## Cyclotron resonance spectroscopy in a high mobility two dimensional electron gas using characteristic matrix methods |

Optics Express, Vol. 20, Issue 28, pp. 29717-29726 (2012)

http://dx.doi.org/10.1364/OE.20.029717

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### Abstract

We develop a new characteristic matrix-based method to analyze cyclotron resonance experiments in high mobility two-dimensional electron gas samples where direct interference between primary and satellite reflections has previously limited the frequency resolution. This model is used to simulate experimental data taken using terahertz time-domain spectroscopy that show multiple pulses from the substrate with a separation of 15 ps that directly interfere in the time-domain. We determine a cyclotron dephasing lifetime of 15.1±0.5 ps at 1.5 K and 5.0±0.5 ps at 75 K.

© 2012 OSA

## 1. Introduction

1. C. K. Sarkar and R. J. Nicholas, “Cyclotron resonance linewidth in a two-dimensional electron gas,” Surf. Sci. **113**, 326–332 (1982). [CrossRef]

4. D. C. Tsui, H. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. **48**, 1559–1562 (1982). [CrossRef]

5. X. Wang, D. J. Hilton, J. L. Reno, D. M. Mittleman, and J. Kono, “Direct measurement of cyclotron coherence times of high-mobility two-dimensional electron gases,” Opt. Express **18**, 12354–12361 (2010). [CrossRef] [PubMed]

6. X. Wang, D. J. Hilton, L. Ren, D. M. Mittleman, J. Kono, and J. L. Reno, “Terahertz time-domain magnetospectroscopy of a high-mobility two-dimensional electron gas,” Opt. Lett. **32**, 1845–1847 (2007). [CrossRef] [PubMed]

7. L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Quantum Electron. **2**, 739–746 (1996). [CrossRef]

*T*

_{2}= 15.1 ± 0.5 ps at 1.5 K and

*T*

_{2}= 5.0 ± 0.5 ps at 75 K in a high mobility 2DEG in an external magnetic field of ±1.25 T using time-domain data that show multiple pulses with a separation of 15 ps.

## 2. Experiment

*L*= 625

*μ*m thick undoped gallium arsenide substrate using molecular beam epitaxy. This substrate is approximately 7.5

*λ*thick at a wavelength of 300

*μ*m (1 THz), which leads to direct interference between the primary and satellite pulses in our terahertz time-domain spectroscopy experiments. The 2DEG layer is a

*d*= 30 nm gallium arsenide quantum well that is modulation doped to an electron sheet carrier concentration of

*n*= 2 × 10

_{e}^{11}cm

^{−2}. The electron mobility, determined using electrical transport measurements, in the 2DEG layer is

*μ*= 3.7 × 10

_{e}^{6}cm

^{2}V

^{−1}s

^{−1}, which corresponds to a scattering lifetime of

*τ*

_{0}= 229 ps.

*ν*= 0.8 THz that is centered near

*ν*

_{0}= 0.5 THz [3, 5

5. X. Wang, D. J. Hilton, J. L. Reno, D. M. Mittleman, and J. Kono, “Direct measurement of cyclotron coherence times of high-mobility two-dimensional electron gases,” Opt. Express **18**, 12354–12361 (2010). [CrossRef] [PubMed]

6. X. Wang, D. J. Hilton, L. Ren, D. M. Mittleman, J. Kono, and J. L. Reno, “Terahertz time-domain magnetospectroscopy of a high-mobility two-dimensional electron gas,” Opt. Lett. **32**, 1845–1847 (2007). [CrossRef] [PubMed]

*x*̂ direction. While not directly measured in our experiments, the typical electric field generated using this method is |

**E**

*| ∼ 100 Vcm*

_{THz}^{−1}, which has a corresponding |

**B**

*| ∼ 33 mT. For the temperature range in our experiments, this terahertz electromagnetic field results in negligible sample heating.*

_{THz}8. Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B **18**, 823–831 (2001). [CrossRef]

*ŷ*) of the detector to be perpendicular to the input polarization and measure this component of the electric field in an external magnetic field at both ±

*B*to remove artifacts that result in the data that are due to the low polarization extinction ratio (100:1) of this detection method.

_{ext}**B**

*, is generated in an Oxford Instruments SpectroMag split coil magnet and aligned so that it is perpendicular to the 2DEG, which is defined to be the*

_{ext}*z*̂ direction. This perpendicular magnetic field results in the formation of a spectrum of Landau levels with an energy spacing, Δ

*E*=

*h̄eB/m*

^{*}. Here,

*e*is the electron charge,

*h*̄ is the reduced Planck constant,

*B*is the magnetic field, and

*m*

^{*}is the electron effective mass [11].

*n*〉, and lowest unfilled, |

*n*+ 1〉, levels results in the formation of a coherent superposition state, |

*ψ*(

*t*)〉 =

*A*|

*n*〉 +

*B*|

*n*+ 1〉. Dephasing of the ensemble of cyclotrons will result in oscillations in the transmitted terahertz electric field that decay with a lifetime that we define as

*T*

_{2}. Figures 2(a) and 2(b) show the perpendicular component of the transmitted terahertz electric field,

*E*(

_{y}*t*), after the 2DEG at

*T*= 75 K in part (a) and

*T*= 1.5 K in part (b) and in an external magnetic field

**B**

*= (1.25 T)*

_{ext}*z*̂. These data show the amplitude and phase of the electric field oscillations directly in the time-domain that result from ensemble dephasing.

7. L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Quantum Electron. **2**, 739–746 (1996). [CrossRef]

5. X. Wang, D. J. Hilton, J. L. Reno, D. M. Mittleman, and J. Kono, “Direct measurement of cyclotron coherence times of high-mobility two-dimensional electron gases,” Opt. Express **18**, 12354–12361 (2010). [CrossRef] [PubMed]

*T*

_{2}, is substantially longer than the width of the temporal window since the time-domain oscillations available to be fitted may not decay within the fitting window to determine

*T*

_{2}without a significant uncertainty, which is limited by the finite signal-to-noise ratio of the experimental data.

## 3. Characteristic matrix method

*T*

_{2}, that are the focus of future devices applications. In addition, as has been recently shown in Ref. [12

12. T. Arikawa, X. Wang, D. J. Hilton, J. L. Reno, W. Pan, and J. Kono, “Quantum control of a Landau-quantized two-dimensional electron gas in a GaAs quantum well using coherent terahertz pulses,” Phys. Rev. B **84**, 241307 (2011). [CrossRef]

*control*of the quantum superposition state, which is needed for future applications based on coherent manipulation of the cyclotron wavefunction and will require more sophisticated methods to properly describe electromagnetic wave propagation.

*isotropic*materials studied using terahertz time-domain spectroscopy in Ref. [13

13. S. E. Ralph, S. Perkowitz, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz spectroscopy of optically thick multilayered semiconductor structures,” J. Opt. Soc. Am. B **11**, 2528–2532 (1994). [CrossRef]

*isotropic*media in Ref. [14].

*x*–

*y*plane, can be generally written in stratified samples in Eq. (3). The electromagnetic fields,

**E**and

**H**, are tangential and will be field matched across each interface in the absence of any surface currents. The full electromagnetic fields in Eq. (3) are the interference between the forward propagating (

*k*̂ = +

*z*̂) waves as well as the reverse propagating (

*k*̂ = −

*z*̂) waves, as will be discussed below.

*U*,

*V*) of the electromagnetic field that is traveling in the direction of stratification,

*k*̂ = ±

*z*̂, propagates independently from the orthogonal circular polarization (

*P*,

*Q*) of the electromagnetic field. Normal incidence propagation results in a simplified mathematical description when compared to the more general case [16

16. S. Teitler and B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. **60**, 830–834 (1970). [CrossRef]

18. H. Wöhler, M. Fritsch, G. Haas, and D. A. Mlynski, “Characteristic matrix method for stratified anisotropic media: optical properties of special configurations,” J. Opt. Soc. Am. A **8**, 536–540 (1991). [CrossRef]

*z*∈ [

*z*

_{0},

*z*

_{1}...,

*z*], where

_{ℓ}*z*

_{0}is the input interface of this sample and

*z*is the final interface, as shown in Fig. 3. The solutions to Eq. (4) within the one layer defined by

_{ℓ}*z*≤

_{j}*z*≤

*z*

_{j}_{+1}are: which define two complex admittances using

*σ*̂

_{+}polarization and

*σ*̂

_{−}polarization component. These solutions also define

*P*(

*z*),

_{j}*Q*(

*z*),

_{j}*U*(

*z*),

_{j}*V*(

*z*)] instead of eight due to the close coupling between these quantities in Eq. (4).

_{j}*z*=

*z*) is a sufficient initial condition to determine these four unknown constants, which can then be propagated to the next interface,

_{j}*z*

_{j+1}, using Eq. (5). Using characteristic matrix notation, the component of the electromagnetic wave with

*σ*̂

_{±}polarization is represented by ℚ

_{±}(

*z*,

*ν*). If both ℚ

_{±}are known at

*z*, the ℚ

_{j}_{±}at

*z*

_{j}_{+1}=

*z*+

_{j}*d*can be found using: where propagation between interfaces is described by a pair of 2 × 2 characteristic matrices for each corresponding polarization,

*σ*̂

_{±}: A stratified sample has a pair of total characteristic matrices, 𝕄̄

_{±,T}, that are given by the product of the matrices for each layer. This total characteristic matrix describes the propagation through the full stratified sample from

*z*

_{0}to

*z*.

_{ℓ}*z*≤

*z*

_{0}), the full electromagnetic field in Eq. (3) is the superposition of the incident field (subscript

*i*) and the reflected waves (subscript

*r*), while beyond the final interface (

*z*≥

*z*), the full electromagnetic wave defines the transmitted component (subscript

_{ℓ}*t*). With these definitions, transmission coefficients for the

*σ*̂

_{±}components are written in Eq. (10). These are normalized using the component of incident amplitude with the same polarization and not the full incident electric field amplitude (

*U*+

_{i}*P*). The field transmission coefficient for the

_{i}*σ*̂

_{±}component of the incident electric field is: which uses the elements of the total characteristic matrix, 𝕄̄

_{±,}

*, for the corresponding polarization component. The admittance,*

_{T}*Y*, describes the isotropic semi-infinite region defined by

_{t}*z*≥

*z*where the transmitted field is measured, while

_{ℓ}*Y*describes the isotropic semi-infinite region defined by

_{i}*z*≤

*z*

_{0}, where the incident field originates.

## 4. Results and discussion

**B**

*= 0, the permittivity of the undoped gallium arsenide substrate at terahertz frequencies is determined by the interband contributions, which are nonresonant at terahertz frequencies and result in a purely real permittivity,*

_{ext}*ε*= 12.96

_{b}*ε*

_{0}[19

19. J. S. Blakemore, “Intrinsic density *n _{i}*(

*T*) in GaAs: Deduced from band gap and effective mass parameters and derived independently from Cr acceptor capture and emission coefficients,” J. Appl. Phys.

**53**, 520–531 (1981). [CrossRef]

*τ*

_{0}= 229 ps or, equivalently, a Drude line width of Δ

*ν*= 2.8 GHz. Since the terahertz pulse in these experiments has a usable lower frequency much greater than 2.8 GHz, we neglect this free carrier response in the absence an external field (

**B**

*= 0) in the 2DEG.*

_{ext}**B**

*, breaks time-reversal symmetry in the 2DEG and results in a permittivity for*

_{ext}*σ*̂

_{+}that differs from the permittivity for

*σ*̂

_{−}. The magnetic field results in the formation of a discrete set of Landau levels whose energy spacing is the cyclotron energy,

*hν*. The interaction of a weak terahertz electromagnetic wave with these states can be modeled using a dipole interaction Hamiltonian, which allows us to neglect excitation to higher Landau levels in a two-level system approximation. The parity of the wavefunctions, |

_{CR}*n*〉and |

*n*+ 1〉, permits coupling of these states with the cyclotron resonance active polarization, defined to be

*σ*̂

_{+}, while this transition cannot occur using the orthogonal circular polarization (

*σ*̂

_{−}). As a result, the cyclotron inactive mode (

*σ*̂

_{−}) has a permittivity element that does not change (

*ε*=

_{mm}*ε*) in the external magnetic field,

_{b}**B**

*.*

_{ext}*σ*̂

_{+}) that is given by: where

*χ*

_{0}is the complex susceptibility at

*ν*, which is determined by the equilibrium populations, dipole matrix element, decoherence time (

_{CR}*T*

_{2}), and the cyclotron resonance frequency (

*ν*) [20].

_{CR}*χ*) has a Lorentzian line shape that describes the circular dichroism induced by the magnetic field. In the quantum limit where the number of filled Landau levels is small,

_{i}*T*

_{2}is a direct measure of the influence of magnetic field on scattering processes in the strongly interacting electrons in the Landau levels near the Fermi surface [21

21. A. Kawabata, “Theory of cyclotron resonance line width,” J. Phys. Soc. Jpn. **23**, 999–1006 (1967). [CrossRef]

23. V. K. Arora and H. N. Spector, “Quantum-limit cyclotron resonance linewidth in semiconductors,” phys. stat. sol. (b) **94**, 701–709 (1979). [CrossRef]

*χ*) is the associated dispersive line shape describing the concomitant circular birefringence in the external magnetic field that determines the frequency and carrier-envelope phase of the oscillations with respect to the decay envelope. Both the circular dichroism and birefringence are centered at cyclotron resonance frequency,

_{r}*ν*, with an absorption line width of Δ

_{CR}*ν*= 2/(

_{FWHM}*πT*

_{2}) in the low excitation limit. The complex permittivity tensor element for the cyclotron active polarization (

*σ*̂

_{+}) is

*ε*=

_{pp}*ε*+ [1 +

_{b}*χ*̃(

*ν*)]

*ε*

_{0}, which describes the response of the 2DEG layer to the cyclotron active (

*σ*̂

_{+}) component of electromagnetic field in this two-level approximation.

*ε̄*

_{1}, and thickness,

*d*, that is grown on a substrate with an isotropic permittivity,

*ε*

_{2}, and thickness,

*L*, as shown in Fig. 3(b). The two layers have permeabilities given by the free space value,

*μ*

_{0}. This two layer model is a significant simplification when compared to the ∼ 600 layers of alternating AlGaAs and GaAs in the 2DEG. This two layer model is, however, sufficient to predict the satellite pulse formation and the existence of cyclotron oscillations in the transmitted data. Since this sample was grown using molecular beam epitaxy, which has precise control over the thickness and composition of each layer, a complete model of the full structure is also possible with our analysis method and will be the focus of future work.

*φ*

_{±}=

*κ*

_{±,1}

*d*and

*θ*=

*κ*

_{2}

*L*. Using these, the transmission coefficient,

*t*

_{±}(

*ν*) for this 2DEG is then given by Eq. (11). The sample is in vacuum with a free space admittance,

*Y*, and transmitted,

_{i}*Y*, semi-infinite media.

_{t}*ν*= 0.8 THz centered at

*ν*

_{0}= 0.5 THz, which results in the formation of one single cycle terahertz pulse (not shown). We use a modeled THz pulse instead of experimental data since it is difficult to acquire the incident terahertz waveform needed for this calculation to demonstrate the validity of this characteristic matrix modeling technique, which would require temporarily removing the sample in this cryogenic magnet system without warming up system. The transmitted terahertz electric field at

**B**

*= 0 is plotted in Fig. 2(c) and shows the formation of a sequence of satellite pulses at time intervals of 15 ps. Fig. 2(d) plots the predicted change to the*

_{ext}*ŷ*electric field at

**B**

*= (1.25 T)*

_{ext}*z*̂ and

*T*= 75 K, while Fig. 2(e) shows the same experimental conditions at

*T*= 1.5 K assuming a longer dephasing time, presumably due to the reduced influence of phonon scattering on dephasing at low temperature.

## 5. Conclusions

*T*

_{2}to elucidate the decoherence mechanisms in this limit using this new analysis method. This will also focus on alternate models for

*χ*̃(

*ν*) that correctly model the coherent interaction of these satellite pulses over a broad range of external magnetic fields and at a range of sample temperatures.

## Acknowledgments

## References and links

1. | C. K. Sarkar and R. J. Nicholas, “Cyclotron resonance linewidth in a two-dimensional electron gas,” Surf. Sci. |

2. | M. Chou, D. Tsui, and G. Weimann, “Cyclotron resonance of high-mobility two-dimensional electrons at extremely low densities,” Phys. Rev. B |

3. | D. J. Hilton, T. Arikawa, and J. Kono, “Cyclotron resonance,” in |

4. | D. C. Tsui, H. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. |

5. | X. Wang, D. J. Hilton, J. L. Reno, D. M. Mittleman, and J. Kono, “Direct measurement of cyclotron coherence times of high-mobility two-dimensional electron gases,” Opt. Express |

6. | X. Wang, D. J. Hilton, L. Ren, D. M. Mittleman, J. Kono, and J. L. Reno, “Terahertz time-domain magnetospectroscopy of a high-mobility two-dimensional electron gas,” Opt. Lett. |

7. | L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Quantum Electron. |

8. | Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B |

9. | D. Mittleman, ed., |

10. | M. C. Nuss and J. Orenstein, |

11. | G. Landwehr, |

12. | T. Arikawa, X. Wang, D. J. Hilton, J. L. Reno, W. Pan, and J. Kono, “Quantum control of a Landau-quantized two-dimensional electron gas in a GaAs quantum well using coherent terahertz pulses,” Phys. Rev. B |

13. | S. E. Ralph, S. Perkowitz, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz spectroscopy of optically thick multilayered semiconductor structures,” J. Opt. Soc. Am. B |

14. | M. Born and E. Wolf, |

15. | J. W. Goodman, |

16. | S. Teitler and B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am. |

17. | D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am. |

18. | H. Wöhler, M. Fritsch, G. Haas, and D. A. Mlynski, “Characteristic matrix method for stratified anisotropic media: optical properties of special configurations,” J. Opt. Soc. Am. A |

19. | J. S. Blakemore, “Intrinsic density T) in GaAs: Deduced from band gap and effective mass parameters and derived independently from Cr acceptor capture and emission coefficients,” J. Appl. Phys. 53, 520–531 (1981). [CrossRef] |

20. | R. W. Boyd, |

21. | A. Kawabata, “Theory of cyclotron resonance line width,” J. Phys. Soc. Jpn. |

22. | P. Argyres and J. Sigel, “Theory of cyclotron-resonance absorption,” Phys. Rev. B |

23. | V. K. Arora and H. N. Spector, “Quantum-limit cyclotron resonance linewidth in semiconductors,” phys. stat. sol. (b) |

**OCIS Codes**

(300.6495) Spectroscopy : Spectroscopy, teraherz

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: October 26, 2012

Revised Manuscript: December 11, 2012

Manuscript Accepted: December 11, 2012

Published: December 20, 2012

**Citation**

David J. Hilton, "Cyclotron resonance spectroscopy in a high mobility two dimensional electron gas using characteristic matrix methods," Opt. Express **20**, 29717-29726 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-28-29717

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### References

- C. K. Sarkar and R. J. Nicholas, “Cyclotron resonance linewidth in a two-dimensional electron gas,” Surf. Sci.113, 326–332 (1982). [CrossRef]
- M. Chou, D. Tsui, and G. Weimann, “Cyclotron resonance of high-mobility two-dimensional electrons at extremely low densities,” Phys. Rev. B37, 848–854 (1988). [CrossRef]
- D. J. Hilton, T. Arikawa, and J. Kono, “Cyclotron resonance,” in Characterization of Materials, E. N. Kaufmann, ed. (John Wiley and Sons, Inc, New York, 2012), p. 2438.
- D. C. Tsui, H. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett.48, 1559–1562 (1982). [CrossRef]
- X. Wang, D. J. Hilton, J. L. Reno, D. M. Mittleman, and J. Kono, “Direct measurement of cyclotron coherence times of high-mobility two-dimensional electron gases,” Opt. Express18, 12354–12361 (2010). [CrossRef] [PubMed]
- X. Wang, D. J. Hilton, L. Ren, D. M. Mittleman, J. Kono, and J. L. Reno, “Terahertz time-domain magnetospectroscopy of a high-mobility two-dimensional electron gas,” Opt. Lett.32, 1845–1847 (2007). [CrossRef] [PubMed]
- L. Duvillaret, F. Garet, and J.-L. Coutaz, “A reliable method for extraction of material parameters in terahertz time-domain spectroscopy,” IEEE J. Quantum Electron.2, 739–746 (1996). [CrossRef]
- Q. Chen, M. Tani, Z. Jiang, and X.-C. Zhang, “Electro-optic transceivers for terahertz-wave applications,” J. Opt. Soc. Am. B18, 823–831 (2001). [CrossRef]
- D. Mittleman, ed., Sensing with Terahertz Radiation (Springer, Berlin, 2002).
- M. C. Nuss and J. Orenstein, Terahertz Time-Domain Spectroscopy, vol. 74 (Springer, Berlin, 1998).
- G. Landwehr, Landau Level Spectroscopy (North-Holland, New York, 1990).
- T. Arikawa, X. Wang, D. J. Hilton, J. L. Reno, W. Pan, and J. Kono, “Quantum control of a Landau-quantized two-dimensional electron gas in a GaAs quantum well using coherent terahertz pulses,” Phys. Rev. B84, 241307 (2011). [CrossRef]
- S. E. Ralph, S. Perkowitz, N. Katzenellenbogen, and D. Grischkowsky, “Terahertz spectroscopy of optically thick multilayered semiconductor structures,” J. Opt. Soc. Am. B11, 2528–2532 (1994). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).
- J. W. Goodman, Introduction To Fourier Optics (McGraw-Hill, New York, 1996).
- S. Teitler and B. W. Henvis, “Refraction in stratified, anisotropic media,” J. Opt. Soc. Am.60, 830–834 (1970). [CrossRef]
- D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,” J. Opt. Soc. Am.62, 502–510 (1972). [CrossRef]
- H. Wöhler, M. Fritsch, G. Haas, and D. A. Mlynski, “Characteristic matrix method for stratified anisotropic media: optical properties of special configurations,” J. Opt. Soc. Am. A8, 536–540 (1991). [CrossRef]
- J. S. Blakemore, “Intrinsic density ni(T) in GaAs: Deduced from band gap and effective mass parameters and derived independently from Cr acceptor capture and emission coefficients,” J. Appl. Phys.53, 520–531 (1981). [CrossRef]
- R. W. Boyd, Nonlinear Optics (Academic Press, 1991).
- A. Kawabata, “Theory of cyclotron resonance line width,” J. Phys. Soc. Jpn.23, 999–1006 (1967). [CrossRef]
- P. Argyres and J. Sigel, “Theory of cyclotron-resonance absorption,” Phys. Rev. B10, 1139–1148 (1974). [CrossRef]
- V. K. Arora and H. N. Spector, “Quantum-limit cyclotron resonance linewidth in semiconductors,” phys. stat. sol. (b)94, 701–709 (1979). [CrossRef]

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