# Modeling of III-Nitride CART LEDs

#### 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 $E^e_{WW}$ 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 $E^e_{WW} = E^e_{AW}+\frac{\hbar^2\pi^2}{2m_e d^2}$ (1)

where $E^e_{AW}$ is the energy position of the bottom of the active quantum well and me 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 $E^h_{WW} = E^h_{AW}+\frac{\hbar^2\pi^2}{2m_hh d^2}$ (2)

where $E^e_{WW}$ and $E^h_{WW}$ are the energy positions of the bottoms of the wide potential well and the active quantum well for holes and mhh 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 $R = \frac{ (1-\eta)m_h}{ \eta m_e} > 1$ ( 3)

where $\eta$ 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 mh, me, and from [2-4] are shown in Table 1.

 Semiconductor alloy Electron mass, m0 Heavy hole mass, m0 $\eta$ R InxGa1-xAs 0.05 0.6 0.6 8 AlxGa1-xAs 0.07 0.65 0.6 6 InxGa1-xN (WZ) 0.15 2 0.68 6 GaxAl1-xN (WZ) 0.25 2 0.68 4 InxAl1-xN (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 [2-4].

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/InxGa1-xN/GaN potential wells

For wurtzite LED structures with GaN/InxGa1-xN/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 built-in electric field in the wells Ein depends on the strain in the GaN barriers and density of misfit dislocations $N_d^{misfit}$ at GaN/InxGa1-x N interface. For GaN epilayers thicker than ~ 1 um we can assume that GaN barriers are strain relaxed and the built-in electric field in the InxGa1-x N well is $E^{in}=\frac{P^{out}_{SP}-P^{in}_{PE}-P^{in}_{SP}}{\epsilon\epsilon_0}$ (4)

where $P^{in(out)}_{SP}$ is the spontaneous polarization inside(outside) the well and $P^{in}_{PE}$ is piezoelectric polarization given by the equation : $P^{in}_{PE} = 2 (\frac{a_{out}}{a_{in}} +a_{in}N_d^{misfit} -1) (\epsilon^{in}_{31} -\epsilon^{in}_{33} \frac{c^{in}_{13}}{c^{in}_{33}})$

here ain(out) are the lattice constants inside (outside) the well; $\epsilon^{in}_{ik}$ and $c^{in}_{ik}$ are the piezoelectic coefficients and the elastic constants in the well; $N_d^{misfit}$ is the density of the misfit dislocations per unit length at the well interface.

The dependence of the built-in electric field inside the InxGa1-xN well embedded in strain relaxed GaN epilayer on the In content x and the misfit dislocation density $N_d^{misfit}$ calculated on the base of Eqs. (4), (5) is shown in Fig.2. Fig.2. The dependence of the built-in electric field inside the InxGa1-xN quantum well embedded in GaN epilayer on the In content x and the misfit dislocation density .

The calculations have been performed with parameters for GaN and InN given in  and use of the linear interpolation for evaluation of the parameters inside the InxGa1-xN well.

For Ga terminated surface the built-in 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/InxGa1-x N interface, the inequality $edE^{in}_{AW}>\hbar\omega_{opt}$  (6)

is satisfied, where $E^{in}_{AW}$ is the electric field inside the active quantum well and $\hbar\omega_{opt}$ 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 built-in electric fields, the resonance tunneling condition given by Eqs.(1) in the infinite wall approximation, is modified in the following way $E_{WW}^{e}+1.86\frac{(e\hbar E_{WW}^{in})^{\frac{2}{3}}}{m_{e}^{\frac{1}{3}}}=E_{AW}^{e}+\frac{\hbar^{2}\pi^{2}}{2m_{e}d^{2}}+eE_{AW}^{in}d/2$ (7)

where $E^{in}_{WW}$ and $E^{in}_{AW}$ are electric fields built in the electron emitting wide potential well and the active quantum well respectively. $E^{e}_{AW}$ is the energy position of the bottom of the active quantum well and me 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 $E_{WW}^{h}+1.86\frac{(e\hbar E_{WW}^{in})^{\frac{2}{3}}}{m_{e}^{\frac{1}{3}}}>E_{AW}^{h}+\frac{\hbar^{2}\pi^{2}}{2m_{hh}d^{2}}+eE_{AW}^{in}d/2$ (8)

The CART condition for wells with built-in electric fields found from Eqs. (7) and (8) is $(\frac{1-\eta}{m_{e}}-\frac{\eta}{m_{hh}})\frac{\hbar^{2}\pi^{2}}{2d^{2}}>1.86\frac{(e\hbar E_{WW}^{in})^{\frac{2}{3}}}{m_{e}^{\frac{1}{3}}}+eE_{AW}^{in}d/2$ (9)

where $\eta$ 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/InxGa1-xN/GaN potential wells. In next section the first experimental realization of such structures will be considered.

#### Numerical modeling ofCART LED structures with GaN/InxGa1-xN/GaN potential wells

Along with analytical investigation of CART condition, ARI has performed self-consisting numerical modeling of CART LED structures using its own developed.software A typical distribution of quasi-Fermi levels for electrons and holes, and the conduction and the valence band bending resulting from self-consistent electrical potential is shown in Fig.3. Fig.3. The distribution of quasi-Fermi levels for electrons and holes, and the conduction and the valence band bending resulting from self-consistent electrical potential for CART LED structure with triangle wide well.

#### Experimental realization of CART LED structures with GaN/InxGa1-xN/GaN potential wells

We have investigated CART phenomenon on structures with two GaN/InxGa1‑xN/GaN potential wells specially grown on AIXTRON-200/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 n-type GaN contact layer and the active quantum well is electrically connected to p-type 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. Fig.4. EL spectra of the InGaN/GaN CART structure at various currents. 1 (dotted curve) – 0.1 mA, 2 (dashed curve) – 0.5 mA, 3 (solid curve) – 1 mA.

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 Fig.5. EL intensity dependence on current through the InGaN/GaN CART structure

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 electron-hole pairs in the active quantum well in contrast with usually observed excitonic recombination .

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