Conductance relaxation in GeBiTe: Slow thermalization in an open quantum system

This work describes the microstructure and transport properties of ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ films with emphasis on their out-of-equilibrium behavior. Persistent-photoconductivity (PPC), previously studied in the phase-change compound ${\mathrm{GeSb}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$, is also quite prominent in this system. Much weaker PPC response is observed in the pure GeTe compound and when alloying GeTe with either In or Mn. Films made from these compounds share the same crystallographic structure, the same $p$-type conductivity, a similar compositional disorder extending over mesoscopic scales, and similar mosaic morphology. The enhanced photoconductive response exhibited by the Sb and Bi alloys may therefore be related to their common chemistry. Persistent photoconductivity is observable in ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ films at the entire range of sheet resistances studied in this work ($\approx {10}^{\text{3}}\mathrm{\Omega }$ to $\approx 55\mathrm{M}\mathrm{\Omega }$). The excess conductance produced by a brief exposure to infrared illumination decays with time as a stretched exponential (Kohlrausch law). Intrinsic electron-glass effects, on the other hand, are observable in thin films of ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ only for samples that are strongly localized just like it was noted with the seven electron glasses previously studied. These include a memory dip which is the defining attribute of the phenomenon. The memory dip in ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ is the widest amongst the germanium-telluride alloys studied to date consistent with the high carrier concentration $N\ge {10}^{\text{21}}{\mathrm{cm}}^{-3}$ of this compound. The thermalization process exhibited in either the PPC state or in the electron-glass regime is sluggish but the temporal law of the relaxation from the out-of-equilibrium state is distinctly different. Coexistence of the two phenomena give rise to some nontrivial effects, in particular, the visibility of the memory dip is enhanced in the PPC state. The relation between this effect and the dependence of the memory-effect magnitude on the ratio between the interparticle interaction and quench disorder is discussed.

Figure 1

Bright field of a 5 nm film of ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ and an associated diffraction pattern. The micrograph shows polycrystalline structure with grain size of the order of 1–2 nm which accounts for the rather uniform rings in the diffraction pattern.

Figure 2

Bright field of a 5 nm film of ${\mathrm{GeMn}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ and an associated diffraction-pattern taken under similar conditions as the sample in Fig. 1 (same selected area for the diffraction pattern). The typical grain size in this case is clearly much larger than in the ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ sample. The relatively large grain size in this case is also reflected in the “spotty” rings of the diffraction pattern.

Figure 3

Histograms of the atomic ratios for the three components of the ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ sample shown in Fig. 1 above. The histograms are based on 35 local EDS measurements across the film, each sampling the contribution from a $40\times 40{\mathrm{nm}}^{\text{2}}$ area. Note that the most likely ratio is close to a composition of 1:1:1, but there is a significant scatter around this value. Note that the average ratio for Te/Ge deviates from unity, while $\text{Te}/(\text{Ge}+\mathrm{Bi})\approx 1$, which may hint that the Bi atoms reside on Ge sites in the crystal.

Figure 4

$G(V_{\text{g}}$) traces for three ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ films showing nearly perfect linear dependence on the gate voltage $V_{\text{g}}$ with a slope that increases with $R_{\square }$. The inset is a schematic depiction of the band structure of the material and the position of the Fermi energy.

Figure 5

Conductance versus gate voltage for a ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ film with $R_{\square }=53\mathrm{M}\mathrm{\Omega }$. This $G(V_{\text{g}}$) trace was taken 24 h after the sample was allowed to equilibrate under $V_{\text{g}}=0\mathrm{V}$. The dashed line stands for the thermodynamic density of states which, consistent with the data in Fig. 4, is assumed to be linear. The inset to the figure shows the memory dip of this sample obtained by subtracting the thermodynamic DOS [the linear part of $G\left(V_{\text{g}}\right)$ taken for this sample]. The memory-dip plot is marked with arrows to define its typical width $\mathrm{\Gamma }$, its magnitude $\delta G_{\text{MD}}$, and the “equilibrium” value of the conductance, $G_{\text{0}}$.

Figure 6

Comparing the magnitude of the memory dip (triangles) and slope of the thermodynamic density of states (circles) for several ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ films. For the latter the value for $\mathrm{\Delta }G/G$ is based on $G(-30\text{V})/G(+30\text{V})$.

Figure 7

Width of the memory dip $\mathrm{\Gamma }$ (defined in Fig. 5) for three germanium-telluride alloys as a function of their carrier concentration $N$. The value for $\mathrm{\Gamma }$ (defined in the inset to Fig. 5) is based on $G(V_{\text{g}}$) measurements at $T=4.1$ K and a ${\mathrm{SiO}}_{\text{2}}$ layer of $0.5\mu \mathrm{m}$ separating the sample for the gate.

Figure 8

Conductance versus temperature for two ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ samples used in the study with thickness of 45 Å and $R_{\square }=8.5\mathrm{M}\mathrm{\Omega }$ and $R_{\square }=30\mathrm{M}\mathrm{\Omega }$ at $T=4.1$ K. Samples are labeled by their activation energies taken from the respective $R\left(T\right)\propto exp[T_{\text{0}}/T]$ data.

Figure 9

Dependence of sample resistance on the longitudinal voltage for ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ films with different disorder. Samples shown had a common geometry of 0.5 mm length and 0.5 mm width.

Figure 10

Conductance evolution during a typical optical excitation protocol. The sample is a 43 Å thick ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ with equilibrium $R_{\square }=53\mathrm{M}\mathrm{\Omega }$ at 4.1 K. It is exposed to the infrared LED operated with 1 mA for 3 s at a distance of $\approx 9\mathrm{mm}$ from the sample. The conductance is monitored for $11000$ s after the LED is turned off during which time it decays slowly with a stretched-exponential law as depicted in the inset. A fit to $\mathrm{\Delta }G\left(t\right)\propto \exp [-{(t/\tau )}^{\beta }]$ with $\tau =1.3\times {10}^{\text{9}}\mathrm{s}$ and $\beta =0.11$ is shown in the inset as a dashed line.

Figure 11

Relative increase of conductance due to a brief infrared exposure $\delta G_{\text{IR}}/G_{\text{0}}$ plotted as function of the sheet resistance of the sample. $G_{\text{0}}$ is the equilibrium value of the conductance and $\delta G_{\text{IR}}$ is arbitrarily chosen as $\delta G_{\text{IR}}=G(t=3\times {10}^{\text{3}})-G_{\text{0}}$, The inset shows the dependence of $\delta G_{\text{IR}}/G_{\text{0}}$ on the infrared intensity for a typical sample. Dashed lines are guides for the eye.

Figure 12

Data for a field-effect measurement (a) and infrared excitation protocol [(b) and (c)] for a diffusive ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ sample with thickness 50 Å and $R_{\square }=15.4\mathrm{k}\mathrm{\Omega }$ at $T=4.1$ K. Note the linear $G(V_{\text{g}}$) trace in (a) consistent with diffusive behavior. Plate (c) illustrates the fit to the relaxation law where the time origin is taken to coincide with the turning off of the infrared LED.

Figure 13

Data for a field-effect measurement (a) and infrared excitation protocol [(b) and (c)] for a strongly localized ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ sample with thickness 48 Å and $R_{\square }=57\mathrm{M}\mathrm{\Omega }$ at $T=4.1$ K. Note the appearance of a memory dip in the $G(V_{\text{g}}$) trace (a) consistent with insulating behavior. Plate (c) illustrates the fit to the relaxation law where the time origin is taken to coincide with the turning off of the infrared LED.

Figure 14

Comparing the result of the infrared excitation protocol for three germanium-telluride alloys. The same protocol conditions were used in each case as specified in Fig. 10.

Figure 15

Conductance evolution during excitation protocol using a sudden change of gate voltage. The sample is a 43 Å thick ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ with “dark” $R_{\square }=53\mathrm{M}\mathrm{\Omega }$ at 4.1 K. It was allowed to relax under $V_{\text{g}}=0\mathrm{V}$ for 40 h and, after recording its “equilibrium” $G_{\text{0}}\mathrm{for}\approx {10}^{\text{2}}\mathrm{s}, V_{\text{g}}$ was swept to $-35$ V within 3.5 s producing the $G\left(t\right)$ data shown in the figure. The inset illustrates that, after the new $V_{\text{g}}=-35\mathrm{V}$ is established, the sample conductance decays with a logarithmic time dependence. Dashed line is best fit for a $G\left(t\right)=G\left(1\right)-a\text{log}\left(t\right)$ law.

Figure 16

Conductance evolution during excitation by a gate protocol using the same parameters as in Fig. 15 except that the sample was first put in its PPC state as per the infrared protocol (Fig. 10). The system was then allowed to relax for 3 h before changing the gate voltage. The inset illustrates that, after the new $V_{\text{g}}=-35\mathrm{V}$ is established, the sample conductance decays with a stretched-exponential time dependence. Dashed line is best fit for a $G\left(t\right)\propto \exp [-{(t/\tau )}^{\beta }]$ with $\beta =0.11$ and $\tau =3\times {10}^{\text{10}}\mathrm{s}$.

Figure 17

Field-effect $G(V_{\text{g}}$) traces for a ${\mathrm{GeBi}}_{\text{x}}{\mathrm{Te}}_{\text{y}}$ sample with “dark” $R_{\square }=54\mathrm{M}\mathrm{\Omega }$ taken 24 h after cooldown to $T=4.1$ K (circles) and 3 h after exposing the film to infrared radiation at 1 mA for 3 s (squares). Both traces were taken with the same sweep rates of 0.023 V/s. The relative slopes $\mathrm{\Delta }G/G$ of the thermodynamic component of $G(V_{\text{g}}$) (see Fig. 6 for definition) were $\approx 30\%\pm 0.1\%$ for both curves, while the magnitude of the memory dip $\delta G_{\text{MD}}/G_{\text{0}}$ is $2\%\pm 0.1\%$ in the dark state and $2.9\%\pm 0.1\%$ in the PPC state.

Figure 18

Field-effect sweeps comparing the memory dip of two ${\mathrm{In}}_{\text{x}}\mathrm{O}$ samples with different values of $N$ (obtained by different In/O composition) but similar $R_{\square }$ and measured under identical conditions (sweep rate, relaxation history). These $G(V_{\text{g}}$) plots are based on the same field-effect structures described in Sec. 2 but with $0.5\mu \mathrm{m}{\mathrm{SiO}}_{\text{2}}$ spacer. Dashed line is the assumed thermal DOS plotted to expose the wide and shallow memory dip of the sample with the larger $N$. The latter has $\delta G_{\text{MD}}/G_{\text{0}}\approx 1.1\%$, while the low $N$ sample exhibits $\delta G_{\text{MD}}/G_{\text{0}}\approx 21\%$.