Engineering coexistence between free and trapped carriers via extrinsic polarons

The transition between delocalized and localized electronic configurations is a characteristic property of all small-polaron systems. In this study, we explore a doping approach by which the coexistence between delocalized and localized electronic states may be precisely tuned within extrinsic polaron systems. Through comprehensive ab initio calculations employing hybrid functionals, for which Ti-doped ${\mathrm{SnO}}_2$ is selected as a model system, it is demonstrated how strain and alloying can be utilized to achieve a high degree of control over the activation energetics separating localized and delocalized electronic states. By means of a phenomenological tight-binding analysis of our ab initio results, it is shown that two physical parameters play a dominant role in the associated transition physics: (1) the dopant electronic offset ${\mathrm{\Delta }}_a$, and (2) the dopant bandwidth $W$. Overall, this study presents an exciting route toward tailoring the coexistence between delocalized and localized states with a potentially much greater degree of control than is currently possible in intrinsic materials.

Figure 1

Illustration of charge trapping and defect polaron formation (carrier self-trapping) mechanisms. In regime I, shown in (a), a carrier is trapped by a defect center with no activation barrier ($E_a$) being present. Regime I is also illustrated in single-particle picture (${\varepsilon }_d$) on the left of (d). Subsequently in regime II, shown in (b), a defect polaron forms with an activation barrier $E_a$ and a negative trapping energy $E^f$. Regime ${\mathrm{II}}_{\text{o}}$ denotes the starting point of regime II, as shown in (e), a special case in which an electron polaron forms almost without an activation barrier directly from the conduction band (${\varepsilon }_C$) of a material as shown in the single-particle picture (${\varepsilon }_p$) given by (d). Finally, in regime III we see in (c) that the defect polaron $E^f$ becomes positive and $E_a$ increases. The distinction between all the cases presented depends on the final change in the electronic relaxation energy $\mathrm{\Delta }{E}_{el}$. The total energy ($E$) summary of these activation ($E_a$) and formation ($E^f$) energy trends is provided in (e). The transition from regimes III through to II and then to I can be accomplished via the strain tuning of the electronic offset ${\mathrm{\Delta }}_a$ [shown in blue in (d) and (e)]. On the other hand, the tuning from case III through II and finally to $\sim {\mathrm{II}}_{\text{o}}$ can be accomplished via alloying of a polaron defect at successively higher concentrations by engineering both ${\mathrm{\Delta }}_a$ and electron kinetic energy contributions to $\mathrm{\Delta }{E}_{el}$ (measured through the impurity bandwidth $W$). This is also shown in blue in (d) and (e).

Figure 2

The rational design of “defect polarons” providing bistability (or coexistence) between delocalized and localized state. The delocalized states are provided by $s$-like orbitals within metal oxides as shown in (a). Localized states are provided transition metal ions with a partially filled $d$ shell when oxidized as shown in (b). The key criteria for engineering coexistence are summarized in (c), where it can be seen that the electron reorganization energy ($U_r$) of the defect polaron must be greater than the “electronic offset” of the defect polaron state (${\mathrm{\Delta }}_a$) and both should be positive ($U_r>{\mathrm{\Delta }}_a>0$). As well the conduction band (CB) of the host lattice should have a low effective mass, such that it is not prone to intrinsic polaron formation. (d) A dopant metal cation with lower electronegativity (${\chi }_{\text{dopant}}$) than that of the host lattice (${\chi }_{\text{host}}$) may also aid the design of a defect polaron. This study is focused on exploring the model “defect polaron” system composed of ${\mathrm{SnO}}_2$ doped with Ti.

Figure 3

(a) A one-dimensional model whose unit cell contains a single-defect $B$ atom and associated parameters. (b) A one-dimensional mode whose unit cell contains $A$ and $B$ atoms and a lattice perturbation $x_m>0$ is applied to one of the $B$ atoms. The matrix elements of the on-site potentials (${\varepsilon }_A$ and ${\varepsilon }_B$) and transfer integral to the nearest and/or the next-nearest neighbors ($V_A$, $V_B$, and $V_{AB}$) are illustrated with arrows. Note that transfer integral $V_{AB}$ may vary within or between unit cells, but for the sake of brevity we use the same value $V_{AB}$ for all transfer integral between atoms $A$ and $B$.

Figure 4

(a) Electronic structure PDOS of neutral Ti-doped ${\mathrm{SnO}}_2$, in the absence of an extra electron. The contribution from metal cation in the host material cation (Sn), the dopant element (Ti), and oxygen atoms (O) are denoted in blue, green, and red, respectively. For visualization purposes, the PDOS on the dopant Ti atom is scaled up by a factor of 10. Formation energy $E^f$ from the delocalized configuration to the fully localized configuration is shown in (b). Band-decomposed (yellow) charge density of a delocalized added extra electron in Ti-doped ${\mathrm{SnO}}_2$ is provided in (c) at an isosurface value of 0.0005 $e$/Å, while the charge density upon formation of a defect polaron is provided in (d) at an isosurface value of $0.005e$/Å${}^3$. The blue, red, and green atoms represent host metal Sn atoms, O atoms, and dopant Ti atoms, respectively.

Figure 5

Ab initio calculations of coexisting states in Ti-doped ${\mathrm{SnO}}_2$ by strain tuning. For various values of compressive strain in the range 0% to 4%: (a) $E^f$ and its decomposed (b) $E_{lat}$ and (c) $E_{el}$ contributions as a function of lattice distortion around the ${\mathrm{MO}}_6$ defect center. ${\mathrm{MO}}_6$ represents the average metal-oxygen bond length within the octahedron hosting the defect center with a localized electron. (d) Ab initio and analytical results of $\mathrm{\Delta }{E}^f$ and $E_a$ as a function of lattice compression. (e) Fit between the defect polaron tight-binding model and ab initio results corresponding to 3% compressive strain (superimposed in green). The analytical $E_{el}$ is also shown in dotted blue line. (f, i–x) PDOS of Ti-doped ${\mathrm{SnO}}_2$ under various compressive and tensile strain conditions ranging from 4% compression to 5% tension. For visualization purposes, the PDOS on the dopant Ti is scaled up by a factor of 10.

Figure 6

Ab initio calculations of coexisting states in ${\mathrm{Ti}}_x{\mathrm{Sn}}_{1-x}{\mathrm{O}}_2$ at various concentrations. For various compositions of $x$ in the range from 0 to 1, (a) $E^f$ and its decomposed (b) $E_{lat}$ and (c) $E_{el}$ as a function of lattice distortions around the localized electron ${\mathrm{MO}}_6$. ${\mathrm{MO}}_6$ represents the average metal-oxygen bond length within the ${\mathrm{MO}}_6$ octahedron with a localized electron. Compositional plots as a function of Ti concentration are provided in (d) of $E^f$ and $E_a$, of lattice parameters in (e), and of the band gaps in (f). (g, i–x) PDOS plots of ${\mathrm{Ti}}_x{\mathrm{Sn}}_{1-x}{\mathrm{O}}_2$ under various compositions of $x$ with respect to the Fermi energy ($E_{\text{F}}$). (h, i–v) Effective band structures of ${\mathrm{Ti}}_x{\mathrm{Sn}}_{1-x}{\mathrm{O}}_2$ with $x=0.014$, 0.056, 0.333, 0.667, and 1.000. The contributions of Sn, Ti, and O atoms to each effective band structure are shown in blue, green, and red, respectively.

Figure 7

Effects of the defect bandwidth on coexistence properties in a two-band tight-binding model. The properties of a polaron forming defect band $B$ in “isolation” are shown in (a, i–iii), with the conduction band minimum of band $B$ placed well below that of band $A$ (${\varepsilon }_{C,A}\gg {\varepsilon }_{C,B}$). A polaron state (${\varepsilon }_p$) forms on site $x_m$ in (a, i) leading to the formation ($E^f$) and electronic relaxation ($E_{el}$) trends in (a, ii) and (a, iii). The properties of a defect polaron forming from a defect band $B$ situated above and then gradually below a host lattice band $B$ (b, i–iii). A defect polaron state (${\varepsilon }_{dp}$) forms on site $x_m$ in (b, i) on the sublattice of $B$ leading to the formation ($E^f$) and electronic relaxation ($E_{el}$) trends in (b, ii) and (b, iii). Each colored line plotted corresponds to a different intersite coupling magnitude ($V_B$) within band $B$. The widths of band $A$ ($W_A$) and band $B$ ($W_B$) are indicated in (a, i) and (b, i). The offset between the band edges prior to any lattice deformation is indicated as ${\mathrm{\Delta }}_a={\varepsilon }_{C,A}-{\varepsilon }_{C,B}$ in (b, ii). For comparative purposes, all results in (a) and (b) are shifted to 0 eV in the absence of any lattice distortion.