more ‘step’ features inthe confinement potential (Figure 1)

One such step, nonlinear phenomena such asbistability, period-doubling bifurcations, andperiod-doubling cascades to chaos were observed inthe simulationsTwo or more steps resulted in a Hopf bifurcation of theperiodic orbit, giving quasi-periodic orbits on a torus,followed by a period-doubling cascade of theassociated invariant tori

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The two-step Quantum Well

Figure: The stationary self-consistent potential of the assymetricGaAs-AlGaAs quantum well from Batista et al. [4]. Theeigenstates’ energy levels are indicated by the horizontal lines.

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Outline






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GaAs/AlGaAs superlattices (SLs)

Dynamics are similarly dominated by the interplaybetween the electronic populations of each site and theself-consistent potentialWell-described by the resonant sequential tunneling(RST) model of Bonilla et al [5]In the RST model, the electronic states are assumed tobe localized at each site of the SL, while the electronicstates in the GBB model simultaneously occupymultiple regions between the stepsIntersubband dynamics at each site of the RST modelare not resolved in time; instead, they are treatedaccording to the Chapman-Enskog method [6]

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Quantum Wells layered together in aSuperlattice

Figure: A mesa-shaped SL of 1.2 mm (square) width and 1.5 µmthickness. (a) shows a dc voltage-biased SL consisting of twocontact regions of about 0.5 µm thickness each and 50 periods,formed by the two semiconductor layers of different bandgaps (b).

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Nonlinear Bifurcations in a Superlattice

We study the usual sequential tunneling model ofelectron transport in a weakly coupled n-doped SL,under a (DC) voltage bias and search for nonlinearbifurcations of coherent electron states, parametrizedby DC voltage V .The model is adapted from M. Alvaro, M. Carretero,and L. Bonilla, [1].Let the electric field, in well i , be Fi , wherei = 1, ...,N(N ≤ 50) is the number of SL periods, andJi→i+1 and J(t) is the tunneling current density fromwell i to well i + 1, and the total current density,respectively.

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Superlattice Equations

εdFi

dt+ Ji→i+1 = J(t), (1)

J−i→i+1 =eni

lv (f )(Fi)−J−i→i+1(Fi ,ni+1,T ), (2)

ni = ND +ε

e(Fi −Fi−1), (3)

J−i→i+1(Fi ,ni+1,T ) =em∗kBT

π h2lv (f )(Fi) ln

[1 + e−

eFi lkbT (e

π h2ni+1m∗kbT −1)

],

(4)J0→1 = σ0F0, JN→N+1 = σ0

nN

NDFN , (5)

N

∑i=1

Fi =Vl. (6)

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Outline






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The Bifurcation Diagram

1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

V (Volts)

32

34

36

38

40

PF

6(a

.u

.)

Figure: The bifurcation diagram for N = 10 and Vbarr = 600 meV.The Hopf bifurcation from the steady state occurs at the kinkabove 1.7V. The period-doubling cascade occurs in the shadedregion above 2.1V.

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The Bifurcations

1.8

V

J(t) F6 vs F5 PF6 vs PF6 P [J ]

2.1

V2.

103

V2.

106

V

0 1 2 3 4 5 6 7 8

t (ns)

2.10

9V

F5 (a.u.) F6 (a.u.)

0 1 2

f (GHz)

Figure: Representative phase portraits. A periodic oscillation isshown in the first row. The period-doubling cascade to a chaoticattractor is shown in the bottom four rows.

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Outline






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Bifurcations and Devices

We simulate SLs that can be used as sources, periodhalvers and squeezers, random sequence generatorsand frequency mixers, at room temperature

Hopf bifurcation: The first bifurcation is the Hopfbifurcation from stationary state to periodic orbit. Thiselectron state is a source; it will emit radiation at thefrequency ω = 2π/T , where T is the fundamentalperiod, and at its superharmonics.Period doubling bifurcation: The second bifurcation thatwe find in the SL is the period-doubling bifurcation. Nowthe periodic orbit has period twice that of the originalperiodic orbit, so the frequency is halved: ω→ ω/2.Can then be used to compress information into adesirable frequency range or to squeeze out of itundesirable noise [9].

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A Strange Attractor

The period doubling cascade leads to a strangeattractor:

Period doubling cascade: The period doubling of theperiodic orbit continues into a period-doubling cascadeto a strange attractor. Based upon the emergence of abroad peak between the two strongest harmonics, weconclude that the invariant manifold is a strangeattractor.A practical application of the dynamics in this regime isultrafast generation of random number sequences [12].This has many applications in areas such as securecommunication and data storage, stochastic modeling,and Monte Carlo simulations.

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The Design of the Superlattice

The design (b) that effectively quenches thebackground noise is taken from Huang et al. [11, 10].

Figure: Two superlattices, respectively GaAs/AlAs (a) and (b)GaAs/Al0.45Ga0.55As.

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Conclusions

This work demonstrates that semiconductorheterostructures, consisting of a superlattice ofweakly-coupled quantum wells under a DC voltagebias, are nonlinear quantum systemsThe coherent electronic dynamics in the superlatticebifurcate with increased voltage.In our simulations, we find a Hopf bifurcation, aperiod-doubling of the resulting periodic orbit and aperiod-doubling cascade to a strange attractorThese bifurcations are observable at roomtemperature. These enable the design of a number ofdevices operating in the GHz range.

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M Alvaro, M Carretero, and LL Bonilla.Noise-enhanced spontaneous chaos in semiconductorsuperlattices at room temperature.EPL (Europhysics Letters), 107(3):37002, 2014.

Tsuneya Ando, Alan B. Fowler, and Frank Stern.Electronic properties of two-dimensional systems.Rev. Mod. Phys., 54:437–672, Apr 1982.

Adriano A. Batista, Bjorn Birnir, P. I. Tamborenea, andD. S. Citrin.Period-doubling and hopf bifurcations in far-infrareddriven quantum well intersubband transitions.Phys. Rev. B, 68:035307, Jul 2003.

Adriano A Batista, PI Tamborenea, Bjorn Birnir, Mark SSherwin, and DS Citrin.

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Nonlinear dynamics in far-infrared driven quantum-wellintersubband transitions.Physical Review B, 66(19):195325, 2002.

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Bryan Galdrikian, Mark Sherwin, and Björn Birnir.Self-consistent floquet states for periodically drivenquantum wells.

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Experimental observation of spontaneous chaoticcurrent oscillations in gaas/al0. 45ga0. 55assuperlattices at room temperature.Chinese Science Bulletin, 57(17):2070–2072, 2012.

Wen Li, Igor Reidler, Yaara Aviad, Yuyang Huang,Helong Song, Yaohui Zhang, Michael Rosenbluh, andIdo Kanter.Fast physical random-number generation based onroom-temperature chaotic oscillations in weakly coupledsuperlattices.Phys. Rev. Lett., 111:044102, Jul 2013.

nonlinear quantum bifurcations in semiconductor superlattices

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