Polarization induced two dimensional confinement of carriers in wedge shaped polar semiconductors

A novel route to achieve two dimensional (2D) carrier confinement in a wedge shaped wall structure made of a polar semiconductor has been demonstrated theoretically. Tapering of the wall along the direction of the spontaneous polarization leads to the development of charges of equal polarity on the two inclined facades of the wall. Polarization induced negative (positive) charges on the facades can push the electrons (holes) inward for a n-type (p-type) material which results in the formation of a 2D electron (hole) gas at the central plane and ionized donors (acceptors) at the outer edges of the wall. The theory shows that this unique mode of 2D carrier confinement can indeed lead to a significant enhancement of carrier mobility. It has been found that the reduced dimensionality is not the only cause for the enhancement of mobility in this case. Ionized impurity scattering, which is one of the major contributer to carrier scattering, is significantly suppressed as the carriers are naturally separated from the ionized centers. A recent experimental finding of very high electron mobility in wedge shaped GaN nanowall networks has been analyzed in the light of this theoretical reckoning.


Self consistent solution of Poisson's and Schrödinger equations
In order to solve the Poisson's and Schrödinger equations self consistently following iterative scheme has been adopted.
(i) A guess potential satisfying the boundary conditions is chosen. Schrödinger equation is solved for this potential to obtain the energy eigen values and eigenfunctions.
(ii) n(x, z) and N + d (x, z) are obtained. At every step, profile for the other half of the trapezium has been obtained through E c (x, z) = E c (x, −z) utilizing the symmetry property of the potential with respect to the z = 0 plane.
(iv) Compare the new potential (conduction band profile) with that is obtained in the previous iteration. If both the profiles match within a specified tolerance limit, self consistency is achieved otherwise continue from step (i) with a revised guess potential formed with a certain linear combination of the newly obtained potential profile and the profile acquired at the previous iteration stage as given in Ref.
(v) Once self consistency is achieved, check the ratio (r ch ) between the total positive and negative charges developed within the wall (exclude the polarization charges). If r ch is very close to 1 (within certain tolerance limit) stop the iteration. If not, then start the exercise all over again from step (i) with a revised value of v m . Note that v m should be increased if r ch < 1 and vice versa. The parameters used in this calculation are listed in Tab. 1.
x (z)| 2 e −q|z−z i | dz, q = k − k with k and k be the initial and final k-state of an electron after scattering, θ the angle between k and k and z i the z-coordinate position of the scatterer. It should be noted that the relaxation time τ II x (E m ) has a week x dependence as a result of the slow variation of the wave function ζ (m) [2][3][4] where the form factor can be expressed as F i j,mn x x ((z ))e −q|z−z | dzdz , i, j,m and n stand for different subbands, q = 2k sin(θ/2) as the ionized impurity scattering is an elastic process.
In order to calculate the neutral impurity scattering rate, we have adopted the general scheme for calculating the rate of any elastic scattering event in 2D system proposed by Stren et al. 5 . In general, momentum cross-section due to any spherically symmetric potential U(r) for the electrons belonging to the m-th subband of a 2D system can be written as: 5 where, r represents the position vector in 2D, J 0 (qr) = 2/πqr cos(qr − π/4). Here we have treated the neutral impurity potential as a spherically symmetric square well 6 ; U(r)=∆ when r ≤ a o and U(r)=0 when r > a o . Where, a o the effective Bohr radius of an electron on a shallow donor.
Finally, the momentum relaxation time can be obtained as n(x, z)/n(x, 0) is the weighted concentration of neutral donors. At high temperatures, the major contribution to the scattering arises due to the lattice vibration.
However it is noteworthy that the rate of scattering due to polar phonon (PO) is expected to be much higher than that for the deformation and piezoelectric potential scattering in GaN because of its strong polar nature. 7 Following Ridley 8 , we write the momentum relaxation time limited by PO phonon scattering as: where, where, the upper and lower signs have been used for absorption and emission respectively. Value of n m , can be evaluated from energy conservation E n m = E m ±hω. From eqn. 4 it may appear that the contribution of emission processes is more than that of absorption processes as the factor (N(ω) + 1) associated with the emission is clearly larger than the factor N(ω) for absorption.
However, it should be noted that PO phonon energy in GaN is ≈ 90 meV, which is much larger than k B T even at room temperature. In a degenerate 2D electron gas, the possibility of transition accompanied by phonon emission is expected to be feeble because of unavailability of states below