Physics Today, volume 56, Number 4, page 24, April 2003
Microwaves Induce Vanishing Resistance in Two-Dimensional Electron Systems
At modest magnetic fields and microwave excitations, the resistance of a 2D semiconductor can oscillate all the way to zero.
Zero resistance is a rare phenomenon in condensed matter systems, and its observation heralds interesting physics. Heike Kamerlingh Onnes was the first to see a transition to a zero-resistance state, when he discovered superconductivity in mercury in 1911. Nearly 70 years later, Klaus von Klitzing observed the quantum Hall effect (QHE) accompanying vanishing longitudinal resistance in two-dimensional electron systems (2DES) at high magnetic fields. So when Ramesh Mani (now at Harvard University) and collaborators submitted a paper to Nature on "zero-resistance states" last June1 and Michael Zudov and Rui-Rui Du (University of Utah) and colleagues posted a preprint with "evidence for a new dissipationless effect" in October2--both groups studying 2DES irradiated with microwaves at lower magnetic fields--many people were intrigued.
At first glance, the new observations bear enticing resemblances to the QHE. For example, the resistance in the two effects approaches zero with similar temperature dependences. But there are significant differences. Most noticeably, not only is the Hall resistance not quantized in the recent experiments, but the microwaves appear to have virtually no effect on the Hall resistance. And while the QHE is typically seen in magnetic fields of several tesla, the new effects are seen in fields a factor of 50 or so lower, about 0.1 T. In addition, the QHE is an equilibrium effect, whereas the low-field state is a nonequilibrium one induced by microwave irradiation.
In 2DES, electrons are typically confined at interfaces between two different semiconductors, or within a quantum well formed in a three-layer semiconductor sandwich. When a magnetic field is applied perpendicular to the 2D plane of such systems, the electrons' orbital motion is quantized, which leads to flat bands--so-called Landau levels--in the energy spectrum. The energy levels are evenly spaced with a separation of wc, where the cyclotron frequency wc = eB/m*. Here, m* is the effective electron mass in the semiconductor (for gallium arsenide, about 0.067 of the bare electron mass). The strength of the magnetic field can be expressed in terms of the filling factor n, which specifies the number of filled Landau levels. The higher the field, the more states there are in each Landau level, and the lower the filling factor for a given electron density.
Resistance of a two-dimensional electron system with very high mobility under microwave irradiation. Shubnikov-de Haas oscillations are seen in the longitudinal resistance Rxx for fields above 0.2 T. Below that, Rxx without microwaves (red) is featureless; with microwaves, Rxx oscillates dramatically (purple), although the transverse Hall resistance Rxy (green) remains essentially unaffected. The positions of the resistance minima are proportional to Bf = wm*/e, where w is the microwave frequency and m* is the effective electron mass. (Adapted from ref. 1.)
Landau-level quantization leads to a variety of effects in 2DES. At large fields--typically several tesla--the resistance is dominated by the QHE: At fields corresponding to filled Landau levels, that is, integer values of n, the longitudinal resistance Rxx--measured in the direction of the applied current--goes to 0 while the transverse or Hall resistance Rxy shows plateaus at h/ne2. And in clean samples, Hall plateaus are also found at fractional values of n. Meanwhile, at lower magnetic fields (corresponding to higher filling factors), so-called Shubnikov-de Haas (SdH) oscillations appear in the resistance as the field-dependent Landau-level energies pass through the Fermi energy of the 2DES.
The QHE and SdH oscillations are well understood and are essentially DC effects. The first report of microwave-induced features was made by Zudov and Du, using samples supplied by Jerry Simmons and John Reno (Sandia National Laboratories).3 Peide Ye (now at Agere Systems) and colleagues subsequently saw similar features.4 Those two early experiments saw radiation-induced oscillations in the resistance at low magnetic fields, below the onset of SdH oscillations. In those experiments, though, the microwave response did not reach zero resistance.
The strength of the induced behavior
appears to depend on the electron mobility, a measure of how "clean"
the sample is. In the earlier experiments, the mobility was about 3 ´106 cm2/Vs--considered
quite a high value. Continuing improvements in sample quality have made the new
work possible: The sample made by Vladimir Umansky (Weizmann Institute of Science) for Mani
and coworkers had a mobility five times higher, and
the sample by Loren Pfeiffer and Ken West (Lucent Technologies' Bell Labs) for
illustrates the effect of microwave irradiation on the low-field resistance, as
measured by Mani and colleagues at the Max Planck
Institute for Solid-State Physics in
Probing the effect
Microwave-induced oscillations in
the resistance are periodic in 1/B,
where B is the applied
magnetic field. (a) Experimenters at Harvard University and the Max
Planck Institute for
When the resistance is plotted in terms of
inverse field (or, alternatively, inverse cyclotron frequency), the
oscillations are regularly spaced, as shown in figure
2, with a period set by the microwave frequency. Much attention has been
paid to the phase of the oscillations. In their early experiments, Zudov and Du and also Ye and colleagues found that, for microwaves of frequency w, the
oscillations were periodic with maxima at º w/wc = j and minima at = j +
1/2, for j an integer. Mani and company also
saw regular oscillations, but with a "1/4-cycle phase shift": The
maxima occurred at = j -
1/4 and the minima at j + 1/4. In the recent
Both the Harvard-Stuttgart group and the Utah group have studied the various dependencies of the resistance. Oscillations were observed at all microwave frequencies they examined, from 27 GHz through 150 GHz. And at the American Physical Society's March meeting in Austin, Texas, Robert Willett (Bell Labs) reported observing oscillations at frequencies down to 3 GHz. The resistance is independent of the bias current applied to the 2DES, and the oscillations grow in amplitude as the microwave power is increased.
The more interesting dependence is that on temperature. Figure 3 shows the temperature dependence of the oscillations, as measured by Zudov and Du for a GaAs/AlGaAs quantum well sample irradiated at 57 GHz. The behavior of the resistance minima resembles that of the QHE: an activated temperature dependence of the form exp(-EA/kBT), where kB is Boltzmann's constant. Surprisingly, the activation energies EA observed by both groups are very high: The Harvard-Stuttgart group found EA/kB up to 10 K and the Utah group up to 20 K--almost an order of magnitude larger than the Landau-level spacing or the microwave photon energy. For fixed microwave frequency, both groups find a roughly linear relationship between the activation energy and the magnetic field at which the minima occur.
Activated behavior is seen in the temperature dependence of the microwave-induced vanishing resistance. When plotted on a logarithmic scale against inverse temperature, the normalized resistance minima (red) for the first four oscillations show a thermally activated dependence of the form exp(-EA/kBT), where kB is Boltzmann's constant. The activation energies are surprisingly large, up to an order of magnitude higher than any other energy scale in the system. The resistance maxima are plotted in blue. (Adapted from ref. 2.)
Some researchers have questioned whether the resistance is truly zero in the microwave-irradiated samples. At the March meeting, Willett reported measuring negative, not zero, voltages between some electrodes placed around the edges of his samples.
Theorists chime in
Word of the observations of vanishing resistance spread quickly, and many theorists began trying to understand the origins of the behavior. The arXiv.org e-print server has been functioning as a clearinghouse of ideas over the past several months. A complete understanding, though, has yet to emerge.
Adam Durst and colleagues (
Impurity scattering is one explanation being proposed for the microwave-induced behavior. In a two-dimensional electron system, an applied voltage V will give a spatial tilt to Landau-level energies (blue). Microwaves of frequency w can excite an electron to a higher Landau level. If the photon energy is slightly higher than an integral multiple of the level spacing wc, energy can still be conserved if the electron scatters laterally in the "upstream" direction. Such scattering would reduce the conductivity and could drive it to zero or even to negative values. (Adapted from ref. 5.)
As illustrated in figure 4, a voltage bias applied to the 2DES
will produce a spatial gradient in the Landau-level energy. An electron can
absorb a microwave photon whose energy is a multiple of the Landau-level
spacing, and the electron will stay in the same place. If the photon energy is
off-resonance, energy can be conserved if impurities or other imperfections in
the 2DES scatter the electron laterally. The "upstream" or
"downstream" motion will reduce or increase the conductivity of the
sample. If scattering events are frequent enough, the conductivity could be
driven to zero or even to negative values when the microwave frequency is above
a multiple of the cyclotron frequency. (In 2DES, if the transverse Hall
conductivity dominates the DC response, the longitudinal resistance is actually
proportional to the longitudinal conductance.) A similar effect has been
observed in a different symbol: James Allen (
The Yale group's calculations using a simplified model for the impurity potential reproduce not only the period of the observed resistance oscillation, but also the phase found by Mani and company. An explicit connection between the calculations and the observed zero resistance was put forward by Anton Andreev (University of Colorado) and colleagues at Columbia University,10 who noted that a negative conductivity makes the 2DES unstable (a point also made by Anderson and Brinkman7 and by Anatoly Volkov of the University of Bochum, Germany11). Andreev and coworkers showed that this instability causes the symbol to develop a domain structure with an inhomogeneous current pattern, for which the measured resistance would be zero. Andreev notes that the resistance oscillations indeed look like they could have swung negative but have instead been truncated at 0. Willett's observation of negative voltages may lend support to the idea of inhomogeneous current flow.
Other explanations have also been
proposed. James Phillips (
1. R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Johnson, V. Umansky, Nature 420, 646 (2002); http://arXiv.org/abs/cond-mat/0303034.
2. M. A. Zudov, R. R. Du, L. N. Pfeiffer, K. W. West, http://arXiv.org/abs/cond-mat/0210034; Phys. Rev. Lett. 90, 046807 (2003).
3. M. A. Zudov, R. R. Du, J. A. Simmons, J. L. Reno, http://arXiv.org/abs/cond-mat/9711149 ; Phys. Rev. B. 64, 201311 (2001).
4. P. D. Ye, L. W. Engel, D. C. Tsui, J. A. Simmons, J. R. Wendt, G. A. Vawter, J. L. Reno, Appl. Phys. Lett. 79, 2193 (2001).
5. A. C. Durst, S. Sachdev, N. Read, S. M. Girvin, http://arXiv.org/abs/cond-mat/0301569.
6. V. I. Ryzhii, R. A. Suris, B. S. Shchamkhalova, Sov. Phys. Semicond. 20, 1299 (1986); V. I. Ryzhii, Sov. Phys. Solid State 11, 2078 (1970).
7. P. W. Anderson, W. F. Brinkman, http://arXiv.org/abs/cond-mat/0302129.
8. J. Shi, X. C. Xie, http://arXiv.org/abs/ cond-mat/0302393;cond-mat/0303141.
9. See, for example, B. J. Keay et al., Phys. Rev. Lett. 75, 4102 (1995).
10. A. V. Andreev, I. L. Aleiner, A. J. Millis, http://arXiv.org/abs/cond-mat/0302063.
11. A. Volkov, http://arXiv.org/abs/cond-mat/0302615.
12. J. C. Phillips, http://arXiv.org/abs/cond-mat/0212416;cond-mat/0303181.
13. A. A. Koulakov, M. E. Raikh, http://arXiv.org/abs/cond-mat/0302465.
14. S. A. Mikhailov, http://arXiv.org/abs/cond-mat/0303130.