Degeneracy and effective mass in the valence band of two-dimensional (100)-GaAsquantum well systemsVinicio Tarquini, Talbot Knighton, Zhe Wu, Jian Huang, Loren Pfeiffer, and Ken West Citation: Applied Physics Letters 104, 092102 (2014); doi: 10.1063/1.4867086 View online: http://dx.doi.org/10.1063/1.4867086 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electrical and structural characterization of a single Ga Sb ∕ In As ∕ Ga Sb quantum well grown on GaAs usinginterface misfit dislocations J. Appl. Phys. 104, 074901 (2008); 10.1063/1.2982277 High mobility two-dimensional hole system in Ga As ∕ Al Ga As quantum wells grown on (100) GaAs substrates Appl. Phys. Lett. 86, 162106 (2005); 10.1063/1.1900949 High mobility of a three-dimensional hole gas in parabolic quantum wells grown on GaAs(311)A substrates J. Appl. Phys. 97, 076107 (2005); 10.1063/1.1888041 Magnetotransport properties of two-dimensional electron gas in AlSb ∕ InAs quantum well structures designed fordevice applications J. Appl. Phys. 96, 6353 (2004); 10.1063/1.1792385 Structural and transport characterization of AlSb/InAs quantum-well structures grown by molecular-beam epitaxywith two growth interruptions J. Vac. Sci. Technol. B 20, 1174 (2002); 10.1116/1.1468658

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Degeneracy and effective mass in the valence band of two-dimensional(100)-GaAs quantum well systems

Vinicio Tarquini,1,a) Talbot Knighton,1 Zhe Wu,1 Jian Huang,1 Loren Pfeiffer,2 and Ken West21Department of Physics and Astronomy, Wayne State University, 666 W. Hancock, Detroit,Michigan 48201, USA2Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

(Received 27 December 2013; accepted 16 February 2014; published online 4 March 2014)

Quantum Hall measurement of two-dimensional high-mobility [l � 2� 106 cm2=ðV � sÞ] hole

systems confined in a 20 nm wide (100)-GaAs quantum well have been performed for charge

densities between 4 and 5� 1010 cm�2 in a temperature range of 10–160 mK. The Fourier analysis

of the Shubnikov-de Haas (SdH) oscillations of the magnetoresistance vs. the inverse of the

magnetic field 1/B reveals a single peak, indicating a degenerate heavy hole (HH) band. The hole

density p ¼ ðe=hÞ � f agrees with the Hall measurement result within 3%. The HH band degeneracy

is understood through the diminishing spin-orbit interaction due to the low charge density and the

nearly symmetric confinement. SdH oscillations fitted for 0.08 T�B� 0.24 T to the Dingle

parameters yield an effective mass between 0.30 and 0.50 me in good agreement with previous

cyclotron resonance results. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4867086]

The study of two-dimensional hole (2DH) systems

confined in GaAs quantum wells (QWs) is of fundamental

importance for the investigation of strongly correlated

phenomena1–3 as well as for spintronics4 and device applica-

tions. 2DH systems are characterized by a much larger effec-

tive mass (m*) than electrons which leads to an enhanced

interaction effect as shown through the interaction parameter

rs ¼ ðm�e2Þ=ð4p�h2�ÞðppÞ�1=2, the ratio of Coulomb and

Fermi energy.3 However, the determination of exact values

for m* is difficult, especially in the low density, strongly cor-

related regime where splitting and mixing of the bands and

interaction effects yield a complex band structure with non-

parabolic dispersion.5 Also the shape and the relative posi-

tion of the sub-bands depend on parameters that are inherent

to individual samples: depth, width, direction, and symmetry

of the QW. For many years, the high density regime has

been explored through the celebrated quantum Hall measure-

ments.6,7 However, to better understand correlation effects, it

is necessary to investigate more dilute systems. Performing

experiments in a low density regime is challenging due to

the localization effects rising from the disorder. We have

realized a high purity 2DH system that allows us to maintain

very high carrier mobility, l � 2� 106 cm2=ðV � sÞ, while

reaching a low density range: 4.3� 1010 cm�2� p� 4.8

� 1010 cm�2. This makes it possible to analyze the low-field

Shubnikov-de Haas (SdH) oscillations and directly obtain in-

formation on m* and the band structure. The results comple-

ment previous studies of low density m* measured via

cyclotron resonance method in magnetic fields around

�0.5 T.8

The samples used for this study are 20 nm wide p-GaAs

quantum wells grown in the (100) direction. The wells sit

400 nm below the surface with a double-sided carbon dop-

ing. The devices are 3� 1 mm Hallbars defined by photoli-

thography and the Ohmic contacts are made with a thin film

deposition of AuBe (1%) annealed at 470 �C. The contact re-

sistance for all leads is consistently �400 X at 10 mK. The

measurement is performed in a dilution refrigerator

where the sample is thermally anchored to a cold finger. The

magnetoresistance (qxx) and the Hall resistance (qxy) are

obtained with a four-probe AC lock-in technique with a

low-frequency (�13 Hz) excitation � 10 nA. The mobility is

l¼ 2.0� 106 cm2/(V � s) for a density p¼ 4.3� 1010 cm�2

for sample A and l¼ 1.9� 106 cm2/(V � s) for p¼ 4.8

� 1010 cm�2 for sample B. l and p are calculated using

l ¼ 1=ðqxxpeÞ and p ¼ @Rxy=@B� ��1jej.

Figure 1 shows the SdH oscillations for both samples at

various T between 10 and 160 mK. The oscillations are well

defined even in the low field starting from 0.06 T, indicating

the high quality of both samples. SdH oscillations are peri-

odic vs. 1/B with a period of D 1=Bð Þ ¼ e=h � gs=pð Þ, where

FIG. 1. T-dependence of SdH oscillations for samples A and B, respectively,

with p¼ 4.3� 1010 cm�2 and p¼ 4.8� 1010 cm�2.a)Electronic address: vinicio.tarquini@wayne.edu

0003-6951/2014/104(9)/092102/3/$30.00 VC 2014 AIP Publishing LLC104, 092102-1

APPLIED PHYSICS LETTERS 104, 092102 (2014)

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gs being the spin degeneracy. Fig. 2, where the longitudinal

resistance qxx is plotted against 1/B, shows a distinctive peri-

odic pattern between 0.06 T and 0.3 T. The Fourier analysis

is performed for each curve in the region where B< 0.11 T,

field beyond which the Zeeman splitting becomes observ-

able. The spectra are shown in the insets of Fig. 2. The spec-

tra reveal single peaks located at f¼ 0.94 T and f¼ 1.03 T,

respectively, for samples A and B. To interpret this result, it

is important to discuss the structure of the valence band. It is

well known that in GaAs/AlGaAs heterostructures the lack

of inversion symmetry of the underlying zinc-blende struc-

ture of the GaAs gives rise to Spin-Orbit interaction (Bulk

Inversion Asymmetry, BIA) so that the holes populate the

higher valence band with total angular momentum j¼ 3/2.9

As discussed in the previous theoretical10 and experimental

papers,11–13 the confinement potential unfolds the degenerate

j¼ 3/2 band into Heavy hole (HH) and Light hole (LH) sub-

bands, respectively, with z component of the total angular

momentum jz¼63/2 and jz¼61/2; the terms heavy and

light refer to values of m*. The interaction between the spin

of the carriers and electric field created by the asymmetry of

the confinement potential further lifts the spin degeneracy

and creates the HHh (heavier HH) and HHl (lighter)

sub-bands. This effect, called structural inversion asymmetry

(SIA), increases with increasing hole density and the asym-

metry of the confinement potential.9,14 The extensive study

of heterostructures and quantum wells grown on (311)-GaAs

demonstrated that, because of the lack of symmetry, the SIA

effect is strong enough to produce an observable HHh-HHl

splitting for p � 7� 1010 cm�2.15 The lift of the degeneracy

is deduced from the Fourier spectra of qxx�1/B which shows

four distinct peaks: the two main frequencies fþ, f–, and their

average and sum, fave, fsum. fþ and f– correspond to the den-

sities pþ and p– that are the populations of the HHh and

HHl.12 The SOI-induced spin splitting is reduced if the con-

finement potential becomes more symmetric as confirmed by

the previous studies16 of 2DHS in nearly symmetric (100)

QWs, reporting an onset of the splitting at much higher

carrier densities �2� 1011 cm�2. In the density range that

we are exploring, a degenerate scenario is expected. This

anticipation is confirmed by the Fourier analysis, as shown

in Fig. 2. The characteristic frequency f for each curve corre-

sponds to a carrier density through the formula p¼ (gse/h)f.Consistently for both samples, the densities calculated,

4.5� 1010 cm�2 and 4.9� 1010 cm�2, agree very well to the

ones measured through the Quantum Hall method assuming

gs¼ 2. So, the HH band is still two-fold degenerate, indicat-

ing that the nearly symmetric confinement potential in this

low p range, provided through symmetrical double-sided

doping, further weakens the effects of the SOI. This result

addresses the lack of documentation for the (100) case for

the low p QW systems which was technologically challeng-

ing due to the sample quality. These SdH results also com-

plements well the cyclotron method, which was used to

measure m* for dilute 2DHs, in terms of providing both m*

and the band information.

m* can be obtained by studying the T-dependence of the

SdH oscillations. In Fig. 1, the T-dependence is clear even at

small fields and the systems have a metallic behavior at zero

field, qxx � 70 X/�. Ando’s formula defines the relation

between the variation of qxx and T (Ref. 17)

qxxðBÞ ¼ qxxð0Þ 1� 4 cosEF

�hxc

� �� Dðm�; TÞ � Eðm�; sqÞ

� �;

(1)

where qxx(0) is the longitudinal resistivity at zero field;

E(m*, sq)—an exponential term that depends on the quantum

scattering time sq and the cyclotron frequency xc¼ eB/m*,

Eðm�; sqÞ ¼ expð�p=xcsq); and D(m*, T)—the Dingle fac-

tor that in the low magnetic field limit D(m*, T)¼ n/sinh nwith n ¼ 2p2kBT=�hxc.18 In the temperature region that we

are exploring, it is possible to make the approximation for

which lnðsinh nÞ � n, allowing simplification to Eq. (1).

Now, m* can be extrapolated from the linear relation

between the logarithm of the ratio of low field oscillation

amplitude and T itself

lnDqxx

T

� �¼ C� 2p2kB

�heBm�T: (2)

The data are fitted using Eq. (1) with an R2 0.995. m*

is calculated from the slope of the lines in Fig. 3 for each

value B. We note that the calculation is performed by using

the approximation to Eq. (1) for T from 25 mK up to 200 mK

beyond which the SdH oscillation are smeared by the ther-

mal energy. Fig. 3 shows that the linear relationship in

Eq. (2) have slightly different slopes at different magnetic

fields, demonstrating that the strength of the field affects the

value of m* as shown in Fig. 4. The causes of this effect, al-

ready observed previously,12,19 arise from the complex rela-

tion between B, kinetic energy, and many body interaction in

the HH band that we do not intend to address in this work.

The values of m* for B> 0.107 T are also affected by the

Zeeman effect, which can be better studied in parallel fields.

For sample A, we found that m* varies from a value of

0.30 me at B¼ 0.076 T to 0.47 me at B¼ 0.250 T and for

sample B the mass ranges from 0.31 to 0.50 me. The

FIG. 2. Observed SdH oscillations for the two samples, A and B, in the

range between 8.5 T�1 and 15 T�1. The insets shows the Fourier spectra of

the oscillations, the dominant peaks, and their frequencies values.

092102-2 Tarquini et al. Appl. Phys. Lett. 104, 092102 (2014)

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observation of a single set of m* for each sample confirms

that the HH band is still degenerate in our range of densities.

It is evident that m* in sample A (lower density) is con-

stantly smaller than the one measured for sample B. The

increase of m* with increasing density has already been

observed using the cyclotron resonance method and our

results are in good agreement with the previous findings8 in

the inset. The dependence of m* on p can be explained by

considering that in our range of densities the e – e repulsion

(Hartree potential) does not contribute much to the

Hamiltonian. So we can safely say that the change in density

only shifts the position of the Fermi energy of the 2DHS,

while the magnitude of the spin-orbit splitting stays nearly

constant. For in-plane motion, the HH band presents a

smaller m* than LH band. This demonstrates that the density

decrease causes the Fermi energy to move away from the

anti-crossing point, consequently making m* decrease.8

These findings agree with the m* values calculated in earlier

papers.5,8

In summary, the Shubnikov-de Haas oscillations of two

dilute 2DHS confined in 20 nm wide (100)-GaAs quantum

wells have been observed. Through the analysis of the SdH

Fourier spectrum has been found that the topmost valence

band (HH) of a 2DHS with p¼ 4.3–4.8� 1010 cm�2 is still

degenerate (to our measurement resolution). This finding

leads to the conclusion that in such conditions, the SOI are

not strong enough to induce an observable band splitting.

Furthermore, m* has been measured for both systems by

studying the T-dependence of the SdH oscillations. It

has been found to slightly increase with increasing B and pand its value ranges between 0.30–0.50 me for 0.08�B� 0.250 T. These results are essential for the investigation of

strongly interacting systems where inter-particle Coulomb

energy becomes increasingly important.

We acknowledge the support of this work from NSF

under No. DMR-1105183. The work at Princeton was par-

tially funded by the Gordon and Betty Moore Foundation

through Grant No. GBMF2719 and by the National Science

Foundation No. MRSEC-DMR-0819860 at the Princeton

Center for Complex Materials.

1A. Isihara, Solid State Phys. 42, 271 (1989).2B. Spivak, S. V. Kravchenko, S. A. Kivelson, and X. P. A. Gao, Rev.

Mod. Phys. 82, 1743 (2010).3J. Huang, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 85, 041304(R)

(2012).4S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von

Moln�ar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science

294, 1488 (2001).5H. Zhu, K. Lai, D. C. Tsui, S. P. Bayrakci, N. P. Ong, M. Manfra, L.

Pfeiffer, and K. West, Solid State Commun. 141, 510 (2007).6K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).7R. Willett, J. P. Eisenstein, H. L. St€ormer, D. C. Tsui, A. C. Gossard, and

J. H. English, Phys. Rev. Lett. 59, 1776 (1987).8T. M. Lu, Z. F. Li, D. C. Tsui, M. J. Manfra, L. N. Pfeiffer, and K. W.

West, Appl. Phys. Lett. 92, 012109 (2008).9R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electronand Hole Systems (Springer, Berlin, 2003), Vol. 191.

10U. Ekenberg and M. Altarelli, Phys. Rev. B 32, 3712 (1985).11H. Stormer, Z. Schlesinger, A. Chang, D. C. Tsui, A. Gossard, and W.

Wiegmann, Phys. Rev. Lett. 51, 126 (1983).12B. Habib, M. Shayegan, and R. Winkler, Semicond. Sci. Technol. 24,

064002 (2009).13Y. Yaish, O. Prus, E. Buchstab, S. Shapira, G. B. Yoseph, U. Sivan, and

A. Stern, Phys. Rev. Lett. 84, 4954 (2000).14J. Lu, J. Yau, S. Shukla, M. Shayegan, L. Wissinger, U. R€ossler, and R.

Winkler, Phys. Rev. Lett. 81, 1282 (1998).15Y. Chiu, M. Padmanabhan, T. Gokmen, J. Shabani, E. Tutuc, M.

Shayegan, and R. Winkler, Phys. Rev. B 84, 155459 (2011).16Z. Q. Yuan, R. R. Du, M. J. Manfra, L. N. Pfeiffer, and K. W. West, Appl.

Phys. Lett. 94, 052103 (2009).17T. Ando and Y. J. Uemura, Phys. Soc. 36, 959 (1974).18R. Dingle, Proc. R. Soc. London, Ser. A 211, 517 (1952).19J. P. Eisenstein, H. L. Stormer, V. Narayanamurti, A. C. Gossard, and W.

Wiegmann, Phys. Rev. Lett. 53, 2579 (1984).

FIG. 3. Variation of the logarithm of the longitudinal resistance with the

temperature and its fit with the Dingle factor for different magnetic fields

for, respectively, sample A (a) and sample B (b).

FIG. 4. In the graph, the values of m* in units of me in the range

0.080 T�B� 0.236 T for both samples are reported. In the inset, our results

are compared to the previous data obtained through cyclotron resonance

measurements,8 using the m* values corresponding to B¼ 0.25 T, that is the

closest to the field applied in the cited experiment.

092102-3 Tarquini et al. Appl. Phys. Lett. 104, 092102 (2014)

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