International Journal of Scientific & Engineering Research Volume 2, Issue 9, September-2011 1
ISSN 2229-5518
Low Temperature Magnetotransport in 2D GaN Quantum Wells
Arindam Biswas, Aniruddha Ghosal, Hasanujjaman, Sahnawaj Khan
Index Terms— Hall Mobility, Magnetoresistance, Scattering Theory .Bolzman transport equation, 2D quantum wells.
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ecent work on III-V nitride family, InN, GaN, and AlN, have led to significant progress in improving material quality. Alloys and heterostructures based on these materials are already exits in literature through
some theoretical and experimental investigation [1–7].However, there has been a revival of interest in magnetic-field-induced transitions in the integer quantum Hall effect [8-11]. According to the scaling theory of loca- lization, at zero magnetic field, all states of a noninteract- ing two-dimensional electron system (2DES) are localized. On the other hand, in the presence of a perpendicular magnetic field, the theoretical understanding of the integ- er quantum Hall effect (IQHE) requires the existence of extended states in a 2DES. In order to explain the evolu- tion from extended states at finite magnetic field to loca- lized states at zero magnetic field, Laughlin [12] and Khmelnitskii [13] showed the picture of extended states at Landau-levels centers and at localized states between Landau levels. It is argued that to be consistent with the scaling theory, as the magnetic field is decreased, the energy of the extended states will float up and exit through the Fermi level of the 2DES. Furthermore, a 2DES is in an insulating phase when all the states below the Fermi level become localized at zeromagnetic field.
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Arindam Biswas is an associate professor for department of ECE,Dumkal
Institute of Engineering & Technology,W.B.U.T,West Bengal,India.
mail: mailarindambiswas@yahoo.co.in
Aniruddha Ghosal is an associate professor for department of Radio
Physics and Electronics, University of Calcutta,India.
Hasanujjaman and Sahnawaj Khan both are assistant professor for de-
partment of ECE, Dumkal Institute of Engineering. and Technology.
Recent investigations have revealed that GaN material has widespread applications in optoelectronic devices, such as blue light emitting diodes (LEDs), laser diodes and high frequency field effect transistors [14]. Theoreti- cal study of magneto-transport characteristics of 2DEG in GaN quantum wells will be relevant in understanding the carrier transport mechanism. The aim of the present pa- per is to study some aspects of magneto-transport proper- ties, namely Hall mobility in GaN quantum wells in non- quantizing magnetic fields. We have considered Fermi- Dirac statistics and the relevant scattering mechanisms like deformation potential acoustic, piezoelectric, back- ground and remote ionized impurity scatterings in the low temperature range 1K-14K. Based on numerical itera- tive technique we have solved the Boltzmann transport equation considering the above mentioned scattering me- chanisms both individually and in combination with the help of Matthiessen’s rule. The intersubband scattering has not been incorporated in our calculations because of its insignificant contribution in the low temperature range of interest here. Our calculations of Hall mobility at such low temperatures have agreed with the experimental re- sults [15].
In Al0.13Ga0.87N / GaN structure, the conduction band offset is about 2.26eV [16]. The maximum Fermi energy of the electrons considered here is found to be 0.013 eV. So ΔEc is about 174 times Ef . Hence the GaN square well can be considered to be infinite. Moreover, we assume that the electrons occupy only the lowest sub-band, since the next upper sub-band is higher than 2Ef times in ener- gy than the lowest sub-band.
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International Journal of Scientific & Engineering Research Volume 2, Issue 9, Septembar-2011 2
ISSN 2229-5518
In our model we consider a rectangular Cartesian co- ordinate system with z-axis perpendicular to the interfa- cial planes so that the 2D transport occurs parallel to the xy plane. The electric field ε is assumed to be along x- axis
xy
eB
N2 D 2
f
2 ( E )
and non-quantizing magnetic field B along z- axis. The
carrier distribution function can be written as
(
0
0
E 1 2 2 ( E )
E dE
f (k ) 
f ( E ) ( e )
0 m*
and the drift mobility xx(0) is the value of xx for B=0.
f0
[ k
( E ) k
( E )]
(1)
E x x B y y
Where k is the 2D wave vector of electrons with energy, f0 (E) is the equilibrium Fermi-Dirac function, e is the elec- tronic charge, ħ is Planck’s constant divided by 2π, m* is the electron effective mass.kx and ky are the x and y com-
ponent of k, is the cyclotron resonance fre- quency, and and are the perturbation functions.
The perturbation functions obtained from the Boltzmann
Transport equations are,
2 2
We have used the following data in our calculations: effective mass of electron m*= 0.218 mo, where mo is the rest mass of the electron, 2D electron concentration is tak- en as N2D = 4.8 x 1015/ m2. The background ionized im- purity concentration is taken as Ni = 8.6 x 1022/m3 to fit the experiment [15]. The well width (Lz) is taken as 67nm. The other parameter values for GaN are taken from Ref.[19]. Fig. 1 shows the variations of Hall mobility as a function of magnetic field B. The Hall mobility variation is exhibited for temperature T=1.38K.
x ( E )
( E ) / 1
B
(2)
x ( E )
2 2 2
( E ) / 1
B
(3)
where, τ (E) is the combined relaxation time for all the
scatterings. The expressions for relaxation times of the acoustic scattering via deformation potential and piezoe- lectric couplings and that for the background and remote ionized impurity scatterings have been taken from Refs.[16,17]. The Hall mobility and the longitudinal mag- neto-resistivity is calculated with the help of the expres- sions given in Refs. [18].
xx (0) | xy |
H
B( 2
xy
(4)
Fig.1: Variation Hall mobility ( μ H) with magnetic field B for N2D = 4.8 x 1015/ m2, Ni = 8.6 x 1022/ m3 and Lz = 67nm
and Rm = HBxx/ xy - 1 (5)
Where,
The Hall mobility has an inverse dependence on the magnetic field B, as given in Ref.[20]. So It decreases with B.
e f0
( E )
Fig. 2 exhibits the variation of the Hall mobility with tem-
xx
N2 D2
( E )
0
12 2 ( E )
E dE ,
perature.Fig. 2, the Hall mobility increases with temperature due to Coulombic nature of ionized impurity scattering.
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International Journal of Scientific & Engineering Research Volume 2, Issue 9, Septembar-2011 3
ISSN 2229-5518
We have shown the variations of the overall Hall mobil- ity due to acoustic, piezoelectric and ionized (both back- ground and remote) impurity scatterings and individual ionized impurity scattering in GaN quantum wells with magnetic field and temperature. We find that the Hall mo- bility decreases sharply at low magnetic fields and then becomes less sensitive to the field variations. The Hall mo- bility increases with temperature due to the Columbic na- ture of ionized impurity scattering. The decrease of ρ xx with T is in agreement with the experimental results of magnetotransport measurements on two-dimensional elec- tron system in GaN.
Fig 2a. and Hall mobility( μ H) variation with temperature T. The other parameter values are the same as in Fig 1.
2b. Longitudinal magnetoresistivity ( ρ xx) temperature T at magnetic field B = 1T.The other parameter values arehe same as in Fig 1.
The decrease of ρ xx with T is in agreement with the expe- rimental results of magneto transport measurements on two- dimensional electron system in GaN [19].This also furnish the comparison of our theoretical results with the experi- mental ones. The overall Hall mobility due to combined ef- fects of all scatterings , The individual contribution of acous- tic scattering to the mobility is found to be almost same as that of the overall mobility value. It shows the Hall mobility for individual contribution of ionized impurity scattering. Referring to Fig. 2b, the contribution of deformation poten- tial acoustic scattering in case of ρ xx coincides with that due to the total scattering mechanisms, while the ionized impuri- ty scattering has been found to contribute negligibly.
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