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A lattice model exhibiting radiation-induced anomalous conductivity

A lattice-based model exhibits an unusual conductivity when it is subjected to both a static magnetic field and electromagnetic radiation. This conductivity anomaly may explain some aspects of the recently observed"zero-resistance states". PACS: 72.40+w, 73.40-c, 73.63 Keywords: Zero-resistance states, negative conductivity, lattice model

Electrons in large magnetic fields, specially in systems of reduced dimension at low temperatures, exhibit a wide variety of interesting phenomena. Recently, "zero-resistance states" have been reported [1], [2]. The unusual conductivity of these states is observed in two-dimensional electron systems in magnetic fields of less than one Tesla, temperatures around 1K, and in microwave fields with frequencies slightly larger than integral multiples of the cyclotron resonance frequency. Prominent among the proposed zeroresistance-state mechanisms are theories which combine excitation by the microwaves with impurity scattering to promote an electron to an "uphill" (opposing the electric force) excited Landau level [3]. The resulting negative microwave-induced current may explain the observed zero-resistance states [4].
We describe here a quasi-one-dimensional lattice model which can produce a microwave-induced negative contribution to the total current. This counter-flowing current is obtained through the combined actions of an external magnetic field and an imposed current. It does not require impurity scattering. Although our model is too crude to reproduce quantitative aspects of the zero-resistance experiments, we think the possibility of obtaining a negative conductivity without resorting to impurity scattering is important.
The elementary geometry of our model allows us to obtain results from derivations and approximations which are relatively transparent. The model has some appealing aspects. The negative current is produced by the excitation of propagating states moving against the applied electric field. Particles in these propagating states can move a relatively long distance in the "backward" direction. This differs from the theories which invoke backward "jumps" with displacements limited by the size of a Landau orbit.
The backward propagating excited states in our model have their origin in the reduced symmetry of the band structure. An imposed current gives the microwave-excited states a preferred direction of motion because the combination of the magnetic field and the Hall field reduce the band structure symmetry.
Our model has significant limitations. The lattice approximation means there is no clearly identified cyclotron resonance, and the one-dimensional geometry obscures the connection between a negative conductivity and a zero-resistance state. Also, anomalous conductivity is seen only when an appreciable magnetic flux threads each unit cell of the lattice. This implies extremely large fields (or a distance unit which is large compared to the interatomic spacing). Despite these problems, we believe the counter-flowing itinerant states produced by the reduced band structure symmetry may relate to the observed zero-resistance states. A generalization of the results presented here to a more realistic geometry is clearly a desirable goal.
Consider then, a system where electrons hop from site to nearest-neighbor site on the essentially one-dimensional lattice illustrated in Figure 1. This is a simplification of the famous model described by Hofstadter [5] and many others [6], [7]. The restriction to only two rows of lattice sites allows one to add the Hall effect and describe the magnetic properties in greater detail. In order to simplify notation and avoid confusing signs, units are scaled, and the electrons are replaced with positively charged fermions. Taking the lattice constant to be unity means the two rows of sites (x, y) are at y =[any integer] and x is either 1 or 2. The hopping matrix element in zero field is set equal to −1. The magnetic flux (directed along +ẑ) through a unit square in the lattice is φ, and the flux is scaled so φ = π corresponds to one flux quantum penetrating each unit cell of the lattice. The Hall effect is included by applying a potential (−v on the x = 1 row, and +v on the x = 2 row).
The magnetic field changes the phases of the hopping matrix elements [8]. Hopping to larger y (up) on the x = 2 row and hopping to smaller y (down) on the x = 1 row are both multiplied by exp(iφ). (Matrix elements for hopping in opposite directions must be complex conjugates.) Matrix elements for hopping between the x = 1 and x = 2 rows remain equal to −1.
Consider Bloch waves traveling in the y−direction (ψ ≈ exp(iky)). For these states, the Schroedinger equation reduces to where ψ 1 and ψ 2 are the wave amplitudes on the x = 1 and x = 2 rows. The resulting two eigenvalues (for each k) yield the energy bands.
Example band structures of this system are shown in Figures 2a,b. In Figure  2a, φ = π/5 and v = 0. The symmetry ε(+k) = ε(−k) follows from the system's two-fold rotational axis. (There is no time-reversal symmetry in a magnetic field.) Also shown in Figure 2a is the Fermi energy E F for a half-filled band and the Fermi wave numbers k L and k R . When there is no current and v = 0, k L = −k R . Imposing a current in this system generates a transverse Hall voltage which appears as the potential v in the Schroedinger equation (Eq.(1)) and in the energy levels (Eq.(2)). Figure 2b shows the reduced symmetry of the φ = π/5 energy bands which results from v = 0.2. The reduced symmetry means the right and left Fermi wave numbers (k L and k R ) are no longer equal in magnitude.
The current I is produced because particles in the interval k L < k < k L + ∆k are moved to the interval k R < k < k R + ∆k, as represented by the open and filled circles in the lower band in Figure 2b. In scaled units, the current is The Hall potential is determined by requiring the total charge on the x = 1 and x = 2 rows to be essentially the same. This will minimize the electrostatic energy. For a given k, the wave function amplitudes on the two rows (x = 1 and x = 2 in the lower band) are (6) and the difference between the probability density on the two rows is The charge difference ρ 2 (k) − ρ 1 (k) is positive for k > 0 (for small enough v). This is the expected sign for the Hall effect. Positive particles (in the lower band) moving along +y in a z−directed magnetic field will be pushed in the +x direction. The Hall voltage is determined self-consistently. In a linear approximation, the difference between the charge on the x = 1 and x = 2 rows can be written as ρ 2 − ρ 1 = αI − βv, where ρ 2 and ρ 1 are the total charges on the two rows, and α and β are proportionality constants. However, electrostatics mean v = γ (ρ 2 − ρ 1 ), where γ is another constant. Physically, one expects γ to be large because a small charge imbalance leads to a large electrostatic potential. The linearized relation ρ 2 − ρ 1 = αI/ (1 + βγ) implies a vanishing charge imbalance in the limit of large γ. We use this near-neutrality condition and Eq.(7) to obtain the Hall potential v.

2π
For example, when φ = π/5 the Fermi surface displacement of ∆k ∼ = 0.33 is required to make v = 0.2. This case is illustrated in Figure 2b. The asymmetric form of the energy bands resulting from a nonzero Hall potential enables the electromagnetic excitation of a current antiparallel to the applied field. An ac electric field with frequency ω will vertically excite electrons from the lower to the upper energy band. For the ac electric field polarized alongx, the transition probabilities are proportional to the squared matrix elements The perturbation expression for the induced current is proportional to the radiation intensity multiplied by the integral (10) The terms in the integrand represent energy conservation, the squared matrix element, and the difference between the speeds in the upper and lower bands, respectively. Using the delta-function identity and Eq.(6), one concludes that contributions to A(hω) will cancel if the energy-conservation condition is met for two values of k, since ∂(ε + (k) − ε − (k))/∂k will be positive for one k and negative for the other. For the example shown in Figure 2b, non-cancelling contributions to A(hω) lie in the restricted range of k (shaded with the large arrow) shown in Figure 2b. Thus and We emphasize that the negative microwave-induced current obtained from this model is a consequence of the combined effects of an external magnetic field and a Hall voltage produced by the imposed current.