Probing the coexistence of semiclassical transport and localization in a two-dimensional electron gas using microwave radiation

Probing the coexistence of semiclassical transport and localization

in a two-dimensional electron gas using microwave radiation

Chang-Shun Hsu

a,1

, E.S. Kannan

b,1

, J.-C. Portal

b

, C.-T. Liang

a,c,n

, C.F. Huang

c

, S.-D. Lin

d

a

Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan

b_{LNCMI, UPR 3228, CNRS-INSA-UJF-UPS BP166 38042, Grenoble Cedex, France} c

Department of Physics, National Taiwan University, Taipei 106, Taiwan

d

Department of Electronics Engineering, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history:

Received 14 August 2012 Accepted 26 November 2012 by M. Wang

Available online 4 December 2012 Keywords:

A. Semiconductors A. Quantum well

a b s t r a c t

We have performed magnetoresistance measurements on a GaAs/AlGaAs two-dimensional electron gas under microwave heating. Both the magneto-oscillations and magnetoresistance minima are used as electron thermometers to determine the electron effective temperature Te. When there is no clear

cyclotron gap, it is found that Tedetermined from these two methods can be substantially different

such that we can still distinguish localized electrons from the semiclassical conducting ones. The almost constant Teobtained from the magnetoresistance minima reveals the survived equivalence

between different Landau-band tails under insufficient localization.

&_{2012 Elsevier Ltd. All rights reserved.}

1. Introduction

When a magnetic field is applied perpendicular to the plane of a two-dimensional electron gas (2DEG), Landau quantization effect modulates the 2D density of states. At low magnetic fields,

the longitudinal resistance Rxx follows the Shubnikov–de Haas

(SdH) formalism given by[1–3]

RxxR0þFðBÞ

w

sinh

w

cosð

p

ð

n

þ1ÞÞ, ð1Þ

under such quantization effect when quantum localization is negligible. Here FðBÞ ¼ 4R0c exp½

p

=

m

qB,

w

¼4

p

3kBmnT=heB,

n

, kB, e, h, mn, T,

m

q, B, Ro, and c are the filling factor, Boltzmann

constant, electron charge, Plank constant, electron effective mass, temperature, quantum mobility, magnetic field, the resistance at B ¼0, and a constant close to unity, respectively. According to

Eq. (1), at a specific B the function f ðmn

,TÞ  lnð

D

Rxx=TÞ þ lnð1e2w_Þsatisfies f ðmn ,TÞ ¼ f0 4

p

3_k Bmn heB T, ð2Þ

where f0const þlnð4

p

3kBmn=heBÞð

p

=

m

qÞ1=B is a function of B.

Here

D

Rxxdenotes the amplitude of SdH oscillations and equals

FðBÞ 

w

=sinh

w

according to Eq. (1). On the other hand, at

sufficiently high magnetic fields when quantum localization effects become important, an exponential T-dependence of the magnetoresistance minimum Rxx,minis given by[4,5]

Rxx,minRxxð0Þ exp 

D

E 2kBT

 

, ð3Þ

over a suitable temperature range. Here

D

E which can be obtained

from the slope of lnRxx,min1=T, is the activation energy and is

expected to be close to the cyclotron gap. In this case, the 2DEG shows activated behaviour. That is, electrons are thermally activated from the localized states at Landau-band tails to the extended states near the centers of the Landau-bands.

Since the SdH formula is derived semiclassically with no localization effect whereas activated behaviour requires the existence of quantum localization, at first glance, SdH oscillations and activated behaviour do not normally co-exist. However, it has been reported that cyclotron gap indicated by the activated

behavior can be identified when

D

Rxxfollows the SdH formula[6].

Two types of electrons are considered to explain this coexistence.

One type responsible for Rxx,min is composed of the localized

electrons at Landau-band tails whereas the other type responsible

for

D

Rxx consists of the semiclassical conducting electrons near

Landau-band centers. But more studies are necessary to clarify whether localization can coexist with the semiclassical

conduct-ing behavior. It was reported in Ref. [7] that there is no

localization-induced hopping under Eq. (1), which can become

invalid with increasing B before quantum localization induces the

integer quantum Hall effect[8]. On the other hand, Eq.(3)is used

to investigate the activated transport only in an intermediate

temperature range because deviation from Eq.(3)is expected at

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Solid State Communications

0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.11.019

n

Corresponding author at: Graduate Institute of Applied Physics, National Taiwan University, Taipei 106, Taiwan. Tel.: þ 886 2 23697238;

fax: þ 886 2 23639984.

E-mail address: [email protected] (C.-T. Liang).

1_{These authors contributed equally to this work.}

either low[7]or high temperatures. It is possible that there is no

localization, but experimental data can be well fitted to Eq.(2)

over a small temperature range. Therefore, more studies are necessary to clarify whether quantum localization can coexist with the semiclassical SdH formula.

In addition to zero resistance state [9,10] and nonlinear

magnetoresistance oscillations [11], microwave radiation may

introduce electron heating and/or dephasing in semiconductors. Therefore, it is interesting to further study the coexistence of activated behaviour and the SdH formula under microwave radiation. In this communication, we use both damping of SdH oscillations and activated behaviour at SdH minima as indepen-dent electron thermometers. When an electron system is heated appreciably by microwave radiation, the equilibrium between the electrons and phonons can be broken. In this case, the electron

temperature Tecan be substantially higher than the crystal lattice

temperature Tl[12]. Various physical phenomena, such as SdH

oscillations[13], weak localization correction[14], single electron

tunneling [15], thermopower of a one-dimensional constriction

[16], and magneoresistance[17]have been applied as electron

thermometers. Using both

D

Rxxand Rxx,minas thermometers under

the microwave radiation when Eq. (2) remains valid, different

electron temperatures Te,SdHand Te,minare obtained from

D

Rxxand Rxx,min although the value of

D

E obtained from Eq.(3) is much smaller than the cyclotron gap. The existence of the two electron

temperatures shows that Rxx,min and

D

Rxxare not dominated by

the same type of electrons although the small

D

E and the validity

of the SdH formula indicate the weak strength of localization.

2. Experimental

The sample which we studied is an AlGaAs/GaAs heterostruc-ture LM4882. The following layer sequence was grown on a GaAs

semi-insulating substrate: 1

m

m undoped GaAs, 20 nm undoped

Al0:33Ga0:67As, 40 nm Si doped (2  1018cm3) Al0:33Ga0:67As, and a 10 nm undoped GaAs cap layer. Indium was alloyed near the sample edges and was annealed at 450 1C in vacuum. Magneto-transport measurements have been performed by a standard

low-frequency ac technique with a current injection of 1

m

A. Magnetic

field was applied normal to the 2DEG plane and the sample temperature was maintained between 1.4 K and 4.2 K. For the microwave measurements, linearly polarized microwave (MW) irradiation from a ‘‘carcinotron’’ tunable in the 33–50 GHz frequency range was employed. Similar results were obtained at different frequencies and in this paper, data obtained using a frequency of 42.04 GHz will be presented. To provide minimal damping of the MW power, a circular (internal diameter of 10 mm) waveguide to rectangular window WR 22 (5:8 mm 2:6 mm) with a reduction brass piece of 5 cm in length was used to obtain output linear polarization of microwave across the WR 22 windows. MW power was attenuated using a diode attenuator placed at the output of the carcinotron. The sample was placed at a distance of 1–2 mm in front of the waveguide output in a variable temperature cryostat. The sample temperature was monitored using a Allen Bradley resistor mounted close to the sample.

3. Results and discussion

The inset toFig. 1shows the magneoresistance measurements

Rxx(B) with no microwave radiation at various temperatures. With

increasing temperature, damping of the magnetoresistance oscil-lations is observed. In order to further study the observed oscillations, we investigate the T-dependence of the amplitudes

D

Rxxwhich was fitted to Eq.(2). Inserting a trial effective mass mnt into the function f on the left hand side of Eq.(2), as mentioned in

Ref. [18], at a specific B we can obtain mn

s from the slope

4

p

3_k

Bmn=heB of f T. The trial value mnt can be taken as the

effective mass mn at such a specific B if 9mn sm n t9 o

D

m, or we shall reset mn s as m n

t and iterate until such an inequality is satisfied.

Here

D

m denotes the acceptable error. Then we can check the

validity of SdH formula by the value of mn

, how good the fitting of the

D

Rxxto the factor

w

=sinh

w

at any B, and whether the function f0ðBÞlnð4

p

3kBmn=heBÞ is linear in 1=B or not. In this study, we set

D

m ¼ 0:001m0 and we obtain mn¼ ð0:06770:003Þm0 for 0.62 T

rBr1:25 T. Here m0is the rest mass of an electron. The values of

mn

over such a magnetic-field range are reasonable for a GaAs 2D

electron system, and we can see fromFig. 2the fitting of

D

Rxxto

the factor

w

=sinh

w

is good at B ¼0.69 T. Moreover, the fit remains

good for B ¼ 0:6221:25 T. In addition, f0ðBÞlnð4

p

3kBmn=heBÞ

shown in Fig. 3 is linear in 1=B. Hence

D

Rxx follows the SdH

formula for B ¼ 0:6221:25 T.

Fig. 1 shows the magneoresistance measurements Rxx(B) at

continuous microwave radiation as the sample is maintained at T¼1.4 K. With decreasing microwave power attenuation (increasing

Fig. 1. (Color online) The four-terminal resistance measurements Rxx(B) at

different microwave attenuations when the temperature T ¼1.4 K. The inset shows the four-terminal resistance measurements Rxx(B) without microwave at various

temperatures.

Fig. 2. The left (right) panel shows the SdH amplitudes (magetoresistance minima) versus temperature T at B¼ 0.69 T on a semi-logarithmic scale. The solid (open) circles correspond to the data points to determine the fitting curve illustrated by the solid (dash) line. The solid (dashed) horizontal line corresponds to the amplitude (minima) under the continuous microwave radiation of 2 dB power for T¼ 1.4 K. The vertical line indicates the temperature 3 K. Te,minand Te,SdH

indicate the estimated electron effective temperatures determined from the resistance minimum and the amplitudes of the SdH oscillations, respectively. C.-S. Hsu et al. / Solid State Communications 156 (2013) 45–48

MW power), the amplitudes of the SdH oscillations decrease and the resistance values of SdH minima increase, which indicates electron-heating effect. The energy of the microwave is much smaller than the band-gap of GaAs therefore no electron–hole pairs are created. This is evidenced by the fact that the determined carrier density with no microwave is the same as that when the 2DEG is under

continuous microwave radiation[19]. By comparing the SdH

ampli-tudes under microwave irradiation to the T-dependence of

D

Rxx, we

can determine the electron temperature Te,SdH. For example, as

shown inFig. 2, at B¼ 0.69 T we can determine Te,SdH¼3:0 K for the heating under the microwave power 2 dB by comparing the microwave-damped SdH amplitude, indicated by the solid horizon-tal line, to the fitting of Rxxto

w

=sinh

w

.

In Fig. 2, the right panel compares the resistance minimum

under the same microwave heating to the T-dependence of Rxx,min.

We can see fromFig. 2 that the dashed horizontal line, which

denotes the resistance minimum under the microwave heating, is

away from Rxx,min93 Kwhereas the microwave-damped SdH

ampli-tude is close to

D

Rxx9_{3 K}. Here

D

Rxx9_{3 K}and Rxx,min93 K denote the SdH amplitude and resistance minima at T¼ 3 K, respectively. At such a magnetic field, therefore, the electron effective tempera-ture Te,min determined by Rxx,min should be different from Te,SdH

obtained from

D

Rxx. Although

D

E ¼ 0:35 meV revealed by the

slope of the linear fit to ln Rxx,min1=T, which is shown in the

inset toFig. 3is much smaller than the cyclotron gap 1.2 meV, we

can take such a fitting as an empirical curve to obtain Te,min. We

can see from the right panel ofFig. 2that the electron temperature obtained by such an empirical way is 2.6 K and is lower than the value determined by SdH amplitude. Rxx,minalso yields lower Tethan

that obtained from

D

Rxxas we decrease the microwave power.

We now propose a possible physical picture for the difference

in the measured Teusing the two independent methods. The SdH

formalism is semiclassically derived without considering high-field localization which leads to the integer quantum Hall effect. On the other hand, the localization strength oscillates with increasing B and increases quickly near the minimum points of

RxxðBÞ. Although the localization strength can become so weak

that there is no clear cyclotron gap when the SdH formula holds true, there could still be localized effect when such strength

reaches a local maxima at the minima in RxxðBÞ. Therefore, the

localized effect can induce the deviation of Te,minfrom Te,SdH, for which the dominated electrons responsible for SdH amplitudes

are semiclassically conducting. Since

D

E is not of the reasonable

value, our study shows that the localized electrons can be distinguished from the semiclassical conducting one by micro-wave heating even when there is no clear cyclotron gap.

Fig. 4 compares the B-dependence of the effective

tempera-tures determined by Rxx,minand by

D

Rxxunder 2 dB microwave. In

Fig. 4, Te,SdH decreases monotonically from 3.2 K to 2.7 K as B

increases from 0.62 T to 1.25 T whereas Te,minis approximately

B-independent near 2.6 K. We note that all the Landau bands are equivalent under localization such that Landau-level addition

transformation [20–22] works well at high enough B although

such equivalence can become invalid as the localization strength

is weak enough for the validity of SdH formula. Since Rxx(B)

reaches the minima as the Fermi energy just locates at Landau-band tails and the localization strength reaches local maxima

[7,19,23,24], the almost constant Te,minin our study reveals the survival of the equivalence between different localized Landau-band tails when the SdH formula holds true.

4. Conclusion

In conclusion, magnetoresistance measurements are per-formed on a GaAs/AlGaAs two-dimensional electron gas under continuous microwave radiation. Both the SdH amplitudes and magnetoresistance minima are used as electron thermometers to

determine the electron effective temperature Teunder the

micro-wave heating. It is found that Te determined from these two

methods can be substantially different even when there is no clear cyclotron gap, which indicates the difference between electrons at Landau-band tails and those responsible for the semiclassical SdH oscillations. Therefore, localization may occur at Landau-band tails without clear cyclotron gap, under which the localized and semiclassical conducting electrons can be distin-guished by the microwave heating. The effective temperature determined by the resistance minima shows that the tails of different Landau bands are almost equivalent when the validity of the SdH formula indicates the insufficient localization in our system.

Acknowledgment

This work was funded by the NSC, Taiwan, and in part, by National Taiwan University (grant no: 101R7552-2 and 101R890932). References

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