Charge Asymmetric Resonance Tunneling (CART) condition for LED structures with square potential wells
In the simplest case the electron emitter of the CART LED can be made of a wide square potential well and the active layer can be made of a single quantum square well
The electron states in wide potential well are not quantized and the resonance tunneling condition is satisfied when energy position of the bottom of the wide potential well is equal to the electron level energy position in the active quantum well. In the case of a strong electron confinement, the infinite wall approximation is valid for the electron energy level position in the active quantum well, and the resonance tunneling condition takes form
(1)
where is the energy position of the bottom of the active quantum well and m_{e} is the electron effective mass and d is the width of the active quantum well.
From other hand the hole tunneling from active well to the wide potential well without thermal activation is forbidden when
(2)
where and are the energy positions of the bottoms of the wide potential well and the active quantum well for holes and m_{hh} is the heavy hole effective mass.
Charge Asymmetric Resonance Tunneling (CART) takes place when both Eq. (1) and Eq.(2) are satisfied. It can be easily seen from Eq. (1) and Eq.(2) that CART condition can be realized for semiconductors with carrier mass ratio satisfying the condition
( 3)
where is the ratio of the conduction band offset to the band gap variation for the semiconductor alloy. The R values for several semiconductor alloys calculated on the base of semiconductor material parameters m_{h}, m_{e}, and from [24] are shown in Table 1.
Semiconductor alloy 
Electron mass, m_{0} 
Heavy hole mass, m_{0} 
R 

In_{x}Ga_{1x}As 
0.05 
0.6 
0.6 
8 
Al_{x}Ga_{1x}As 
0.07 
0.65 
0.6 
6 
In_{x}Ga_{1x}N (WZ) 
0.15 
2 
0.68 
6 
Ga_{x}Al_{1x}N (WZ) 
0.25 
2 
0.68 
4 
In_{x}Al_{1x}N (WZ) 
0.2 
2 
0.74 
3.5 
Table 1. The R values in the CART condition inequality Eq.(3) calculated with semiconductor material parameters [24].
Since all R values in Table 1 are higher then unity it is possible to realize the Charge Asymmetric Resonance Tunneling structures based on the corresponding semiconductor alloys. However, the simple CART structures with square well can be realized only for cubic semiconductor epilayer grown on (001) surface. For wurtzite semiconductor epilayers or for cubic epilayers grown on (111) surface the electric field built in into the electron injection potential well and the active layer quantum well significantly modifies the CART conditions given in Eqs. (1),(2).
CART condition for wurtzite LED structures with GaN/In_{x}Ga_{1x}N/GaN potential wells
For wurtzite LED structures with GaN/In_{x}Ga_{1x}N/GaN potential wells CART condition should account the electric fields built in into the electron injection potential well and the active layer quantum well. Generally, the builtin electric field in the wells E^{in} depends on the strain in the GaN barriers and density of misfit dislocations at GaN/In_{x}Ga_{1x }N interface. For GaN epilayers thicker than ~ 1 um we can assume that GaN barriers are strain relaxed and the builtin electric field in the In_{x}Ga_{1x }N well is
(4)
where is the spontaneous polarization inside(outside) the well and is piezoelectric polarization given by the equation [5]:
here a_{in(out)} are the lattice constants inside (outside) the well; and are the piezoelectic coefficients and the elastic constants in the well; is the density of the misfit dislocations per unit length at the well interface.
The dependence of the builtin electric field inside the In_{x}Ga_{1x}N well embedded in strain relaxed GaN epilayer on the In content x and the misfit dislocation density calculated on the base of Eqs. (4), (5) is shown in Fig.2.
The calculations have been performed with parameters for GaN and InN given in [6] and use of the linear interpolation for evaluation of the parameters inside the In_{x}Ga_{1x}N well.
For Ga terminated surface the builtin electric field is directed opposite to the growth direction.
It can be seen from Fig.2 that for high In content in active quantum well and low densities of misfit dislocations at GaN/In_{x}Ga_{1x }N interface, the inequality
(6)
is satisfied, where is the electric field inside the active quantum well and is optical phonon energy. This makes the over barrier capture into the quantum well via single optical phonon emission forbidden by energy conservation low. Therefore, in the case when inequality (6) is satisfied, the CART LED design should be especially effective.
For wells with builtin electric fields, the resonance tunneling condition given by Eqs.(1) in the infinite wall approximation, is modified in the following way
(7)
where and are electric fields built in the electron emitting wide potential well and the active quantum well respectively. is the energy position of the bottom of the active quantum well and m_{e} is the electron effective mass.
The corresponding condition for suppression of the hole tunneling from active well to the wide potential well without thermal activation takes form
(8)
The CART condition for wells with builtin electric fields found from Eqs. (7) and (8) is
(9)
where is the ratio of the conduction band offset to the band gap variation for the semiconductor alloy. An analysis of inequality (9) has shown that CART LED structures can be realized on the base of GaN epilayers with GaN/In_{x}Ga_{1x}N/GaN potential wells. In next section the first experimental realization of such structures will be considered.
Numerical modeling of CART LED structures with GaN/In_{x}Ga_{1x}N/GaN potential wells
Along with analytical investigation of CART condition, ARI has performed selfconsisting numerical modeling of CART LED structures using its own developed.software A typical distribution of quasiFermi levels for electrons and holes, and the conduction and the valence band bending resulting from selfconsistent electrical potential is shown in Fig.3.
Experimental realization of CART LED structures with GaN/In_{x}Ga_{1x}N/GaN potential wells
We have investigated CART phenomenon on structures with two GaN/In_{x}Ga_{1‑x}N/GaN potential wells specially grown on AIXTRON200/HT/S/4 MOVPE reactor. The electron emitting wide potential well has width 500 Å and In content x = 0.1 and the active quantum well has width 30 Å and In content x = 0.26. The wells are separated by the tunneling GaN barrier with width of 10 Å. The wide potential well is electrically connected to ntype GaN contact layer and the active quantum well is electrically connected to ptype GaN contact layer. This CART LED structure has shown bright blue electroluminescence. The corresponding electroluminescence spectra at different injection currents are shown in Fig.4.
A small red shift of the electroluminescence pick position with increase of the injectioncurrent has been observed. The dependence of the total electroluminescence intensity on the injection current is shown in Fig.5
It can be seen that the total electroluminescence intensity is approximately proportional to the square of the injection current. This seems to be related with radiative recombination of electronhole pairs in the active quantum well in contrast with usually observed excitonic recombination [7].
References
[1] Y.T.Rebane, Y.G.Shreter and W.N.Wang, phys.stat.sol.(a) vol 180, pp 121126 (2000)
[2] E.A.Albanesi, W.R.L.Lambrecht and B.Segall J.Vac.Sci.Technol. B 12, 2470 (1994)
[3] B.A.Foreman Phys.Rev. B 49, 1757 (1994)
[4] S.Adashi J.Appl.Phys. 58, R1 (1985)
[5] Y.T.Rebane, Y.G.Shreter and W.N.Wang Appl.Surf.Sci.166, 300 (2000)
[6] O.Ambacher, J.Phys.D. 31, 2653 (1998)
[7] S.Chichibu, T.Azuhata, T.Sota and S.Nakamura Appl.Phys.Lett. 69, 4188 (1996)
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