Anisotropic magnetic properties of excitons in GaAs multiple quantum wells

Superlattices and Microstructures 137 (2020) 106332

Contents lists available at ScienceDirect

Superlattices and Microstructures journal homepage: www.elsevier.com/locate/yspmi

Anisotropic magnetic properties of excitons in GaAs multiple quantum wells S. Haldar a,b , A. Banerjee c , Kranti Kumar c , R. Kumar a , Geetanjali Vashisht a,b , T.K. Sharma a,b , V.K. Dixit a,b ,∗ a

Semiconductor Materials Laboratory, Materials Science Section, Raja Ramanna Centre for Advanced Technology, Indore-452013, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai-400094, India c UGC-DAE Consortium for Scientific Research, Khandwa Road, Indore-452001, India b

ARTICLE

INFO

Keywords: Magnetic-susceptibility Quantum well Excitons Anisotropic magnetic-property

ABSTRACT Anisotropic magnetic-behavior of charge carriers in a GaAs/AlGaAs quantum well is experimentally investigated under a magnetic field, perpendicular and parallel to the quantum well plane. A van-Vleck type paramagnetism is observed in the measured magnetic-susceptibility under a very-low magnetic field regime (𝐵 < 0.1 T). However, with an increasing magnetic field, diamagnetic-contribution dominates over the paramagnetic counterpart and carriers in the quantum well exhibit a net-diamagnetic behavior. The magnetic-susceptibility of carriers at 2 and 300 K under a perpendicular magnetic field configuration is also explained by considering Coulomb interaction in a variational approach. On the contrary, paramagnetic and diamagneticsusceptibility of carriers are found to be suppressed under a magnetic field that is parallel to the quantum well plane. The suppression of magnetic-properties under the latter configuration is realized by the coupling of diamagnetic energy with the quantum confinement energy due to barrier layers. Results obtained in the present study would be beneficial in controlling the magnetic-behavior of charge carriers; especially, when the magnetic contributions determine the spin/charge transport efficiency in advanced optoelectronic devices.

1. Introduction Optical and transport properties of charge carriers in III–V semiconductors have been extensively investigated over the past few decades. [1–4] In particular, the excellent electro-optical properties of an ultra-low disordered quantum structure possess the key aspects in realizing highly-efficient optical emitters and nearly dissipation-less charge transportation. [5–8] In addition to this, the magnetic behavior of charge carriers in a low-dimensional system is equally important, which determines the performance of an advanced optoelectronic device, e.g., spintronics and valleytronics applications. [9–11] In view of this, a considerable amount of research has already been performed to investigate magnetization (M) and magnetic-susceptibility (𝜒) of GaAs-based materials. [9,12–16] In these previous reports, diamagnetic and paramagnetic susceptibility of bulk-GaAs was estimated with a variation in temperature (2–300 K) and doping concentration (1015 –1018 cm−3 ). [9,16] Also, it was theoretically understood that the magnetic behavior of carriers can be controlled by confining charge carriers in a low-dimensional system [17–23] or by applying external pressure. [24–26] Nonetheless, an experimental investigation on the anisotropic magnetic properties of charge carriers in a quantum structure and its correlation with previous theoretical reports is yet to be performed.

∗ Corresponding author at: Semiconductor Materials Laboratory, Materials Science Section, Raja Ramanna Centre for Advanced Technology, Indore-452013, India. E-mail addresses: [email protected] (S. Haldar), [email protected] (V.K. Dixit).

https://doi.org/10.1016/j.spmi.2019.106332 Received 29 August 2019; Received in revised form 23 October 2019; Accepted 31 October 2019 Available online 2 November 2019 0749-6036/© 2019 Elsevier Ltd. All rights reserved.

Superlattices and Microstructures 137 (2020) 106332

S. Haldar et al.

Fig. 1. (a) Simulated result of HRXRD data, (b) cross-sectional TEM image and (c) the results of ECV measurements showing the carrier density with the depth of the 50-period GaAs/AlGaAs QW sample.

With this in mind, magnetic properties of charge carriers in a GaAs multiple-quantum well (QW) structure are probed under a magnetic field, perpendicular or parallel to the QW plane. Paramagnetic and diamagnetic contributions in the measured magneticsusceptibility (𝜒) up to 7 T are observed at two different temperatures. It is found that Landau-diamagnetic susceptibility is not sufficient to describe the experimentally estimated 𝜒, as a function of the magnetic field. However, the magnetic-susceptibility calculated by considering Coulomb interaction between electron–hole pairs and spin splitting of energy state is found to be in good agreement with the experimental observations. Furthermore, the anisotropic-magnetic properties of carriers in a QW are estimated by a directional dependent magnetization measurement. The magnitude of diamagnetic and paramagnetic susceptibilities are found to be suppressed under a magnetic field that is parallel to the QW plane. Such an observation is explained by a coupling of diamagnetic energy with the quantum confinement energy due to barrier layers. 2. Experimental details In order to acquire a large magnetization signal from QWs, a 50-period GaAs/Al0.27 Ga0.63 As multiple-QW sample is grown by metal–organic vapor phase epitaxy (MOVPE) technique. Here, the thickness of the QW and barrier layers is kept approximately 75 and 210 Å, respectively. Details of the sample growth in the MOVPE reactor chamber may be found in Refs. [27]. Alloy composition and layer-thicknesses of the grown sample are estimated by high-resolution X-ray diffraction (HRXRD) measurements and crosssectional transmission electron microscopy (Figs. 1a and 1b). Silicon atoms are doped in QW layers by SiH4 precursor, with a typical carrier density 𝑛 ∼ 1018 cm−3 . This is estimated by classical Hall measurements in van der Pauw geometry (B ≤ 0.3 T). Also, electrochemical capacitance–voltage (ECV) measurements are performed to deduce the carrier distribution in the multi-QW structure. [28] The result obtained by ECV measurements is plotted in Fig. 1c. Magnetization measurements under a magnetic field, perpendicular or parallel to the QW plane, are performed in Quantum Design: superconducting quantum interference device (SQUID). [29] Measurements are performed at 300 and 2 K temperatures under the magnetic field up to 7 T. Sample in the experimental arrangement is mounted by a transparent non-magnetic straw, where additional small pieces of straw are used to hold the sample in the desired plane. [9] To minimize the magnetization signal from the GaAs substrate (300 μm), the thickness of the sample is reduced to ∼ 80 μm by back-side lapping and polishing methods. It is noteworthy to mention that the accumulated magnetization signal from the 50-QWs at 7 T (𝑀𝑀𝑄𝑊 𝑠 ∼ 3 × 10−6 EMU) is more than two orders of magnitude higher than the sensitivity of the SQUID magnetometer (∼ 10−8 EMU). Under this condition, anisotropic magnetic behavior of carriers can be probed by SQUID-based measurements. Here, the magnetic moment contributed by the QW regions of the MQW sample is estimated by, 𝑀𝑀𝑄𝑊 𝑠 = 𝜒𝐺𝑎𝐴𝑠 𝐻𝜌𝐺𝑎𝐴𝑠 × 𝑉𝑀𝑄𝑊 𝑠 ∕𝜇𝐺𝑎𝐴𝑠 , where magnetic susceptibility 𝜒𝐺𝑎𝐴𝑠 ≈ 3 × 10−5 cm3 /mole, density 𝜌𝐺𝑎𝐴𝑠 =5.32 gm/cm3 , volume of the QWs 𝑉𝑀𝑄𝑊 𝑠 = (50 × effective QW width × sample area), and 𝜇𝐺𝑎𝐴𝑠 = 144.6 g/mole It is also ensured that the magnetic contribution from the straw (used for holding the sample) is feeble as compared to the magnetization of the MQW sample (see the supplementary note). 3. Theoretical background Anisotropic magnetic properties of charge carriers in a QW and its temperature-dependent variation are in-general understood by defining a partition function with the help of the energy eigen value. In order to estimate the energy of charge carriers, Schrödinger equation of the above system can be solved by using a magnetic vector potential A(r) = − 12 [r × B]. [

(p𝐞 + 𝑒A(r))2 (p𝐡 − 𝑒A(r))2 𝑒2 + − + 𝑒𝑉𝑧 ∗ ∗ 2𝑚𝑒 2𝑚ℎ 4𝜋𝜖0 𝜖𝑟 |𝑟𝑏 |

] 𝜓(𝑟) = 𝐸𝑥,𝑦,𝑧 𝜓(𝑟)

(1)

Here, r and p represent the position co-ordinate and momentum of charge carriers in a relative frame of reference. The confinement potential due to barrier layer is denoted by 𝑉𝑧 and the separation of electron–hole pairs (i.e., exciton Bohr radius) is symbolized by 2

Superlattices and Microstructures 137 (2020) 106332

S. Haldar et al.

𝑟𝑏 =

√

𝑟2𝑥 + 𝑟2𝑦 + 𝑟2𝑧 . The contribution of center of mass (CM) motion and relative motion (RM) of charge carriers in Eq. (1) can be

understood by canonical transformations, which are given by, [30] ( [ ] ) 𝑖 𝜓(𝑅, 𝑟) = exp P − 𝑒A(r) R 𝛷(𝑟) ℏ

(2a)

R = (𝑚∗𝑒 r𝐞 + 𝑚∗ℎ r𝐡 )∕(𝑚∗𝑒 + 𝑚∗ℎ )

(2b)

P = (𝐩𝐞 − 𝐩𝐡 )

(2c)

where, R and 𝐏 symbolize the position coordinate and momentum of electron-hole as a quasi-particle (i.e., with respect to the CM frame). Using the above transformations, Eq. (1) can be expressed by, [31] [ ] ) ( 𝑝2 𝑒2 𝐴(𝑟)2 1 2𝑒 𝑃2 𝑒2 1 − A(r) ⋅ p − + + − 𝑒 A(r) ⋅ 𝐏 − + 𝑒𝑉 𝛷(𝑟) = 𝐸𝛷(𝑟) (3) 𝑧 2𝑀 ∗ 2𝜇 ∗ 2𝜇 ∗ 𝑚∗𝑒 𝑚∗ℎ 𝑀∗ 4𝜋𝜖0 𝜖𝑟 |𝑟𝑏 | Here, the reduced mass and total effective mass of electron–hole pairs are denoted by 𝜇 ∗ = 𝑚∗𝑒 𝑚∗ℎ ∕(𝑚∗𝑒 + 𝑚∗ℎ ) and 𝑀 ∗ = (𝑚∗𝑒 + 𝑚∗ℎ ), respectively. 3.1. Perpendicular magnetic field configuration Under a magnetic field perpendicular to the QW plane, charge carriers in a QW are confined in x–y plane with 𝑟𝑥 = 𝑟𝑦 ≠ 𝑟𝑧 . Therefore, a cylindrical symmetry can be used to describe the in-plane contribution of Eq. (3) by using symmetric vector potential A(r) = 1/2[-yB𝑧 , xB𝑧 , 0]. Considering the contribution of spin splitting, Schrödinger equation of carriers is expressed in cylindrical co-ordinate as follows, ] [ ( 2 ) ( ) 𝑒2 𝐵𝑧2 𝜌2 𝑒𝐵𝑧 𝑒𝐵𝑧 𝜕 1 𝜕 𝛬𝑒2 𝑃2 ℏ2 − ∗ + + + (𝐿 + 𝑔 𝑠 ) − − (−𝑦𝑃 + 𝑥𝑃 ) 𝛷(𝜌) = 𝐸 − 𝛷(𝜌) (4) 𝑧 ⟂ 𝑧 𝑥 𝑦 𝑥,𝑦 ∗ ∗ 2𝜇𝑥,𝑦 𝜕𝜌2 𝜌 𝜕𝜌 8𝜇𝑥,𝑦 2𝜇𝑥,𝑦 4𝜋𝜖0 𝜖𝑟 |𝜌𝑏 | 𝑀 ∗ 2𝑀 ∗ Here, |𝜌| = (𝑥2 + 𝑦2 )1∕2 stands for the cyclotron radius in the x-y plane, 𝑔⟂ is the lande-g factor of electron–hole under 𝐵𝑧 , and 𝐿𝑧 /𝑠𝑧 denotes the orbital/spin angular momentum of charge carriers. In Eq. (4), the in-plane component of spherically symmetric Coulomb interaction is extracted by introducing a variational parameter 𝛬. Note that, under a magnetic field perpendicular to the QW plane, the contribution of CM motion in Eq. (4) vanishes, i.e., 𝑦𝑃𝑥 = 𝑥𝑃𝑦 , which is because of the impact of the magnetic field along x and 𝑦-directions is same for this magnetic field orientation. Under this condition, Eq. (4) can be written as: [ ( 2 ) ] 𝑒2 𝐵𝑧2 𝜌2 𝑒𝐵𝑧 ℏ2 𝜕 𝛬𝑒2 1 𝜕 + + (𝐿 + 𝑔 𝑠 ) − 𝜙(𝜌) = 𝐸𝑥,𝑦 𝜙(𝜌) (5) + 𝑧 ⟂ 𝑧 ∗ ∗ ∗ 2𝜇𝑥,𝑦 8𝜇𝑥,𝑦 2𝜇𝑥,𝑦 4𝜋𝜖0 𝜖𝑟 |𝜌𝑏 | 𝜕𝜌2 𝜌 𝜕𝜌 Now, the energy eigen value of charge carriers in a QW can be easily estimated by solving Eq. (5), which is given by, 𝐸𝑥,𝑦 (𝐵𝑧 ) ≃

𝑒2 𝐵𝑧2 ⟨𝜌⟩2 ∗ 2𝜇𝑥,𝑦

±

1 𝛬𝑒2 𝑔 𝜇 𝐵 − 2 ⟂ 𝐵 𝑧 4𝜋𝜖0 𝜖𝑟 |𝜌𝑏 |

(6)

Here, the Coulomb separation parameter 𝛬 is estimated by a variation method using a trial wave function 𝛷(𝜌) ∼ exp(−2 𝛬𝜌∕𝛼1 ) + 𝑐 exp(−𝜌2 ∕4𝛼22 ) with the help of following relations, [32] ⟨𝛷(𝜌, 𝑧)|𝐻(𝛬)|𝛷(𝜌, 𝑧)⟩ = 0 𝜕 ⟨𝛷(𝜌)|𝐻(𝛬)|𝛷(𝜌)⟩ = 0 𝜕𝛼𝑖

(7a) (7b)

where, 𝛼𝑖 is a variational parameter in wave function and 𝐻(𝛬) is the energy Hamiltonian of the system. Thereafter, the temperature dependent magnetic-properties of carriers can be realized by defining a partition function: ( ) [ 2 2 2 ] ∑ 𝑔⟂ ℏ𝑒𝐵𝑧 𝑒 𝐵 ⟨𝜌⟩ 𝛬𝑒2 𝑍(𝑇 , 𝐵𝑧 ) = exp(−𝐸∕𝑘𝐵 𝑇 ) ≃ cosh exp − + (8) ∗ 𝑘 𝑇 ∗ 𝑘 𝑇 4𝜇𝑥,𝑦 2𝜇𝑥,𝑦 4𝜋𝜖0 𝜖𝑟 |𝜌𝑏 |𝑘𝐵 𝑇 𝐵 𝐵 Notably, the kinetic and potential energy terms related to the 𝑧-direction of motion remain invariant under 𝐵𝑧 , and therefore not-considered in Eq. (8). Thereafter, the magnetic-susceptibility of valence/conduction band charge carriers can be calculated by 𝜒𝑣,⟂ = 𝑀∕𝐵𝑧 = 𝑘𝐵 𝑇 ∕𝐵𝑧 × 𝜕 ln 𝑍∕𝜕𝐵𝑧 . 3.2. Parallel magnetic field configuration On the contrary, Schrödinger equation of charge carriers in a QW cannot be described by cylindrical symmetry under 𝐵𝑦 . Using symmetric vector potential A(r)=1/2[𝑧𝐵𝑦 , 0, −𝑥𝐵𝑦 ], Eq. (3) for this configuration can be expressed by, [ ] ) 𝑒2 𝐵𝑦2 ( 𝑧2 𝑒𝐵𝑦 ℏ2 𝜕 2 ℏ2 𝜕 2 𝑒2 𝑥2 − ∗ − ∗ + + − − (𝑧𝑃 − 𝑥𝑃 ) + 𝑒𝑉 𝛷(𝑥, 𝑦, 𝑧) (9) 𝑥 𝑧 𝑧 ∗ 2𝜇𝑧 𝜕𝑧2 2𝜇𝑥,𝑦 𝜕𝑥2 8 𝜇𝑥,𝑦 𝜇𝑧∗ 4𝜋𝜖0 𝜖𝑟 |𝑟𝑏 | 𝑀 ∗ ( ) 𝑝2𝑦 𝑃2 𝛷(𝑥, 𝑦, 𝑥) = 𝐸𝑥,𝑦,𝑧 − ∗ − 2𝜇𝑥,𝑦 2𝑀 ∗ 3

Superlattices and Microstructures 137 (2020) 106332

S. Haldar et al.

Fig. 2. (a) Magnetic susceptibility of GaAs/AlGaAs multi-QWs which is measured by SQUID magnetometer at 300 and 2 K under perpendicular magnetic field configuration, 𝜒⟂ . (b) Theoretically estimated magnetic susceptibility of valence/conduction band carriers (by Eqs. (4)) at two temperatures is shown.

Here, CM motion of electron–hole pairs along 𝑧-direction is forbidden due to the quantum confinement by barrier potential, i.e. 𝛥𝑃𝑧 (𝐵𝑦 ) ≈ 0. However, the contribution of 𝑧𝑃𝑥 (center of mass motion in the 𝑥-direction) can be altered with an increasing 𝐵𝑦 , i.e., 𝛥(𝑧𝑃𝑥 ) ≠ 0. Note that the magnetic confinement of charge carriers along 𝑧-direction competes with the quantum confinement due to barrier layers, because of the carriers in QW are already confined along the growth direction. Under this condition, an √ analytical solution of Eq. (9) is possible, when 𝑥 = 8𝑧𝑃𝑥 𝜇𝑧∗ ∕𝑒𝐵𝑦 𝑀 ∗ . The above transformation indicates that impact of a magnetic field along the 𝑧-direction is now utilized for relative motion of electron and hole in 𝑥-direction. Considering a parabolic potential due to barrier layer, 𝑒𝑉𝑧 = 12 𝜇𝑧∗ 𝜔2𝑧 𝑧2 , energy eigen value can be estimated by solving Eq. (9), 𝐸𝑧 (𝐵𝑦 ) ≈

1 2

( 𝑒2 𝐵 2 ℏ2 𝑦

∗ 𝜇∗ 4𝜇𝑥,𝑦 𝑧

+ 𝐸02

)1∕2 ±

1 𝛬∗ 𝑒2 𝑔∥ 𝜇𝐵 𝐵𝑦 − √ 2 4𝜋𝜖0 𝜖𝑟 𝑥2 + 𝑧2

(10)

Also, using Eq. (5), parallel magnetic field driven cyclotron-radius of carriers along the growth direction of the QW is derived as, [ ] 𝑝2𝑧 2ℏ 𝑟𝑧 (𝐵𝑦 )2 = ≃ (11) ∗ ∗ 2 2 ∗ 𝜇 ∗ + 4𝐸 ∕ℏ2 )1∕2 (𝜇𝑧 𝜔𝑐 ) 𝜇𝑧∗ (𝑒2 𝐵𝑦2 ∕𝜇𝑥,𝑦 𝑧 0 ∗ + 𝐸 2 ∕ℏ2 )1∕2 represents the cyclotron frequency of carriers under a magnetic field 𝐵 . Thereafter, the Here, 𝜔𝑐 (𝐵𝑦 ) = (𝑒2 𝐵𝑦2 ∕4𝜇𝑧∗ 𝜇𝑥,𝑦 𝑦 0 magnetic-susceptibility under parallel field (𝜒𝑣,∥ ) can be estimated by using Eq. (10) with the help of a partition function method as described in Section 3.1. It is noteworthy to mention that the contribution of Coulomb interaction on the magnetic properties of charge carriers would be feeble in this configuration when the electron–hole pairs are confined in the 𝑧-direction and separated along the 𝑥-direction, 𝛥[𝑥(𝐵𝑦 )2 + 𝑧(𝐵𝑦 )2 ] ∼ 0 (see the last term of Eq. (10)).

4. Results and discussion Fig. 2 shows the experimentally estimated magnetic susceptibility of charge carriers at 2 and 300 K, which is estimated by magnetization measurements under a perpendicular field configuration (𝜒⟂ = 𝑀∕𝐵𝑧 ) [33]. The inset of the figure illustrates the sample orientation with respect to the magnetic field in the SQUID magnetometer. Observation shows that the measured 𝜒⟂ exhibits a paramagnetic behavior (𝜒 𝑝𝑎𝑟𝑎 ) under 𝐵𝑧 ≲ 0.1 T. Moreover, the magnitude of 𝜒 𝑝𝑎𝑟𝑎 is found to be suppressed during a lowtemperature measurement, as highlighted by 𝛥𝜒 𝑝𝑎𝑟𝑎 in Fig. 2a. The paramagnetic behavior observed in the low-field regime is mainly contributed by a virtual magnetic-dipole-moment between the carriers in conduction and valence bands (i.e., van Vleck paramagnetism 𝜒𝑉𝑝𝑎𝑟𝑎 ). [9,12] 𝜒𝑉𝑝𝑎𝑟𝑎 =

∑ 𝑛(𝑇 )𝑒2 𝑎 ⟨𝑠𝑗 |𝐼𝑖𝑗 |𝑏𝑖 ⟩2 = 𝑛(𝑇 ) 𝐸𝑔 (𝑇 ) 𝑎 − 𝐸𝑏 )

(12)

2𝑚∗2 (𝐸

Here, 𝐸𝑔 (𝑇 ) = 𝐸𝑎 − 𝐸𝑏 stands for the band-gap of the QW, n(T) is the carrier density in bands and 𝐼𝑖𝑗 is the angular momentum operator for the carriers in conduction/valence band. In general, the band-gap of a GaAs QW monotonically decreases with a rise in temperature. This temperature dependent band-gap variation is often described by a semi-empirical Varshini relation, 𝐸𝑔 (𝑇 ) ≃ 𝐸𝑔 (0) − 𝛼𝑇 2 ∕(𝛽 + 𝑇 ), [34,35] where 𝛼 and 𝛽 are related to the entropy and Debye temperature. Also, the carrier density in a QW increases at a high temperature due to the thermal activation of donor atoms. Under an equilibrium condition, two-dimensional electron density in the conduction band is expressed by, 𝑛(𝑇 ) = 𝑚∗𝑒 𝑘𝐵 𝑇 ∕𝜋ℏ2 × ln[1 + exp{(𝐸𝑓 − 𝐸𝑐 )∕𝑘𝐵 𝑇 }]. Here, 𝑚∗𝑒 represents the effective mass of electrons and 𝐸𝑓 is the Fermi energy. Therefore, according to Eq. (12), an enhancement in carrier density and a decrease in band-gap at a high temperature can be responsible for a greater 𝜒𝑉𝑝𝑎𝑟𝑎 at 300 K (𝛥𝜒 𝑝𝑎𝑟𝑎 , Fig. 2a). Another factor that may affect the van-Vleck paramagnetism is the temperature dependent effective mass of carriers, which generally decreases with a rise in temperature. [36,37]. Furthermore, a temperature-dependent band gap variation may deviate from the usual Varshini’s relation 4

Superlattices and Microstructures 137 (2020) 106332

S. Haldar et al.

Fig. 3. (a) Experimentally estimated magnetic susceptibility of the GaAs multi-QWs under two different magnetic field orientations (i.e., 𝜒∥ and 𝜒⟂ ) at 2 K. (b) The magnetic susceptibility of charge carriers is theoretically computed with the help of Eqs. (3)–(5) for the two magnetic field configurations.

due to the localization of carriers at a low-temperature (T ≲ 50 K). [33,38,39]. However, a correction on the temperature-dependent 𝜒𝑉𝑝𝑎𝑟𝑎 due to this anomalous band gap variation in a low temperature regime can be feeble for a high-quality GaAs QW. It is observed that the diamagnetic-susceptibility of charge carriers significantly increases with a rise in magnetic field, which becomes dominant over the paramagnetic counterpart under 𝐵𝑧 ≥ 0.1 T (Fig. 2a). Also, the carriers in QWs are found to exhibit a higher diamagnetism at 300 K than the 2 K measurements. The experimentally estimated 𝜒⟂ as a function of the magnetic field (Fig. 2a) cannot be explained by the previously described Landau-diamagnetic susceptibility of carriers. This is because, according to a simple theoretical expression |𝜒 𝑑𝑖𝑎 | = 𝑒2 𝜌2 ∕6𝑚∗ 𝜖0 𝜖𝑟 , [17,18,33] diamagnetic susceptibility should decrease with the magnetic field driven confinement of charge carriers, which is inconsistent with our experimental results Fig. 2a. The disagreement between experimental and theoretical results might be explained by considering Coulomb interaction between electrons and positive charge centers or holes (contributed by acceptor-type carbon impurities). Now, using Eq. (8), magnetic-susceptibility of conduction and valence band charge carriers is numerically estimated which is then plotted in Fig. 2(b). In this √ analysis, lande-g factor of electron– hole is considered to be ≈ −0.1, [40,41] cyclotron radius in high field regime is ⟨𝜌⟩ = ℏ∕𝑒𝐵, [31] and numerical values of 𝛬 are estimated by a variation method (described in Section 3.1). [32] Theoretically calculated magnetic susceptibility (Fig. 2b) shows that carriers in the QW exhibit paramagnetic behavior in low field regime, which is contributed by Coulomb interaction and spin splitting of energy state. However, with an increasing magnetic field, diamagnetic contribution becomes dominant over the paramagnetic susceptibility and 𝜒𝑣,⟂ exhibits a net-diamagnetic behavior. The rate of change of 𝜒𝑣,⟂ as a function of 𝐵 is found to be feeble under 𝐵𝑧 ≥ 3.5 T. The calculated result in Fig. 2b also shows that the diamagnetic susceptibility is slightly increased at a high temperature, which explains the experimentally observed higher diamagnetism at 300 K (see Fig. 2a). In addition to this, diamagnetic-susceptibility due to the orbital motion of core-electrons can be added with 𝜒𝑣,⟂ for a quantitative comparison with the experimentally estimated 𝜒⟂ (𝜒⟂ = 𝜒𝑣,⟂ + 𝜒𝑐𝑜𝑟𝑒,⟂ ). [9,12,14] Nonetheless, the theoretically estimated magnetic susceptibility of valence and conduction band carriers are found to be in good agreement to explain the experimentally estimated 𝜒⟂ as a function of the field. On the other hand, anisotropic magnetic-behavior of carriers in the MQW sample is also estimated by the magnetization measurements under a parallel magnetic field configuration (𝜒∥ = 𝑀∕𝐵𝑦 ). In order to compare the magnetic susceptibility under perpendicular and parallel magnetic field orientations (𝜒∥ and 𝜒⟂ ), the results obtained under both the configurations are plotted in the same graph, Fig. 3a. Insets of the figure show the sample orientations with respect to the external magnetic field. Fig. 3a shows that the paramagnetic and diamagnetic contributions are considerably suppressed under a magnetic field that is parallel to the QW plane. This directional-dependent magnetic-properties indicate that the measured magnetic-signal by SQUID magnetometer is mainly contributed by the QW regions of multi-QW structure. In order to understand the suppression of magnetic behavior under this parallel field configuration, the magnetic susceptibility for this configuration is also calculated with the help of Eq. (10). The quantum confinement energy of electron and hole (𝐸0 = 𝐸𝑒 + 𝐸ℎ ) is numerically estimated by solving one-dimensional Schrödinger equation in the finite difference method, which is found to be ∼ 55.7 meV for a 75 Å thick GaAs QW. The in-plane and out-of-plane ∗ = 0.042𝑚 and 𝜇 ∗ = 0.056𝑚 . [31] With the carriers reduced mass, used in this calculation, are estimated by Luttinger parameters 𝜇𝑥,𝑦 0 0 𝑧 in QW are confined along 𝑧-direction and separated in 𝑥-direction under 𝐵𝑦 , the contribution of Coulomb interaction in 𝜒𝑣,∥ would be feeble, and therefore not considered in this analysis. The anisotropic magnetic-properties of carriers can be understood by comparing the results obtained in Figs. 3b and 2b. It is found that the diamagnetic-susceptibility and the magnetic field dependent variation in 𝜒𝑣,∥ are significantly reduced under 𝐵𝑦 , which explains the experimental result (Fig. 3a). Here, the suppression in magnetic behavior can be understood by the fact that carriers in QW are already confined along growth direction. Therefore, the impact of 𝐵𝑦 on the out-of-plane cyclotron radius [𝑟𝑧 (𝐵𝑦 )] becomes feeble, particularly when the quantum confinement is stronger than magnetic perturbation, see Eq. (11). In summary, the theoretical results obtained for the two different configurations are found to be in good agreement to explain the experimental observations, i.e., 𝜒⟂ and 𝜒∥ as a function of magnetic field and temperature. In the future, charge screening mechanism and the contribution of core-electrons can be included in the theoretical model presented in this report, which is beyond our theoretical expertise. This will help for a quantitative explanation of 𝜒⟂ and 𝜒∥ of a low dimensional system, particularly in a low magnetic field regime. In our previous report, it was demonstrated that the optical and transport properties 5

Superlattices and Microstructures 137 (2020) 106332

S. Haldar et al.

of charge carriers can be controlled by applying a magnetic field with different orientations [42]. In particular, the carriers in a QW based device were transported across the GaAs/AlGaAs hetero-interfaces under a parallel magnetic field, without applying an electrical bias. In the present work, an experimental investigation on the anisotropic magnetic behavior of charge carriers in a low dimensional system and a theoretical framework to explain the experimental results are presented. Results obtained in the present study would be helpful in acquiring further insight of our previous work, where the magnetic moment of carriers can be controlled by applying a magnetic field. However, an application of this work in spintronic devices is beyond the scope of our present discussion. 5. Conclusions In conclusion, anisotropic magnetic-properties of charge carriers in GaAs multiple-QWs are experimentally probed, under a perpendicular or parallel magnetic field configuration, for the first time. Under both the magnetic field orientations, paramagnetic and diamagnetic contributions in the measured magnetic susceptibility up to 7 T is observed at two different temperatures. It is observed that the variation in the magnetic-susceptibility of carriers with an increasing magnetic field cannot be explained by a simple relation of Landau-diamagnetic susceptibility. In particular, the contribution of in-plane Coulomb interaction in this analysis provides a good agreement with the experimentally estimated magnetic-susceptibility under a perpendicular field. On the other hand, the magnetic behavior of carriers in a QW is found to be suppressed under a magnetic field, parallel to the QW plane. The suppression of paramagnetic and diamagnetic contributions in the latter configuration is explained by the coupling of diamagnetic energy with the confinement energy of carriers in a QW. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments Authors acknowledge Dr. A.K. Srivastava for TEM measurements and Dr. P.A. Naik, Director, RRCAT for his constant support during this work. S. Haldar thank HBNI-India for providing research fellowship. Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.spmi.2019.106332. References [1] L. Teng, T. Jiang, X. Wang, T. Lai, Spin-polarization dependent carrier recombination dynamics and spin relaxation mechanism in asymmetrically doped (110) n-GaAs quantum wells, Phys. Lett. A 382 (17) (2018) 1181–1184, http://dx.doi.org/10.1016/j.physleta.2018.02.035. [2] L. Sun, L. Wang, J. Liu, Z. Ma, Y. Jiang, H. Jia, W. Wang, H. Chen, Anomalous enhancement of the absorption coefficient of GaAs in a pn junction, Superlattices Microstruct. 122 (2018) 80–84, http://dx.doi.org/10.1016/j.spmi.2018.08.019. [3] S. Haldar, G. Vashisht, S. Porwal, T.K. Sharma, V.K. Dixit, Simultaneous magneto-electro-optical measurements in modulation-doped GaAs/AlGaAs quantum well: an investigation on the magneto-photoluminescenec intensity oscillations., J. Appl. Phys. 125 (20) (2019) 205701, http://dx.doi.org/10.1063/1. 5092873. [4] K. Ohtani, D. Turčinková, C. Bonzon, I.-C. Benea-Chelmus, M. Beck, J. Faist, M. Justen, U.U. Graf, M. Mertens, J. Stutzki, High performance 4.7 THz GaAs quantum cascade lasers based on four quantum wells, New J. Phys. 18 (12) (2016) 123004, http://dx.doi.org/10.1088/1367-2630/18/12/123004. [5] D. Liang, Y. Kang, Y. Huo, Y. Chen, Y. Cui, J.S. Harris, High-efficiency nanostructured window GaAs solar cells, Nano Lett. 13 (10) (2013) 4850–4856, URL https://pubs.acs.org/doi/abs/10.1021/nl402680g. [6] L. Li, L. Chen, J. Zhu, J. Freeman, P. Dean, A. Valavanis, A.G. Davies, E.H. Linfield, Terahertz quantum cascade lasers with >1 w output powers, Electron. Lett. 50 (4) (2014) 309–311, http://dx.doi.org/10.1049/el.2013.4035. [7] B.A. Bernevig, S. Zhang, Toward dissipationless spin transport in semiconductors, IBM J. Res. Dev. 50 (1) (2006) 141–148, http://dx.doi.org/10.1147/rd. 501.0141. [8] X. Ge, J. Zou, W. Deng, X. Peng, W. Wang, S. Jiang, X. Ding, Z. Chen, Y. Zhang, B. Chang, Theoretical analysis and modeling of photoemission characteristics of GaAs nanowire array photocathodes, Materials Research Express 2 (9) (2015) 095015, http://dx.doi.org/10.1088/2053-1591/2/9/095015. [9] A. Ney, J. Harris, S. Parkin, Temperature dependent magnetic properties of the GaAs substrate of spin-LEDs, J. Phys. Condens. Matter 18 (17) (2006) 4397–4406, http://dx.doi.org/10.1088/0953-8984/18/17/025. [10] I. Žutić, J. Fabian, S.D. Sarma, Spintronics: Fundamentals and applications, Rev. Modern Phys. 76 (2) (2004) 323, URL https://link.aps.org/doi/10.1103/ RevModPhys.76.323. [11] H. Yu, W. Yao, Valleytronics: Magnetization without polarization, Nature Mater. 16 (9) (2017) 876, http://dx.doi.org/10.1038/nmat4979. [12] S. Hudgens, M. Kastner, H. Fritzsche, Diamagnetic susceptibility of tetrahedral semiconductors, Phys. Rev. Lett. 33 (26) (1974) 1552, URL https: //link.aps.org/doi/10.1103/PhysRevLett.33.1552. [13] K. Ohno, F. Mauri, S.G. Louie, Magnetic susceptibility of semiconductors by an all-electron first-principles approach, Phys. Rev. B 56 (1997) 1009–1012, http://dx.doi.org/10.1103/PhysRevB.56.1009, URL https://link.aps.org/doi/10.1103/PhysRevB.56.1009. [14] D. Chadi, R. White, W. Harrison, Theory of the magnetic susceptibility of tetrahedral semiconductors, Phys. Rev. Lett. 35 (20) (1975) 1372, URL https://link.aps.org/doi/10.1103/PhysRevLett.35.1372. [15] T. Frey, M. Maier, J. Schneider, M. Gehrke, Paramagnetism of the manganese acceptor in gallium arsenide, J. Phys. C: Solid State Phys. 21 (32) (1988) 5539–5545, http://dx.doi.org/10.1088/0022-3719/21/32/008. [16] T. Hesjedal, U. Kretzer, A. Ney, Magnetic susceptibility of n-type GaAs, Semicond. Sci. Technol. 27 (5) (2012) 055018, http://dx.doi.org/10.1088/02681242/27/5/055018. 6

Superlattices and Microstructures 137 (2020) 106332

S. Haldar et al.

[17] E. Kilicarslan, S. Sakiroglu, M. Koksal, H. Sari, I. Sokmen, The effects of the magnetic field and dielectric screening on the diamagnetic susceptibility of a donor in a quantum well with anisotropic effective mass, Physica E 42 (5) (2010) 1531–1535, http://dx.doi.org/10.1016/j.physe.2009.12.031. [18] P. Nithiananthi, K. Jayakumar, Diamagnetic susceptibility of hydrogenic donor impurity in low-dimensional semiconducting systems, Solid State Commun. 137 (8) (2006) 427–430, http://dx.doi.org/10.1016/j.ssc.2005.12.025. [19] A.J. Peter, Diamagnetic susceptibility of a laser dressed donor in a quantum well, Superlattices Microstruct. 47 (3) (2010) 442–451, http://dx.doi.org/10. 1016/j.spmi.2009.12.001, URL http://www.sciencedirect.com/science/article/pii/S0749603609002730. [20] N. Irulayee, K. Krishnan, Diamagnetic susceptibility of a shallow donor in a quantum well, Physica E 41 (2) (2008) 193–195, http://dx.doi.org/10.1016/ j.physe.2008.06.028. [21] L. Wang, R. O’connell, Magnetic susceptibility of a two-dimensional electron gas in the strong magnetic field limit and for non-zero temperatures, Phys. Status Solidi b 144 (2) (1987) 781–786, http://dx.doi.org/10.1002/pssb.2221440238. [22] M. Wilde, J. Springborn, C. Heyn, D. Heitmann, D. Grundler, Magnetization of GaAs quantum wires with quasi one-dimensional electron systems, Physica E 22 (1–3) (2004) 729–732, http://dx.doi.org/10.1016/j.physe.2003.12.110. [23] H. Akbas, P. Bulut, C. Dane, Y. Skarlatos, The diamagnetic susceptibility of hydrogenic donor in two-dimensional semiconductors with anisotropic effective mass of carriers, Superlattices Microstruct. 51 (4) (2012) 455–461, http://dx.doi.org/10.1016/j.spmi.2011.11.020, URL http://www.sciencedirect.com/ science/article/pii/S0749603611002345. [24] B. Sukumar, K. Navaneethakrishnan, Diamagnetic susceptibility of a donor in a GaAs/Ga1- x Alx As quantum well heterostructure and its pressure dependence, Phys. Status Solidi (b) 158 (1) (1990) 193–199, http://dx.doi.org/10.1002/pssb.2221580118. [25] P. Nithiananthi, K. Jayakumar, Effect of 𝛤 –X band crossover and impurity location on the diamagnetic susceptibility of a donor in a quantum well, Solid State Commun. 138 (6) (2006) 305–308, http://dx.doi.org/10.1016/j.ssc.2006.02.038. [26] G. Safarpour, A. Jamasb, M. Dialameh, S. Yazdanpanahi, The effect of hydrostatic pressure on the binding energy and diamagnetic susceptibility of a laser dressed donor impurity in a GaAs/GaAlAs nanowire superlattice, Superlattices Microstruct. 76 (2014) 442–452, http://dx.doi.org/10.1016/j.spmi.2014.10. 030, URL http://www.sciencedirect.com/science/article/pii/S0749603614004005. [27] V.K. Dixit, S.D. Singh, T.K. Sharma, T. Ganguli, R. Jangir, S. Pal, B.Q. Khattak, A.K. Srivastava, H. Srivastava, S.M. Oak, Studies on GaAs/AlGaAs based (p and n-type) quantum well infrared photodetector structures grown using movpe, in: International Workshop on the Physics of Semiconductor Devices, IEEE, 2007, pp. 355–358. [28] K.M. Yu, W. Walukiewicz, T. Wojtowicz, W.L. Lim, X. Liu, Y. Sasaki, M. Dobrowolska, J.K. Furdyna, Determination of free hole concentration in ferromagnetic Ga1- x Mnx As using electrochemical capacitance–voltage profiling, Appl. Phys. Lett. 81 (5) (2002) 844–846, http://dx.doi.org/10.1063/1. 1496143. [29] A. Pramanik, A. Banerjee, Finite-size effect on evolution of griffiths phase in manganite nanoparticles, J. Phys.: Condens. Matter. 28 (35) (2016) 35LT02, http://dx.doi.org/10.1088/0953-8984/28/35/35LT02. [30] S. Harrison, M. Young, P.D. Hodgson, R.J. Young, M. Hayne, L. Danos, A. Schliwa, A. Strittmatter, A. Lenz, H. Eisele, et al., Heterodimensional charge-carrier confinement in stacked submonolayer InAs in GaAs, Phys. Rev. B 93 (8) (2016) 085302, URL https://link.aps.org/doi/10.1103/PhysRevB.93.085302. [31] N. Miura, Physics of Semiconductors in High Magnetic Fields, Vol. 15, Oxford University Press, 2008. [32] X.L. Zheng, D. Heiman, B. Lax, Magnetoexciton ground state in a quantum well: A variational and perturbation approach, Phys. Rev. B 40 (1989) 10523–10528, http://dx.doi.org/10.1103/PhysRevB.40.10523, URL https://link.aps.org/doi/10.1103/PhysRevB.40.10523. [33] R. Khordad, H. Bahramiyan, Electron-phonon interaction effect on the energy levels and diamagnetic susceptibility of quantum wires: Parallelogram and triangle cross section, J. Appl. Phys. 115 (12) (2014) 124314, http://dx.doi.org/10.1063/1.4869918. [34] Y.P. Varshni, Temperature dependence of the energy gap in semiconductors, Physica 34 (1) (1967) 149–154, http://dx.doi.org/10.1016/0031-8914(67) 90062-6. [35] I. Vainshtein, A. Zatsepin, V. Kortov, Applicability of the empirical Varshni relation for the temperature dependence of the width of the band gap, Phys. Solid State 41 (6) (1999) 905–908, http://dx.doi.org/10.1134/1.1130901. [36] S. Das Sarma, R. Jalabert, S.-R.E. Yang, Band-gap renormalization in quasi-two-dimensional systems induced by many-body electron-electron and electronphonon interactions, Phys. Rev. B 39 (1989) 5516–5519, http://dx.doi.org/10.1103/PhysRevB.39.5516, URL https://link.aps.org/doi/10.1103/PhysRevB. 39.5516. [37] A. Sharma, N. Ravindra, S. Auluck, V. Srivastava, Temperature-dependent effective masses in III-V compound semiconductors, Phys. Status Solidi b 120 (2) (1983) 715–721, http://dx.doi.org/10.1002/pssb.2221200231. [38] Q. Li, S. Xu, M. Xie, S. Tong, Origin of the ‘s-shaped’ temperature dependence of luminescent peaks from semiconductors, J. Phys.: Condens. Matter. 17 (30) (2005) 4853, http://dx.doi.org/10.1088/0953-8984/17/30/011. [39] V.K. Dixit, S. Porwal, S.D. Singh, T.K. Sharma, S. Ghosh, S.M. Oak, A versatile phenomenological model for the s-shaped temperature dependence of photoluminescence energy for an accurate determination of the exciton localization energy in bulk and quantum well structures, J. Phys. D: Appl. Phys. 47 (6) (2014) 065103, http://dx.doi.org/10.1088/0022-3727/47/6/065103. [40] M.J. Snelling, E. Blackwood, C.J. McDonagh, R.T. Harley, C.T.B. Foxon, Exciton, heavy-hole, and electron g factors in type-I GaAs/Alx Ga1- x As quantum wells, Phys. Rev. B 45 (7) (1992) 3922, URL https://link.aps.org/doi/10.1103/PhysRevB.45.3922. [41] P. Le Jeune, D. Robart, X. Marie, T. Amand, M. Brousseau, J. Barrau, V. Kalevich, D. Rodichev, Anisotropy of the electron Landé g factor in quantum wells, Semicond. Sci. Technol. 12 (4) (1997) 380. [42] S. Haldar, A. Banerjee, G. Vashisht, S. Porwal, T. Sharma, V. Dixit, A parallel magnetic field driven confinement versus separation of charges in GaAs quantum well investigated by magneto-photovoltage and magneto-photoluminescence spectroscopy, J. Lumin. 206 (2019) 342–347, http://dx.doi.org/10. 1016/j.jlumin.2018.10.076.

7