## Abstract

A superlattice (SL) is an artificial crystal in which alternating nanometre-thick layers of two or more different semiconductor materials provide a periodic potential for conduction electrons. Strong magnetic and electric fields applied to this type of structure provide a means of exploring novel regimes of electron dynamics. The applied fields lower the dimensionality of the electronic states and lead to qualitative changes in the electronic conduction. This discovery is of fundamental interest and highly relevant to the properties of other low-dimensional conductors, such as nanowires and quantum dot SLs, which are presently attracting the attention of the physics and device communities. In addition, a rare type of chaotic electron dynamics, called non-Kolmogorov–Arnold–Moser (KAM) chaotic motion, which has been theoretically studied for several decades, is observed experimentally in SLs. The onset of chaos at discrete values of the applied electric and magnetic fields is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns in phase space. Therefore, non-KAM chaos could provide a new mechanism for controlling the electrical conductivity of the electronic devices with extreme sensitivity.

## 1. Introduction

In their pioneering paper on superlattices (SLs), Esaki & Tsu (1970) stated ‘The study of superlattices and observation of quantum mechanical effects on a new physical scale may provide a valuable area of investigation…’ Since then, after more than three decades, many advances in the physics and device applications of SLs have been made. The SLs have opened up new, challenging and exciting research activities. So what is an SL and why has it proved to be so ‘valuable’?

An SL is an artificial crystal that consists of alternating thin layers (approx. 10^{−8} m) of two or more different semiconductor materials. The unit cell of this structure is larger than the typical spatial separation between the atoms (less than 10^{−9} m). This creates an artificial periodic potential consisting of a series of quantum wells and potential barriers for electron motion along the SL axis, *x* (figure 1*a*,*b*).

The preparation of SLs requires advanced epitaxial growth techniques, such as molecular beam epitaxy, which allow the crystal grower to control accurately the thickness of the layers on a length scale as small as a single layer of atoms, thus enabling the precise control and reproducibility of the composition and spatial periodicity along the SL axis.

The periodic potential of the SL breaks up the continuous energy band of electrons into a series of narrow bands, called minibands, separated from each other by energy gaps called minigaps (figure 1*b*). The miniband width, *Δ*, and extent in momentum space can be controlled in a flexible way. The miniband width is determined by the rate at which an electron tunnels between adjacent quantum wells and can be adjusted by changing the thickness and composition of the wells and tunnel barriers. This also determines the extent of carrier confinement along the SL axis, *x*. In the limit of small *Δ*, the electrons undergo free motion in the *y–z* plane in a series of uncoupled quantum wells. The extent of the minizone is simply 2*π*/*d*, where *d* is the SL period (figure 1*c*).

These artificial crystals have provided the condensed matter physicist with a system in which the electronic properties can be ‘made to order’, thus making possible the observation of a unique regime of electron dynamics, called Bloch oscillations, which cannot be observed in ‘natural’ crystals. The basic physics of Bloch oscillations is shown in figure 2. This figure shows the dependence of the energy, *ϵ*, and group velocity, *v*_{g}, on *k*-vector for electron motion along *x*. The velocity has a periodic dependence on *k*_{x} with a maximum at the inflection point of the *ϵ*(*k*_{x}) curve. In the absence of scattering, a dc electric field, ** F**, applied along the SL axis, accelerates an electron along the

*ϵ*(

*k*

_{x}) curve so that it performs Bloch oscillations due to repeated Bragg reflections of the electron wave at the boundaries of the SL minizone at

*k*

_{x}=±

*π*/

*d*. The angular frequency of the Bloch oscillations is

*ω*

_{B}=

*edF*/

*ℏ*and at relatively modest electric fields, it can be tuned to values as high as 2

*π*10

^{12}rad s

^{−1}(corresponding to a temporal frequency of 1 THz). In the presence of scattering, electrons perform Bloch oscillations only when the average scattering time,

*τ*, is larger than . When this condition is satisfied, an electron can gain sufficient energy (approx.

*Δ*/2) to reach the middle of the SL minizone, whereafter an increase of

*k*

_{x}decreases the velocity. In this regime, an electron slows down as it acquires more energy. When the electron reaches the top of the miniband, it undergoes Bragg reflection, which reverses the electron's direction of motion. The repeated Bragg reflections correspond to the onset of Bloch oscillations, which are localized in real space. This effect cannot so far be observed in natural crystals because electrons scatter long before they traverse the large Brillouin zone (of width

*π*/

*a*>10

^{9}m

^{−1}) in a conventional crystal with lattice periodicity

*a*<10

^{−9}m (Grahn 1995).

Since the applied electric field controls the Bloch frequency of the electrons, SLs could be used for tuneable high-frequency (0.1–10 THz) detectors and emitters of electromagnetic radiation (Savvidis *et al*. 2004). This frequency range, referred to as the THz gap due to the technological need for better THz devices, is of topical interest for a wide range of applications including the emerging field of THz imaging in medicine and biology (Ferguson & Zhang 2002).

The flexible control of the electronic properties of an SL makes these artificial crystals key components in many other innovative devices. The recently proposed SL-based thermoelectric devices are based on the concept of filtering the electron momentum in the direction of the SL axis, thus enhancing the thermoelectric figure of merit (Harman *et al*. 2002). The SL is also used in quantum cascade lasers (QCLs; Faist *et al*. 1994), whose output is in the infrared spectrum. The QCL has also the potential to be used at THz frequencies (Köhler *et al*. 2002).

In this paper, we focus on fundamental aspects of the electron motion in an SL and review novel regimes of electron dynamics that arise from the unique electronic band structure of SLs and the effect of externally applied magnetic and electric fields. A magnetic field ** B** applied along the SL axis,

*x*, induces cyclotron motion in the

*y–z*plane and provides a means of creating a one- or zero-dimensional band structure. We show that this reduced dimensionality restricts the range of inelastic scattering processes available to the conduction electrons, leading to a pronounced decrease of the electrical conductance (Patanè

*et al*. 2004; Fowler

*et al*. 2006). Our study reveals the fundamental link between current flow and energy dissipation in low-dimensional conductors, which is of relevance to the properties of other low-dimensional systems, such as nanowires (Lauhon

*et al*. 2002) and quantum dot SLs (Harman

*et al*. 2002), which are presently attracting great interest in the physics community.

While a magnetic field ** B** applied along the SL axis,

*x*, provides a means of creating a low-dimensional band structure, the application of a magnetic field tilted at an angle relative to

*x*, leads to an unusual type of chaotic electron motion (figure 3). In classical mechanics, chaotic orbits are characterized by extreme sensitivity to changes in their initial conditions and have an irregular appearance.

The behaviour of systems in which the dynamics becomes chaotic under the influence of a perturbation is of fundamental interest to a wide range of problems in science and applied mathematics. The transition to chaos in Newtonian mechanics usually occurs by the gradual destruction of stable orbits in parameter space, in accordance with the Kolmogorov–Arnold–Moser (KAM) theorem, which is a cornerstone of nonlinear dynamics (Lichtenberg & Leiberman 1992). This type of chaos has been observed in a wide variety of systems and, in semiconductor physics, it has been studied, for example, in tunnelling diodes (Wilkinson *et al*. 1996). Here, we consider a much rarer type of chaos, non-KAM chaos, which switches on and off abruptly when the perturbation, in our experiment an electric or magnetic field, reaches certain critical values. The onset of chaos originates from the *z*-component of the tilted ** B** field, which produces nonlinear coupling of the Bloch oscillations along

*x*and the cyclotron motion in the

*y–z*plane (Fromhold

*et al*. 2004). At discrete values of

**, the coupling transforms the localized stable Bloch trajectories into unbounded chaotic electron paths, which propagate through intricate web patterns in phase space. The unique feature of this type of electron dynamics is that the transition to chaos is accompanied by a large increase in the current flow through the SL. Therefore, non-KAM chaos could, in principle, provide a mechanism for controlling the electrical conductivity in a new generation of electronic devices with high sensitivity.**

*F*The remainder of the paper is organized as follows. In the next section, we describe experimental and theoretical investigation of the one- and zero-dimensional regimes of conduction in SLs. This is followed in §3 by the description of non-KAM chaos. The paper concludes with a discussion of prospects for further research.

## 2. Electronic conduction in low-dimensional superlattices

### (a) Electronic conduction in a periodic potential

Figure 4 shows one of the SLs used in our experiments. A unique feature of this structure is the InAs monolayer at the centre of each GaAs/AlAs quantum well. This provides further degrees of freedom for tailoring the electronic properties with respect to the case of the commonly used GaAs/(AlGa)As SLs. The effect of the InAs monolayer is to lower the energy of the first miniband and increase the energy of the minigap, thereby facilitating electron injection from the emitter contact into the lower miniband and reducing inter-miniband Zener tunnelling (Patanè *et al*. 2002).

The current, *I*, through the SL has a nonlinear dependence on the applied bias, *V* (∼*F*; figure 4*c*). This differs markedly from the ohmic behaviour generally observed in a natural crystal. The current increases monotonically with increasing *V* up to a critical value, but thereafter it decreases and remains almost constant over an extended bias region. This behaviour is observed clearly at room temperature, which is an important property for device applications.

The peak in the *I*(*V*) curve is associated with a negative differential velocity effect, i.e. the decrease of the electron mean velocity with increasing *F* above a critical field *F*_{p}=*ℏ*/*edτ* (Ignatov *et al*. 1993). For *F*>*F*_{p} corresponding to *ω*_{B}>*τ*^{−1}, an electron can perform a Bloch oscillation, i.e. it can make one complete cycle through the SL minizone. The electrons undergoing Bloch oscillations are localized in real space and in a quantum mechanical picture are described by Wannier–Stark (WS) states. Each WS state has its electron wavefunction centred on a single quantum well within the SL. Its energy along *x* is quantized in units of *ℏω*_{B} and is given by *ϵ*_{p}=−*pℏω*_{B}, where *p* is an integer (Voisin *et al*. 1988). For *F*>*F*_{p}, the electronic conduction can be described in terms of hopping transitions of electrons between WS states (figure 5*b*). Increasing values of *F* lead to an increasing localization of the electron wavefunction. This reduces the coupling between WS states thus decreasing the hopping rate and corresponding current flow.

### (b) One- and zero-dimensional electrical conduction

An SL provides a flexible system for tailoring the electronic band structure. However, note that this flexibility only applies to the electron motion along a specific direction, i.e. the SL axis. But to what extent can we tailor the electron motion in the SL plane? Nanometre-scale quantum pillars fabricated by lithography and etching techniques could, in principle, provide a means of confining the carrier motion in the SL plane, but this approach is presently limited by technological constraints on the control of the pillar width at length scales (100 nm or less) necessary for observing quantum confinement effects in this type of structure. Alternatively, we can use large quantizing magnetic and electric fields. A magnetic field, ** B**, applied along the SL axis, quantizes the electron in-plane motion into one-dimensional Landau-level (LL) minibands. These are labelled by the Landau index,

*n*, and are separated in energy by

*ℏω*

_{c}, where

*ω*

_{c}=

*eB*/

*m*is the cyclotron frequency and

*m*is the in-plane electron effective mass (figure 5

*c*). The corresponding WS states split into LLs, creating a ladder of zero-dimensional WS–Landau (WSL) states with energies (figure 5

*d*). Therefore, the combination of electric and magnetic fields applied to an SL provides a means of creating either a one- or zero-dimensional band structure, which can then be exploited to investigate the role of the lowered dimensionality on electrical conduction.

As shown in figure 6*a*, the lowered dimensionality causes a strong suppression of the SL current at all bias and a shift to lower voltage of the current peak in *I*(*V*). These effects are qualitatively the same for SLs with different unit cell structures and miniband widths, *Δ*. When the intensity of the current peak in *I*(*V*) is plotted versus the ratio *ℏω*_{c}/*Δ*, the data for SLs with different *Δ* fall to a good approximation on a single ‘universal’ curve (figure 6*b*).

The quenching of the current flow with increasing *B* arises from a qualitative change in electron dynamics caused by increasing magneto-quantum confinement of carriers. For *ℏω*_{c}<*Δ*, the energies of the one-dimensional Landau minibands overlap with each other, and elastic and inelastic scattering processes involving a change of the Landau index *n* can occur. In contrast, for *ℏω*_{c}>*Δ*, the minibands become energetically decoupled from each other (figure 6*c*). Since *Δ* is smaller than the longitudinal optical (LO) phonon energy, *ℏω*_{LO}, for all our SLs, optical phonon scattering processes are forbidden within an LL miniband and the inelastic scattering time . In this regime, electrons can only undergo elastic processes and/or quasi-elastic scattering by acoustic phonons within one Landau miniband. This leads to a continuous reversal of the electron velocity and to a corresponding suppression of the current flow.

We estimate the increase of inelastic scattering time *τ*_{i} induced by *B* by noting that the power dissipated per electron at the peak of the velocity–field curve is *P*=*eFv* =*Δ*/4*τ*_{i} (Fowler *et al*. 2006). We use this relation and the measured values of *F* (∼ *V*) and *v* (∼ *I*) at different *B*, to determine *τ*_{i}. Our data reveal a common dependence of *τ*_{i} on *ℏω*_{c}/*Δ* for SLs with different *Δ* and indicate the existence of a universal quenching behaviour of the current with increasing *τ*_{i} (figure 6*d*,*e*).

The quenching of the SL current is observed both at low and high *F*. However, at high *F*, beyond the main current peak in *I*(*V*), the current is modulated by additional resonant features. These can be seen as two clear peaks, labelled *α* and *β*, in the differential conductance, *G*(*V*)=d*I*/d*V*, plots (see grey-scales plots of *G* versus *V* and *B* in figure 7*a*). Features *α* and *β* are strongly enhanced around *B*=21 T, which corresponds to the condition *ℏω*_{c}=*ℏω*_{LO}.

To understand the origin of these features, we consider a WS hopping model. WSL states with different Landau and WS indices become isoenergetic when the Stark-cyclotron resonance (SCR) condition, *Δpω*_{B}=*Δnω*_{c}, is satisfied. Here, the integer *Δp* (*Δn*) gives the change in WS or LL index in a hopping transition. Resonant hopping between WSLs of different index *n* is permitted by elastic scattering or quasi-elastic scattering via acoustic phonons. Our data indicate that these processes, of themselves, do not cause the resonant increase of current at peaks *α* and *β* (see diagonal lines in figure 7*a* representing the SCR condition). This is because elastic and/or quasi-elastic scattering lead to a continuous reversal of the electron velocity and the electrons have effectively equal probability of hopping up the potential as down the potential. A resonant enhancement of current is possible only when the SCR condition and the magnetophonon resonance (MPR) condition, *Δnω*_{c}=*ω*_{LO}, are *simultaneously* satisfied. These conditions correspond to the hopping of electrons between the WSL states accompanied by irreversible dissipation of energy through LO phonon emission. Features *α* and *β* correspond to 2*ω*_{B}=*ω*_{c}=*ω*_{LO} (*Δp*=2, *Δn*=1) and to *ω*_{B}=*ω*_{c}=*ω*_{LO} (*Δp*=*Δn*=1), respectively (figure 7). Only low values of *Δp* are observed as the elastic scattering rate between isoenergetic WS states falls exponentially with increasing spatial separation.

Therefore, a reduced dimensionality leads to suppression of the electrical current and a resonant enhancement of the electrical conduction when quantum hopping transitions between low-dimensional states are accompanied by an efficient LO-phonon emission process. This study reveals the fundamental link between current flow and energy dissipation, which should be considered carefully in the exploitation of low-dimensional conductors for novel electronic devices.

## 3. Non-KAM chaos and electrical conduction in superlattices

### (a) Non-KAM chaotic electron motion

Here, we consider non-KAM chaos in an SL, which switches on and off abruptly when an electric or magnetic field reaches certain critical values. The onset of chaos is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns in phase space (see the real space trajectories in figure 8 and the Poincaré sections of phase space in figure 9).

To understand the essential features of this non-KAM chaos in SLs, first we consider the semiclassical dynamics of electrons in the presence of an electric field, ** F**, applied along the SL axis (

*x*) and a tilted magnetic field,

**, which lies in the**

*B**x–z*plane at an angle

*θ*to

*x*(inset of figure 8). The equations of motion are

**=∂**

*v**H*/∂

*p*and d

**/d**

*p**t*=−

*e*(

**+**

*F***x**

*v***), where**

*B***and**

*v***represent, respectively, the velocity and crystal momentum of the electron,**

*p**H*=

*ϵ*(

*p*

_{x})−

*eFx*+(

*q*

_{y}

^{2}+

*p*

_{z}

^{2})/2

*m*is the effective Hamiltonian of the SL,

*ϵ*(

*p*

_{x}) is the energy–momentum dispersion of the SL for electron motion along

*x*, and

*p*

_{z}and

*q*

_{y}are the in-plane mechanical momentum components. From these equations, it can be shown that

*p*

_{z}satisfies the same equation as a one-dimensional simple harmonic oscillator driven by a time (

*t*)-dependent plane wave (Fromhold

*et al*. 2001), i.e.(3.1)Here,

*C*is a constant and

*K*=−

*d*tan

*θ*/

*ℏ*. Since the solution,

*p*

_{z}, of this one-dimensional problem determines uniquely the electron orbits in real space, the motion of the miniband electron is equivalent to that of a time-dependent driven harmonic oscillator: a classical example of a non-KAM chaotic system.

As shown in figure 8*a*, when *θ*=0, the electrons undergo cyclotron motion about the ** B**-axis and Bloch oscillations along the

*x*-direction. Consequently, the electron orbits in the

*x–z*plane are separable and stable. When

**is tilted, electrons starting from rest follow trajectories that are chaotic and extended compared with the orbits at**

*B**θ*=0, but still localized, as shown in figure 8

*b*. However, at discrete

*F*values for which the ratio

*r*=

*ω*

_{B}/

*ω*

_{c}cos

*θ*is an integer, the chaotic trajectories become unbounded along

*x*and

*z*(figure 8

*c*).

It is interesting to note that although this resonance condition is very similar to the SCR condition (§2), the underlying physics is fundamentally different: non-KAM chaos originates from Hamiltonian dynamics only and involves no energy level quantization, whereas the SCR is a purely quantum-mechanical effect, which only occurs when ** F** and

**produce quantized WS and LLs. The non-KAM chaos has interesting quantum-mechanical manifestations. It induces an abrupt transition from completely localized to highly extended energy eigenfunctions. The quantized energy eigenfunctions of the SL,**

*B**ψ*(

*x*,

*z*), change discontinuously from completely localized (figure 8

*a*) to highly extended when the resonance condition is satisfied (figure 8

*c*).

The resonant delocalization of the electron trajectories can be understood by considering the Poincaré sections and the Wigner functions (quantum-mechanical analogues of the Poincaré sections) shown in figure 9.

To understand these plots, we should note that as an electron's *x*-coordinate increases, its in-plane kinetic energy also increases, thus producing points further from the centre of the section. When *θ*=0 (figure 9*a*), each electron orbit is confined to a single ring in phase space and therefore to a particular region of the *x*-axis, where it performs Bloch oscillations. Tilting ** B** generates a chaotic sea (figure 9

*b*), which is enclosed by isolated stable islands resembling the ring pattern at

*θ*=0. Orbits within the chaotic sea are extended, but remain bounded by the encircling stable islands. However, when 0<

*θ*<90°, the phase space structure changes dramatically at discrete

*F*values for which

*r*=

*ω*

_{B}/

*ω*

_{c}cos

*θ*is an integer. When this resonance condition is satisfied, the electron orbits map out intricate ‘stochastic web’ patterns (figure 9

*c*). Quasi-linear web filaments link the rings that are isolated from one another in the off-resonance case. This delocalizes the electrons. Moving

*F*off resonance destroys the stochastic web by removing the quasi-linear filaments, thereby localizing the electrons.

### (b) Non-KAM chaos and electrical conduction

Stochastic web formation gives rise to an unusual type of metal–insulator-like transition at a series of discrete voltages, where the electron trajectories change from localized (insulating character) to unbounded (conducting character). In turn, this has strong effects on the drift velocity, *v*, and electrical current, *I* (figure 10).

When *θ*=0°, the *v*(*F*) curve (black trace in figure 10*a*) exhibits a single Esaki & Tsu (1970) peak. Tilting ** B** produces additional peaks in

*v*at

*F*values for which

*r*=1 and 2 (green arrows in figure 10

*a*), when the electrons undergo rapid diffusive motion through stochastic webs. In turn, this induces resonant features in

*I*(

*V*). For 15°<

*θ*<55°, the slope of each measured

*I*(

*V*) curve increases sharply at

*V*<0.3 V. This generates strong resonant peaks in the measured

*G*(

*V*)=d

*I*/d

*V*characteristics (figure 10

*b*,

*c*), in good agreement with those calculated (figure 10

*d*,

*e*). Each peak occurs at the voltage for which

*r*=1 throughout most of the SL layers, so that the phase space is threaded by an infinite stochastic web. Electrons undergo rapid diffusive motion through the web, and hence through the SL itself, thereby generating the resonant peak in

*G*(

*V*). The current resonances shown in figure 10 for a lattice temperature of 4.2 K are also clearly visible at room temperature, although they are slightly broadened. This indicates that in our SL devices, raising the temperature has little effect on the inherent sensitivity of non-KAM chaos.

## 4. Conclusion

In this paper, we have discussed remarkable regimes of electron motion, i.e. one- and zero-dimensional motion, and non-KAM chaos, which cannot be observed in natural crystals. These arise from the unique miniband dispersion of an SL and the effect of externally applied electric and magnetic fields. Non-KAM chaos provides a new method for modulating the electrical conductivity of a semiconductor SL with the potential for novel switching devices that exploit the intrinsic sensitivity of chaos.

We have realized non-KAM chaos for electrons in an SL with stationary applied electric and magnetic fields that act like a THz plane wave (equation 3.1). Therefore, we can expect that the electrical conductivity could be modulated through excitation of the SL with THz electromagnetic radiation. In addition, lateral confinement of carriers in the SL plane should significantly suppress electron scattering and therefore help us to satisfy the condition for Bloch oscillations (*ω*_{B}>*τ*^{−1}) needed to create a THz emitter or detector.

The exciting fundamental physics and novel effects found in SLs could encourage similar studies in new engineered crystals. The discovery of new material systems and advances in lithographic patterning techniques has pushed further the boundary of the parameter space available to the condensed matter physicist, thus leading to the development of novel structures. These include nanowire one-dimensional SLs (Lauhon *et al*. 2002) and binary nanoparticle SLs (Shevchenko *et al*. 2006) and many others. In the near future, we may be able to tailor with even higher precision the shape of the energy dispersion curves of conduction electrons (Patanè *et al*. 2005; Rourke *et al*. 2005) and control precisely the dimensionality of the electronic states to generate other, still unexplored, regimes of electron motion.

Therefore, this paper may provide further stimulus for new discoveries in a vibrant and challenging field of research that continues to provide many scientific surprises even after three decades of intensive research.

## Acknowledgments

This work is supported by the Engineering and Physical Sciences Research Council and the Royal Society. We are very grateful to L. Eaves, N. Mori, M. Henini, A. Ignatov, D. K. Maude, D. Fowler, R. Airey, S. Bujkiewicz, P. B. Wilkinson, F. W. Sheard and A. A. Krokhin, who were our collaborators in the work described here.

## Footnotes

One contribution of 23 to a Triennial Issue ‘Mathematics and physics’.

- © 2006 The Royal Society