# Off-resonance magnetoresistance spike in irradiated ultraclean 2D electron systems

- Jesus Iñarrea
^{1}Email author

**8**:241

https://doi.org/10.1186/1556-276X-8-241

© Iñarrea; licensee Springer. 2013

**Received: **11 November 2012

**Accepted: **6 April 2013

**Published: **16 May 2013

## Abstract

We report on the theoretical studies of a recently discovered strong radiation-induced magnetoresistance spike obtained in ultraclean two-dimensional electron systems at low temperatures. The most striking feature of this spike is that it shows up on the second harmonic of the cyclotron resonance. We apply the radiation-driven electron orbits model in the ultraclean scenario. Accordingly, we calculate the new average advanced distance by the electron in a scattering event which will define the unexpected resonance spike position. Calculated results are in good agreement with experiments.

## Keywords

## Background

Transport excited by radiation in a two-dimensional electron system (2DES) has been always [1–3] a central topic in basic and especially in applied research. In the last decade, it was discovered that when a high mobility 2DES in a low and perpendicular magnetic field (*B*) is irradiated, mainly with microwaves (MW), some striking effects are revealed: radiation-induced magnetoresistance (*R*_{
x
x
}) oscillations and zero resistance states (ZRS) [4, 5]. Different theories and experiments have been proposed to explain these effects [6–18], but the physical origin is still being questioned. An interesting and challenging experimental results, recently obtained [19] and as intriguing as ZRS, consists in a strong resistance spike which shows up far off-resonance. It occurs at twice the cyclotron frequency, *w*≈2*w*_{c}[19], where *w* is the radiation frequency, and *w*_{c} is the cyclotron frequency.

Remarkably, the only different feature in these experiments [19] is the use of ultraclean samples with mobility *μ* ∼ 3 × 10^{7} cm^{2} V s^{-1} and lower temperatures *T*∼0.4 K. Yet, for the previous ‘standard’ experiments and samples [4, 5], mobility is lower (*μ* < 10^{7} cm^{2} V s^{-1}) and *T* higher (*T* ≥ 1.0 K).

In this letter, we theoretically study this radiation-induced *R*_{
xx
} spike, applying the theory developed by the authors, the *radiation-driven electron orbits model*[6–10, 20–25]. According to the theory, when a Hall bar is illuminated, the electron orbit centers perform a classical trajectory consisting in a classical forced harmonic motion along the direction of the current at the radiation frequency, *w*. This motion is damped by the interaction of electrons with the lattice ions and with the consequent emission of acoustic phonons.

We extend this model to an ultraclean sample, where the Landau levels (LL), which in principle are broadened by scattering, become very narrow. This implies an increasing number of states at the center of the LL sharing a similar energy. In between LL, the opposite happens: the density of states dramatically decreases. This will eventually affect the measured stationary current and *R*_{
x
x
}.

We obtain that in the ultraclean scenario, the measured current on average is the same as the one obtained in a sample with full contribution to *R*_{
x
x
} but delayed as if it were irradiated with a half MW frequency (*w*/2). Accordingly, the cyclotron resonance is apparently shifted to a new *B*-position around *w* ≈ 2*w*_{c}.

## Methods

*R*

_{ x x }response of an irradiated 2DEG at low magnetic field [6–10, 20–25]. The corresponding time-dependent Schrödinger equation can be exactly solved. Thus, we first obtain an exact expression of the electronic wave vector for a 2DES in a perpendicular

*B*, a DC electric field, and radiation:

*ϕ*

_{ n }is the solution for the Schrödinger equation of the unforced quantum harmonic oscillator.

*x*

_{ cl }(

*t*) is the classical solution of a forced and damped harmonic oscillator:

where *E*_{0} is the MW electric field, and *γ* is a damping factor for the electronic interaction with the lattice ions.Then, the obtained wave function is the same as the standard harmonic oscillator, where the center is displaced by *x*_{
cl
}(*t*). Next, we apply time-dependent first-order perturbation theory to calculate the elastic charged impurity scattering rate between the two *oscillating* Landau states, the initial *Ψ*_{
n
}, and the final state *Ψ*_{
m
}[6–10, 20–24]: *W*_{n,m} = 1 / *τ*, with *τ* being the elastic charged impurity scattering time.

We find that the average effective distance advanced by the electron in every scattering jump [6–10, 20–24],

*Δ*

*X*

^{MW}=

*Δ*

*X*

^{0}+

*A*cos

*w*

*τ*, where

*Δ*

*X*

^{0}, is the advanced distance in the dark [26]. Finally, the longitudinal conductivity

*σ*

_{ xx }is given by,

with *E* being the energy [26], and $\frac{\Delta {X}^{\text{MW}}}{\tau}$ the average electron drift velocity. To obtain *R*_{
xx
}, we use the usual tensor relationships ${R}_{\mathit{\text{xx}}}=\frac{{\sigma}_{\mathit{\text{xx}}}}{{\sigma}_{\mathit{\text{xx}}}^{2}+{\sigma}_{\mathit{\text{xy}}}^{2}}\simeq \frac{{\sigma}_{\mathit{\text{xx}}}}{{\sigma}_{\mathit{\text{xy}}}^{2}}$.

*R*

_{ xx }∝

*σ*

_{ xx }. Thus, finally, the dependence of the magnetoresistance with radiation is given by:

## Results and discussion

*γ*is very small; for experimental magnetic fields [19], $\mathrm{\Gamma}<\hslash {w}_{\mathrm{c}}$. This condition will dramatically affect the average advanced distance by electron in every scattering process. In contrast with standard samples where electrons always find available empty states where to be scattered, in ultraclean samples, we can clearly find two different scenarios that are described in Figure 1.

In the four panels of energy versus distance, the grey stripes are LL tilted by the action of the DC electric field in the *x* direction. Here, LL are narrow ($\mathrm{\Gamma}<\hslash {w}_{\mathrm{c}}$) and hardly overlap each other, leaving regions with a low density of states in between (white stripes). Therefore, we can observe regularly alternating grey (many states) and white (few states) stripes equally spread out. The first scenario corresponds (see Figure 1b) to an electron being scattered to the central part of a LL. As a result, the scattering can be completed with empty states to be occupied; we obtain full contribution to the conductivity and *R*_{
x
x
}. In Figure 1c, we describe the second scenario where the electron scatters to a region in between LL with a very low density of states. Obviously, in this case, there is no much contribution to the average or stationary current. In Figure 1d, the scattering is not efficient because the final Landau state is occupied. Both regimes, ‘in-between LL’ and ‘center of LL’, are distributed equally and alternately along one cycle of the MW-driven electron orbit motion; then, only in one-half of the cycle, we would obtain a net contribution to the current or *R*_{
x
x
}.

*w*. On the other hand, it is well known that for a simple harmonic motion, it is fulfilled that averaging in one cycle, $\left|\frac{A}{2}cos\mathit{\text{wt}}\right|=\left|Acos\frac{w}{2}t\right|$. Adapting this condition to our specific case, our MW-driven (forced) harmonic motion can be perceived on average as a forced harmonic motion of whole amplitude (full scattering contribution during the whole cycle) and half frequency:

*A*≃

*A*

_{2}, which is a good approximation according to the experimental parameters [19], (

*T*= 0.4 K,

*B*≤ 0.4 T,

*w*=101 GHz and MW power

*P*∼ 0.4-1 mW). With these parameters, we obtain that the amplitudes

*A*and

*A*

_{2}are similar and of the order of 10

^{-6}to 10

^{7}m. The consequence is that the

*ultraclean*harmonic motion (electron orbit center displacement) behaves as if the electrons were driven by the radiation of half frequency. Therefore, applying next the theory [6–10] for the ultraclean scenario, it is straightforward to reach an expression for magnetoresistance:

According to it, now the resonance in *R*_{
x
x
} will take place at *w* ≈ 2*w*_{c}, as experimentally obtained [19]. The intensity of the *R*_{
xx
} spike will depend on the relative value of the frequency term, (${w}_{\mathrm{c}}^{2}-{\left(\frac{w}{2}\right)}^{2}$), and the damping parameter *γ* in the denominator of the latter *R*_{
xx
} expression. When *γ* leads the denominator, the spike is smeared out. Yet, in situations where *γ* is smaller than the frequency term, the resonance effect will be more visible, and the spike will show up.

*γ*is given, after some lengthy algebra, by [27]:

where *w*_{ac} is the frequency of the acoustic phonons for the experimental parameters [19].For ultraclean samples *γ* is small [19], and according to the last expression, this makes also the term inside the brackets and *γ* smaller [28–30]. In other words, it makes the damping by acoustic phonon emission and the release of the absorbed energy to the lattice increasingly difficult. Therefore, we have a *bottleneck effect* for the emission of acoustic phonons. Now, it is possible to reach a situation where ${({w}_{\mathrm{c}}^{2}-{\left(\frac{w}{2}\right)}^{2})}^{2}\gtrsim {\gamma}^{4}$, making a resonance effect visible and, therefore, giving rise to a strong resonance peak at *w* ≈ 2*w*_{c}.

*R*

_{ xx }vs. static magnetic field for a radiation frequency of

*f*= 101 GHz. The curve or a dark situation is also presented. For a temperature

*T*= 0.4 K, we obtain a strong spike at

*w*≈ 2

*w*

_{c}as in the experiments by [19].

Finally, we obtain the usual radiation-induced *R*_{
x
x
} oscillations and ZRS as in standard samples.

## Conclusions

In this letter, we have presented a theoretical approach to the striking result of the magnetoresistance spike in the second harmonic of the cyclotron frequency. According to our model, the strong change in the density of Landau states in ultraclean samples affects dramatically the electron impurity scattering and eventually the conductivity. The final result is that the scattered electrons perceive radiation as of half frequency. The calculated results are in good agreement with experiments.

## Authors’ information

JI is an associate professor at the University Carlos III of Madrid. He is currently studying the effect of radiation on two-dimensional electron systems.

## Declarations

### Acknowledgements

This work is supported by the MCYT (Spain) under grant MAT2011-24331 and ITN grant 234970 (EU).

## Authors’ Affiliations

## References

- Iñarrea J, Platero G: Photoinduced current bistabilities in a semiconductor double barrier.
*Europhys Lett*1996, 34: 43–47. 10.1209/epl/i1996-00413-7View ArticleGoogle Scholar - Iñarrea J, Platero G: Photoassisted sequential tunnelling through superlattices.
*Europhys Lett*1996, 33: 477–482. 10.1209/epl/i1996-00366-3View ArticleGoogle Scholar - Iñarrea J, Aguado R, Platero G: Electron-photon interaction in resonant tunneling diodes.
*Europhys Lett*1997, 40: 417–422. 10.1209/epl/i1997-00481-1View ArticleGoogle Scholar - Mani RG, Smet JH, von Klitzing K, Narayanamurti V, Johnson WB, Umansky V: Zero-resistance states induced by electromagnetic-wave excitation in GaAs/AlGaAs heterostructures.
*Nature (London)*2002, 420: 646–650. 10.1038/nature01277View ArticleGoogle Scholar - Zudov MA, Du RR, Pfeiffer LN, West KW: Evidence for a new dissipationless effect in 2D electronic transport.
*Phys Rev Lett*2003, 90: 046807.View ArticleGoogle Scholar - Iñarrea J, Platero G: Theoretical approach to microwave-radiation-induced zero-resistance states in 2D electron systems.
*Phys Rev Lett*2005, 94: 016806.View ArticleGoogle Scholar - Iñarrea J, Platero G: From zero resistance states to absolute negative conductivity in microwave irradiated two-dimensional electron systems.
*Appl Phys Lett*2006, 89: 052109. 10.1063/1.2335408View ArticleGoogle Scholar - Iñarrea J, Platero G: Polarization immunity of magnetoresistivity response under microwave excitation.
*Phys Rev B*2007, 76: 073311.View ArticleGoogle Scholar - Iñarrea J: Hall magnetoresistivity response under microwave excitation revisited.
*Appl Phys Lett*2007, 90: 172118. 10.1063/1.2734506View ArticleGoogle Scholar - Iñarrea J, Platero G: Temperature effects on microwave-induced resistivity oscillations and zero-resistance states in two-dimensional electron systems.
*Phys Rev B*2005, 72: 193414.View ArticleGoogle Scholar - Durst AC, Sachdev S, Read N, Girvin SM: Radiation-induced magnetoresistance oscillations in a 2D electron gas.
*Phys Rev Lett*2003, 91: 086803.View ArticleGoogle Scholar - Mani RG, Smet JH, von Klitzing K, Narayanamurti V, Johnson WB, Umansky V: Demonstration of a 1/4-cycle phase shift in the radiation-induced oscillatory magnetoresistance in GaAs/AlGaAs devices.
*Phys Rev Lett*2004, 92: 146801.View ArticleGoogle Scholar - Mani RG, Smet JH, von Klitzing K, Narayanamurti V, Johnson WB, Umansky V: Radiation-induced oscillatory magnetoresistance as a sensitive probe of the zero-field spin-splitting in high-mobility GaAs/AlxGa1-xAs devices.
*Phys Rev B*2004, 69: 193304.View ArticleGoogle Scholar - Yuan ZQ, Yang CL, Du RR, Pfeiffer LN, West KW: Microwave photoresistance of a high-mobility electron gas in a triangular antidot lattice.
*Phys Rev B*2006, 74: 075313.View ArticleGoogle Scholar - Mani RG, Gerl C, Schmult S, Wegscheider W, Umansky V: Nonlinear growth in the amplitude of radiation-induced magnetoresistance oscillations.
*Phys Rev B*2010, 81: 125320.View ArticleGoogle Scholar - Mani RG: Narrow-band radiation sensing in the terahertz and microwave bands using the radiation-induced magnetoresistance oscillations.
*Appl Phys Lett*2008, 92: 102107. 10.1063/1.2896614View ArticleGoogle Scholar - Mani RG, Ramanayaka AN, Wegscheider W: Observation of linear-polarization-sensitivity in the microwave-radiation-induced magnetoresistance oscillations.
*Phys Rev B*2011, 84: 085308.View ArticleGoogle Scholar - Mani RG, Hankinson J, Berger C, Wegscheider W: Observation of resistively detected hole spin resonance and zero-field pseudo-spin splitting in epitaxial graphene.
*Nature Comm*2012, 3: 996–1002.View ArticleGoogle Scholar - Dai Y, Du RR, Pfeiffer LN, West KW: Observation of a cyclotron harmonic spike in microwave-induced resistances in ultraclean GaAs/AlGaAs quantum wells.
*Phys Rev Lett*2010, 105: 246802.View ArticleGoogle Scholar - Iñarrea J, Platero G: Magnetoresistivity modulated response in bichromatic microwave irradiated two dimensional electron systems.
*Appl Phys Lett*2006, 89: 172114. 10.1063/1.2364856View ArticleGoogle Scholar - Iñarrea J, Lopez-Monis C, MacDonald AH, Platero G: Hysteretic behavior in weakly coupled double-dot transport in the spin blockade regime.
*Appl Phys Lett*2007, 91: 252112. 10.1063/1.2828029View ArticleGoogle Scholar - Iñarrea J: Anharmonic behavior in microwave-driven resistivity oscillations in Hall bars.
*Appl Phys Lett*2007, 90: 262101. 10.1063/1.2751585View ArticleGoogle Scholar - Iñarrea J, Platero G: Driving Weiss oscillations to zero resistance states by microwave radiation.
*Appl Phys Lett*2008, 93: 062104. 10.1063/1.2969796View ArticleGoogle Scholar - Iñarrea J: Effect of frequency and temperature on microwave-induced magnetoresistance oscillations in two-dimensional electron systems.
*Appl Phys Lett*2008, 92: 192113. 10.1063/1.2920170View ArticleGoogle Scholar - Kerner EH: Note on the forced and damped oscillator in quantum mechanics.
*Can J Phys*1958, 36: 371. 10.1139/p58-038View ArticleGoogle Scholar - Ridley BK:
*Quantum Processes in Semiconductors*. UK: Oxford University Press; 1993.Google Scholar - Ando T, Fowler A, Stern F: Electronic properties of two-dimensional systems.
*Rev Mod Phys*1982, 54: 437–672. 10.1103/RevModPhys.54.437View ArticleGoogle Scholar - Iñarrea J, Platero G: Microwave-induced resistance oscillations and zero-resistance states in two-dimensional electron systems with two occupied subbands.
*Phys Rev B*2011, 84: 075313.View ArticleGoogle Scholar - Iñarrea J, Mani RG, Wegscheider W: Sublinear radiation power dependence of photoexcited resistance oscillations in two-dimensional electron systems.
*Phys Rev B*2010, 82: 205321.View ArticleGoogle Scholar - Iñarrea J, Platero G: Effect of an in-plane magnetic field on microwave-assisted magnetotransport in a two-dimensional electron system.
*Phys Rev B*2008, 78: 193310.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.