Structural and thermal transport properties of ferroelectric domain walls in GeTe from first principles

Ferroelectric domain walls are boundaries between regions with different polarization orientations in a ferroelectric material. Using first-principles calculations, we characterize all different types of domain walls forming on ($11\overline{1}$), (111), and ($1\overline{1}0$) crystallographic planes in thermoelectric GeTe. We find large structural distortions in the vicinity of most of these domain walls, which are driven by polarization variations. We show that such strong strain-order parameter coupling will considerably reduce the lattice thermal conductivity of GeTe samples containing domain walls with respect to a single crystal. Our results thus suggest that domain engineering is a promising path for enhancing the thermoelectric figure of merit of GeTe.

Figure 1

An example of the domain structure in a ferroelectric material. Different polarization directions in a single domain are color coded (top) and the boundaries between these domains correspond to different types of domain walls (bottom). Domain wall boundary for this configuration of domains is one of the faces of the cube, (001) plane. Tail-to-tail, head-to-head, and tail-to-head domain walls are labeled as T-T, H-H, and T-H, respectively.

Figure 2

Geometry of GeTe domain structure containing ${39}^{\circ }$ or ${141}^{\circ }$ ($11\overline{1}$) domain walls for the case where domains are one unit cell long. Blue lines are the unit cell vectors of the domain structure, while green vectors represent the third primitive lattice vectors of individual domains. Red lines represent polarization directions in different domains. The positions of Ge atoms are labeled as ${\mathrm{Ge}}_1$ and ${\mathrm{Ge}}_2$. The lattice constants of the two primitive unit cells that constitute this supercell are labeled as $a_1$ and $a_2$.

Figure 3

Polarization directions inside each domain for GeTe structures containing ${39}^{\circ }$ or ${141}^{\circ }$ ($11\overline{1}$) domain walls. Polarization directions in two neighboring domains are labeled as primed and nonprimed. $P_{\left|\right|}$ is the direction along the trigonal axis. $P_B$ is the direction normal to the plane of the trigonal axes in neighboring domains and corresponds to the Bloch character of polarization. The third direction, $P_{\perp }$, is perpendicular to the other two directions and quantifies the Néel character of polarization.

Figure 4

Polarization profiles for GeTe structures containing (a) ${39}^{\circ }$, (b) ${141}^{\circ }$, and (c) ${180}^{\circ }$ ($11\overline{1}$) domain walls (DWs). Red line represents polarization in the direction of trigonal axes of each domain (${\mathrm{P}}_{\left|\right|}$), and blue line red shows the Néel component of polarization (${\mathrm{P}}_{\perp }$). DW boundaries are indicated by black vertical lines and labeled T-T for tail-to-tail, H-H for head-to-head, H-T for head-to-tail, and T-H for tail-to-head DWs. For ${39}^{\circ }$ T-T DW, ${\mathrm{P}}_{\left|\right|}$ values in one of the domains are plotted as negative (orange line) to aid visualization.

Figure 5

Local lattice constant (blue line) and local rhombohedral angle (red line) for GeTe structures containing (a) ${39}^{\circ }$, (b) ${141}^{\circ }$, and (c) ${180}^{\circ }$ ($11\overline{1}$) domain walls (DWs). DW boundaries are indicated by black vertical lines and labeled T-T for tail-to-tail, H-H for head-to-head, H-T for head-to-tail, and T-H for tail-to-head DWs.

Figure 6

Polarization profiles for GeTe structures containing (a) (111) and (b) ($1\overline{1}0$) domain walls (DWs). Red line represents polarization in the direction of the trigonal axis (${\mathrm{P}}_{\left|\right|}$), while blue and black lines represent the Néel (${\mathrm{P}}_{\perp }$) and Bloch ($P_B$) components of polarization (multiplied by constant values to make them visible on the graph). DW boundaries are indicated by black vertical lines and labeled T-T for tail-to-tail, H-H for head-to-head, H-T for head-to-tail, and T-H for tail-to-head DWs.

Figure 7

Local lattice constant (blue line) and local rhombohedral angle (red line) for GeTe structures containing (a) (111) and (b) ($1\overline{1}0$) domain walls (DWs). DW boundaries are indicated by black vertical lines and labeled T-T for tail-to-tail, H-H for head-to-head, H-T for head-to-tail, and T-H for tail-to-head DWs.

Figure 8

(a) Lattice thermal conductivity of bulk GeTe and GeTe structure with ${39}^{\circ }$ head-to-head (H-H) domain walls (DWs) and the domain size of 160 nm at 300 K, showing the contribution of each scattering mechanism due to the DWs. Contributions from lattice orientation mismatch and the change of angle to the conductivity reduction are negligible and hence these lines are on top of the single crystal result. (b) Dependence of the lattice thermal conductivity of GeTe with ${39}^{\circ }$ H-H DWs on the domain size at 300 K.