GaN wurtzite heterostructure¶

Header¶

Files for the tutorial located in nextnano++\examples\education

piezo-pyro-charges_wz_GaN-AlN_1D_nnp_offsets.in
piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain.in
piezo-pyro-charges_wz_GaN-AlN_1D_nnp_pyro.in
piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-pyro.in
piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-piezo.in
piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-piezo-pyro.in

Scope of the tutorial:

defining wurtzite heterostructure
piezo- and pyroelectricity in wurtzite

Main adjustable parameters in the input file:

parameter $Strain
parameter $Polarity

Relevant output files:

bias_00000\bandedges.dat
bias_00000\potential.dat

Introduction¶

This tutorial presents how to define wurtzite heterostructure and explains how piezo- and pyroelectric polarization constants influence respective charges on interfaces on a n example of GaN/AlAn/GaN heterostructure bringing insight into piezoelectricity and pyroelectricity in wurtzite. More detailed explanation of piezoelectricity in wurtzite can be also found in Piezoelectricity in wurtzite.

Crystallographic orientation¶

Input files for this tutorial simulate a GaN/AlN/GaN wurtzite structure grown pseudomorphically on GaN, i.e., the AlN is tensely strained, whereas the GaN is unstrained. The growth direction [$0001$] is set along which corresponds to the growth on Ga-polar GaN (0001) surface (Ga-face polarity).

As the wurtzite structure belongs to the hexagonal crystal system, one should take additional care about defining Miller indices of the growth plane.

global{ }
    simulate1D{}

    ## This is along [0001] direction: Ga-face polarity
    crystal_wz{
        x_hkl = [ 0, 0, 1 ]   # hkil = (0, 0,  0, 1) Miller-Bravais indices
        y_hkl = [ 1, 0, 0 ]   # hkil = (1, 0, -1, 0) Miller-Bravais indices

    substrate{
        name = "GaN"
    }
}

Although the four-digit Miller-Bravais indices ($h k i l$) are usually used in a wurtzite structure, you have to omit $i$ in nextnano++ because $i = h - k$ holds. x_hkl refers to a plane and perpendicular to the crystal growing direction. See Crystal Coordinate Systems for more details. As the wurtzite structure lacks symmetry plane perpendicular to the c-axis, the c-plane is polarized. The $0001$ plane in GaN is the Ga-polar plane, while the opposite $000\overline{1}$ plane is the N-polar plane. All the examples in this tutorials are prepared for the growth on the Ga-polar plane. The N-polar polarity is discussed at the end.

Strain-induced energy shift¶

Energy profiles without the strain effects¶

Figure 2.4.46 shows the energy band offsets (conduction and valence band edges) of the heterostructure. It is done by neglecting all polarization and strain effects. Poisson equation is solved to bring the offsets already near the Fermi level set to zero. Clearly AlN forms the barrier for both electrons and holes.

../../../../_images/1D_AlGaN_wall_no_strain.svg — Figure 2.4.46 Calculated conduction and valence band structures without strain effects. (a) Full energy profile. (b) Valence band edges of AlN. (Run piezo-pyro-charges_wz_GaN-AlN_1D_nnp_offsets.in to reproduce.)¶

It is visible that without strain the CH (crystal hole) band lies above the HH (heavy hole) and LH (light hole) bands in AlN while the situation is opposite for GaN. This mechanism is explained in [Chuang1996]. Note that heavy and light hole are not degenerate under no-strain condition, unlike in zincblende crystals.

Including energy shift due to pseudomorphic strain¶

As the substrate in the simulation is set to GaN, the GaN remains unstrained also when the strain model is turned on. Since AlN has the lattice constant, $a_\mathrm{AlN} = 0. 3112\;\mathrm{nm}$, smaller than the one of GaN, $a_\mathrm{GaN} = 0. 3189\;\mathrm{nm}$, it becomes strained as follows.

The biaxial (in-plane) strain is tensile.

\[\varepsilon_{\parallel} = (a_\mathrm{substrate} - a) / a = 0.0247429\]

The uniaxial (growth direction) strain is compressive.

\[\varepsilon_{\perp} = -2 (c_\mathrm{13} / c_\mathrm{33}) \varepsilon_{\parallel} = -0.0143283\]

The hydrostatic strain is positive, which corresponds to an increase in volume for AlN.

\[\varepsilon_\mathrm{hy} = Tr(\varepsilon_\mathrm{ij}) = (2 \varepsilon_{\parallel} + \varepsilon_{\perp}) = 0.0351575\]

Introduction of the strain leads to an energy shift of both conduction and valence band edges.

The crystal anisotropy leads to two distinct conduction band deformation potentials for the $\Gamma$ point in wurtzite. The one is parallel, defpot_absolute_l, and the other one is perpendicular, defpot_absolute_t, to the c axis. These values are taken from the database_nnp.in.

binary_wz{
    name = AlN

    ...

    conduction_bands{
        Gamma{
            defpot_absolute_l = -20.5       # Vurgaftman2 (a1) along c axis
            defpot_absolute_t = -3.9        # Vurgaftman2 (a2) perpendicular to c axis
        }
    }
}

Denoting defpot_absolute_l as $a_\mathrm{c,c axis}$ and defpot-absolute_t as $a_\mathrm{c,a axis}$, the conduction band minimum energy including the hydrostatic shift is given by

\[\begin{split}\begin{aligned} E_\mathrm{c}' &= E_\mathrm{c} + a_\mathrm{c,c axis}\varepsilon_{\perp} + 2a_\mathrm{c,a axis}\varepsilon_{\parallel} \\ &= 4.712 + (-20.5 \times (-0.0143283)) + 2 (-3.9) \times 0.0247429 \\ &= 4.712 + 0.10073553 \\ &= 4.81274\;\mathrm{eV} \end{aligned}\end{split}\]

Therefore, the barrier for electrons is increased in this particular example.

Note

Data for uniaxial deformation potentials of other minima than $\Gamma$ are not available yet. The uniaxial deformation potential is zero for the conduction band at the $\Gamma$ point.

There are six valence band deformation potentials ($D_1$, $D_2$, $D_3$, $D_4$, $D_5$, and $D_6$) which arise from a full treatment of the effect of strain on the six-band Hamiltonian. These values are also specified in database_nnp.in.

binary_wz{
    name = AlN

    ...

    valence_bands{
        defpotentials = [ -17.1, 7.9, 8.8, -3.9, -3.4, -3.4 ] # D_1, D_2, D_3, D_4, D_5, D_6, respectively, Vurgaftman2
    }
}

In contrast to zincblende, an absolute deformation potential for the valence band is not needed. The shifts of the valence bands are obtained by diagonalizing the Bir-Pikus strain Hamiltonian, which is a general approach giving correct shifts for arbitrary crystallographic orientations. Note that this holds only for the valence bands.

In our example, the tensile strain in AlN shifts all holes upwards, - the heavy hole by $0.32847\;\mathrm{eV}$, - the light hole by $0.32877\;\mathrm{eV}$ and - the crystal field split-off hole by $0.64726\;\mathrm{eV}$, thus strongly reducing the barrier for holes.

../../../../_images/1D_vb_AlGaN_wall_strain_comparison.svg — Figure 2.4.47 Calculated conduction and valence band structures with strain effects. (a) Full energy profile. (b) Valence band edges of AlN. (Run piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain.in to reproduce.)¶

Polarization Effects¶

Polarization charges are simply computed basic formula from classical electrodynamics once proper Polarization fields are defined.

\[\nabla \circ \mathbf{P} = - \rho\]

Note that polarization effects are addittive, i.e., if $P = P_1 + P_2$ then

\[\nabla \circ \mathbf{P} = \nabla \circ \left[\mathbf{P}_1 + \mathbf{P}_2\right] = \nabla \circ \mathbf{P}_1 + \nabla \circ \mathbf{P}_2 = -\rho_1 -\rho_2\]

Pyroelectric polarization (spontaneous polarization)¶

The wurtzite material GaN, AlN, and InN are pyroelectric materials and thus show the pyroelectric polarization. The pyroelectric polarization field $\mathbf{P}_\mathrm{py}\left(\mathbf{x}\right)$ is antiparallel to the c-axis, [0001], of the hexagonal unit cell (x-direction of exemplary simulations). Therefore, only non-zero component of the pyroelectric polarization vectors is parallel to the x-axis of the exemplary simulation: -0.034 C/m² for GaN and -0.090 C/m² for AlN.

Once the pyroelectric polarization is defined, the pyroelectric charge density can be computed as.

\[\rho_\mathrm{py}\left(\mathbf{x}\right) = - \nabla \circ \mathbf{P}_\mathrm{py}\left(\mathbf{x}\right)\]

If the c-axis is oriented along the x-axis as in our example, this equation reduces to

\[\rho_\mathrm{py}\left(x\right) = - \frac{\partial}{\partial x} P_\mathrm{py}\left(x\right).\]

As the derivative is non-zero only at the discontinuity of the polarization at the interfaces, all polarization charges will be located at these interfaces for this example. The surface densities of the polarization charges can be determined based on the Polarizations of GaN, $P_\mathrm{py,x}\left(\mathrm{GaN}\right)$, and AlN, $P_\mathrm{py,x}\left(\mathrm{AlN}\right)$, as follows:

The 1^st interface (GaN/AlN) at 100 nm:: \[-\left[P_\mathrm{py,x}\left(\mathrm{AlN}\right) - P_\mathrm{py,x}\left(\mathrm{GaN}\right)\right] = P_\mathrm{py,x}\left(\mathrm{GaN}\right) - P_\mathrm{py,x}\left(\mathrm{AlN}\right) = -0.034 + 0.090 = 0.056\;\mathrm{C/m^2}\]
2^nd interface (AlN/GaN) at 117 nm:: \[-\left[P_\mathrm{py,x}\left(\mathrm{GaN}\right) - P_\mathrm{py,x}\left(\mathrm{AlN}\right)\right] = P_\mathrm{py,x}\left(\mathrm{AlN}\right) - P_\mathrm{py,x}\left(\mathrm{GaN}\right) = -0.090 + 0.034 = -0.056\;\mathrm{C/m^2}\]

The interface charge of $-0.056\;\mathrm{C/m}^2$ corresponds to $34.952 \times 10^{12}\;\mathrm{electrons/cm}^2$.

Piezoelectric polarization¶

Piezoelectric polarization appears due to presence of strain. In the exemplary simulation the AlN layer is strained, while GaN is not. Therefore, the piezoelectric polarization is non-zero only in the AlN layer.

\[P_\mathrm{pz,x}\left(\mathrm{AlN}\right) = e33\,\varepsilon_{\perp} + e31\left[\varepsilon_{\parallel} + \varepsilon_{\parallel}\right] = 1.79 \times \left[-0.0143283\right] - 0.50 \times 2 \cdot 0.0247429 = -0.050390\;\mathrm{C/m}^2\]

The piezoelectric constants are specified in database_nnp.in.

binary_wz{
    name = AlN

    piezoelectric_consts{
        e31 = -0.50  e33 = 1.79   # Vurgaftman1 (Vurgaftman2 lists d_ij (/= e_ij !) parameters.)
        e15 = -0.48               # [Tsubouchi1985] (experiment) and [Momida2016] and O. Ambacher
    }
}

Note

The e15 is not relevant for [$0001$] growth direction.

Similarly as for the pyroelectric polarization the piezoelectric charge density can be computed as

\[\rho_\mathrm{pz}\left(\mathbf{x}\right) = - \nabla \circ \mathbf{P}_\mathrm{pz}\left(\mathbf{x}\right)\]

and

\[\rho_\mathrm{pz}\left(x\right) = - \frac{\partial}{\partial x} P_\mathrm{pz}\left(x\right),\]

if the c-axis is oriented along the x-axis as in our example.

In this case as well, the derivative is non-zero only at the interfaces yielding the surface densities of the polarization charges based on the Polarizations of GaN, $P_\mathrm{pz,x}\left(\mathrm{GaN}\right)$, and AlN, $P_\mathrm{pz,x}\left(\mathrm{AlN}\right)$.

The 1^st interface (GaN/AlN) at 100 nm:: \[-\left[P_\mathrm{pz,x}\left(\mathrm{AlN}\right) - P_\mathrm{pz,x}\left(\mathrm{GaN}\right)\right] = P_\mathrm{pz,x}\left(\mathrm{GaN}\right) - P_\mathrm{pz,x}\left(\mathrm{AlN}\right) = 0 + 0.050390 = 0.050390\;\mathrm{C/m^2}\]
2^nd interface (AlN/GaN) at 117 nm:: \[-\left[P_\mathrm{pz,x}\left(\mathrm{GaN}\right) - P_\mathrm{pz,x}\left(\mathrm{AlN}\right)\right] = P_\mathrm{pz,x}\left(\mathrm{AlN}\right) - P_\mathrm{pz,x}\left(\mathrm{GaN}\right) = -0.050390 - 0 = -0.050390\;\mathrm{C/m^2}\]

The interface charge of $-0.050390\;\mathrm{C/m}^2$ corresponds to $31.451 \times 10^{12}\;\mathrm{electrons/cm}^2$.

Electrostatic potential of piezo- and pyroelectric charges¶

The electrostatic potential $\phi\left(\mathbf{r}\right)$ is the solution of the nonlinear Poisson equation.

\[\nabla \circ \left[ \epsilon\left( \mathbf{r} \right) \nabla \phi\left( \mathbf{r} \right) \right] = -\rho\left(\mathbf{r},\phi\left(\mathbf{r}\right)\right)\]

The charge density $\rho$ contains the (static) piezo and pyroelectric charge densities as well as the electron and hole charge densities and ionized donors and acceptors.

While the ionization of the impurities and free carriers depend on the electrostatic potential $\phi$, the piezo- and pyroelectric charge densities do not.

The figure Figure 2.4.48 (a) shows electrostatic potential calculated for the heterostructure including:

both pyro- and piezoelectric charges (black)

only piezoelectric charges (turquoise)

only pyroelectric charges (purple)

../../../../_images/1D_AlGaN_wall_strain.svg — Figure 2.4.48 Electrostatic potential and energy profiles for Ga-face polarity. (a) The electrostatic potential with pyroelectric (py) and piezoelectric (pz) charges. (b) Conduction and valence band energy profiles under strain with all polarization charges included. (Run piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-pyro.in, piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-piezo.in, and piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-piezo-pyro.in to reproduce.)¶

The pyro and piezoelectric contributions are comparable in this example. The band structure including the electrostatic potential is plotted in Figure 2.4.48 (b). Note that the conduction band is pulled below and the valence band above the Fermi level near the interfaces.

N-face polarity versus Ga-face polarity¶

The exactly same simulation of the GaN/AlN/GaN wurtzite structure can be performed also for the N-face polarity. The only difference from the previous simulations is implemented in the crystallographic orientation of the system.

Figure 2.4.49 shows again the electrostatic potential and the energy profiles, as before, but for both, Ga-face and N-face polarities.

../../../../_images/1D_AlGaN_wall_n_face_polarity_comparison.svg — Figure 2.4.49 Electrostatic potential and energy profiles for Ga-face (dotted) and N-face polarities (solid). (a) The electrostatic potential with pyroelectric (py) and piezoelectric (pz) charges. (b) Conduction and valence band energy profiles under strain with all polarization charges included.¶

Note that the positions of 2D electron gas (2DEG) and 2D hole gas (2DHG) are reversed.

Exercises¶

Repeat all simulations for N-face polarity-

Explain why the built-in electric field is comparable in all simulations: piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-pyro.in, piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-piezo.in, and piezo-pyro-charges_wz_GaN-AlN_1D_nnp_strain-piezo-pyro.in

Last update: 07/08/2024