Effect of charged dislocation scattering on electrical and electrothermal transport in $n$-type InN

Temperature-dependent thermopower and Hall-effect measurements, combined with model calculations including all of the relevant elastic- and inelastic-scattering mechanisms, are used to quantify the role of charged line defects on electron transport in $n$-type InN films grown by molecular-beam epitaxy. Films with electron concentrations between $4\times {10}^{17}$ and $5\times {10}^{19}$ cm${}^{-3}$ were investigated. Charged point and line defect scattering produce qualitatively different temperature dependences of the thermopower and mobility, allowing their relative contribution to the scattering to be evaluated using charge neutrality at the measured electron concentration. Both charge state possibilities for the dislocations [positively charged (donors) or negatively charged (acceptors)], were considered. The 100–300 K temperature dependence of the mobility and the 200–320 K temperature dependence of the thermopower can be modeled well with either assumption. The dislocation density was independently measured by plan-view and cross-sectional transmission electron microscopy and corresponds well with the values obtained from transport modeling.

Figure 1

Comparison of momentum relaxation times $\tau ={\nu }^{-1}$ for ionized impurity (ii) and charged dislocation (dis) scattering as a function of electron energy $\varepsilon$ for an electron concentration of $4\times {10}^{17}$ cm${}^{-3}$ and a dislocation density of $1\times {10}^9$ cm${}^{-2}$. Under these conditions, the Fermi level at 300 K is 11 meV. At higher energies, the relationship is approximately a power law with $r$ $=$ 1.1 for ii and 1.4 for dis. At lower electron energies, the scattering time becomes less dependent on electron energy, but the slope (effective value of $r$) is always steeper for dis.

Figure 2

Calculated thermopower (a) and mobility (b) for $n$ $=$ $4\times {10}^{17}$ cm${}^{-3}$ for uncompensated InN ($Z=+1$) and for the case of compensation or multiply charged donors $Z=+3$. Assuming that all electrons come from triply charged donors is equivalent to assuming a compensation ratio $N_A/N_D$ of 0.5; i.e., $N_A$ $=$ $4\times {10}^{17}$ cm${}^{-3}$ and $N_D$ $=$ $8\times {10}^{17}$ cm${}^{-3}$ and full ionization.

Figure 3

Calculated dependence of mobility on dislocation density for three different values of electron concentration, 10${}^{17}$–10${}^{19}$ cm${}^{-3}$ (solid lines, donor dislocations; dashed lines, acceptor dislocations). The ionized impurity concentration was determined from the charge neutrality condition in Eq. (2) (see text). The shaded region indicates the range of typical dislocation densities reported in the literature for InN.

Figure 4

Measured values of the room-temperature Seebeck coefficient as a function of electron concentration $n$ for 14 InN films with thicknesses between 128 nm and 12.2 $\mu$m. Circles are for films grown at Cornell; squares are for films grown at UCSB. Samples used in the TEM analysis are indicated by blue symbols. Lines are calculated values for a charged dislocation density $N_{\text{dis}}$ in the range of 10${}^9$–10${}^{11}$ cm${}^{-2}$. Solid lines are for donor dislocations; dashed line assume that they are acceptors.

Figure 5

Measured (points) and modeled (lines) Seebeck coefficient (a) and mobility (b) of selected samples A, B, and C as a function of temperature. Solid lines were obtained assuming that the dislocations are donors; for dashed lines, acceptor-like dislocations were assumed (see text). The dislocation densities used to obtain the best fit are shown in Table . In (b), the thick dashed line indicates the modeled mobility for sample C (lowest $n$ and highest $\mu$ of the films in this study) without dislocation scattering, i.e., with $N_D=n$. Without charged dislocation scattering, the modeled mobility is too large and does not reproduce the observed relatively flat temperature response.

Figure 6

Plan-view TEM image of sample A. Most of the contrast in this image comes from dislocations. The dislocation density is estimated to be on the order of 10${}^{11}$ cm${}^{-2}$.

Figure 7

Plan-view (a) and cross-sectional (b) TEM images of sample B. The surface dislocation density was estimated from several plan images and is $2\times {10}^{10}$ cm${}^{-2}$. The pattern of dislocations is interesting; they form in lines stitching out a patchwork of areas that are largely dislocation free. This implies that the growth began as three-dimensional islands, which later coalesced requiring dislocations to accommodate the low-angle grain boundaries. The dislocation types are estimated to be $\sim$1/3 mixed type, $\sim$2/3 pure edge, and $<$10% screw. The cross-section image in (b) indicates that the dislocation density does not vary substantially through the thickness of this film. With the indicated contrast condition, the concentration of pure edge and/or mixed character dislocations is on the order of 5$\times {10}^{10}$ cm${}^{-2}$.

Figure 8

Surface plan-view (a) and mid-film plan-view (b) TEM images of sample C. Dislocation density at surface, averaged from several images, appears to be as low as $1\times {10}^9$ cm${}^{-2}$ although the roughness of the films and the pits which form in the foil as a result add some uncertainty to this determination. The mid-layer dislocation density determined from the image in (b) and several other images is $\sim 5\times {10}^{10}$ cm${}^{-2}$, similar to the values found throughout sample B. However, in contrast to sample B, the dislocations do not appear to be distributed along grain boundaries.