#
Electron and Hole Mobility of SnO_{2} from Full-Band Electron–Phonon and Ionized Impurity Scattering Computations

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}, which is extensively studied as a practical transparent oxide n-type semiconductor. In experiments, the mobility of electrons in bulk SnO

_{2}single crystals varies from 70 to 260 cm

^{2}V

^{−1}s

^{−1}at room temperature. Here, we calculate the mobility as limited by electron–phonon and ionized impurity scattering by coupling the Boltzmann transport equation with density functional theory electronic structures. The linearized Boltzmann transport equation is solved numerically beyond the commonly employed constant relaxation-time approximation by taking into account all energy and momentum dependencies of the scattering rates. Acoustic deformation potential and polar optical phonons are considered for electron–phonon scattering, where polar optical phonon scattering is found to be the main factor which determines the mobility of both electrons and holes at room temperature. The calculated phonon-limited electron mobility is found to be 265 cm

^{2}V

^{−1}s

^{−1}, whereas that of holes is found to be 7.6 cm

^{2}V

^{−1}s

^{−1}. We present the mobility as a function of the carrier concentration, which shows the upper mobility limit. The large difference between the mobilities of n-type and p-type SnO

_{2}is a result of the different effective masses between electrons and holes.

## 1. Introduction

_{2}) is a critically important n-type semiconductor with a relatively high mobility and a wide band gap (${E}_{\mathrm{g}}$ = 3.6–3.7 eV) [1,2]. Due to its good electrical, optical, and electrochemical properties, SnO

_{2}has been extensively exploited in various state-of-the-art applications: perovskite solar cells [3], as both compact layers and mesoporous layers for transparent electrodes; lithium-ion batteries [4], as promising candidates to serve as the anode material due to their high theoretical capacity; gas sensors [5], as the most commonly used commercial material [6]; photocatalytic applications [7], as photocatalysts in organic pollutant degradation, water splitting, Cr(VI) reduction, CO

_{2}reduction, air purification, and photocatalytic sterilization; thermoelectric materials [8], as ceramic thermocouples to replace noble-metal thermocouples that are unable to withstand the harsh environments inside the hot sections of turbine engines used for power generation and propulsion.

_{2}single crystals vary from 70 to 260 cm

^{2}V

^{−1}s

^{−1}at room temperature [9,10,11], while SnO

_{2}thin films show lower electron mobilities from 25 to 130 cm

^{2}V

^{−1}s

^{−1}[12,13,14,15,16]. The large variation is a result of the many carrier scattering processes that take place beyond the intrinsic electron–phonon, such as scattering by ionized impurities [17], neutral impurities [18], grain boundaries [19], and dislocations [20]. To properly evaluate the intrinsic mobility of the material, as well as that of the doped material, we need to calculate its electronic transport using full band electronic structure details, but also consider scattering processes that include the entire energy and momentum dependence of the scattering rates, beyond the constant relaxation time approximation. The latter is one of the earliest and most common approaches [21,22,23], especially in the context of high-throughput computational searches targeting electronic transport properties [24]. However, it introduces an arbitrary uncertainty upon the choice of the scattering time [25].

^{2}V

^{−1}s

^{−1}) compared to the highest reported experimental values, as well as hole mobility (14.1 cm

^{2}V

^{−1}s

^{−1}) considering only electron–phonon scattering using density functional theory calculations [28]. Other works include ionized impurity scattering using the empirical Brooks–Herring–Dingle formula, but use a fixed value (260 cm

^{2}V

^{−1}s

^{−1}) for the phonon-limited mobility of n-type SnO

_{2}[16]. For proper mobility evaluation of the doped material, it is necessary to use a full-band numerical approach to compute the intrinsic mobility for both electron–phonon and ionized impurity scattering.

_{2}. We first compute the band structures from density functional theory, from which we also extract the density of states’ effective mass and the conductivity effective mass. Then, we calculate the acoustic deformation potential and polar optical phonon scattering rates. Finally, we use those rates within the linearized Boltzmann transport equation, which is solved numerically beyond the constant relaxation-time approximation to calculate the mobility of SnO

_{2}as a function of the carrier concentration.

## 2. Computational Methods

## 3. Results and Discussions

_{2}is a Rutile structure and crystallizes in the tetragonal P4${}_{2}$/mnm space group, as shown in Figure 2a. The calculated lattice parameters are $a=b$ = 4.81 Å, c = 3.23 Å, indicating a slight 1.5% overestimation with respect to the available experimental value of $a=b$ = 4.74 Å, c = 3.19 Å [33], which is the general tendency of GGA [34]. Both the valance band maximum (VBM) and conduction band minimum (CBM) are located at the $\mathsf{\Gamma}$ point, as shown in Figure 2b. The band gap ${E}_{\mathrm{g}}$ is calculated to be 0.734 eV, lower than experimental values (${E}_{\mathrm{g}}$ = 3.6–3.7 eV) [1,2], but in good agreement with previous calculations (${E}_{\mathrm{g}}$ = 0.832 eV) using GGA [35]. There is a known problem with the underestimation of the band gap using the GGA pseudopotentials [36]. This shortage can, in principle, be overcome by using Heyd–Scuseria–Ernzerhof (HSE) hybrid functionals [37], $GW$ method [38], GGA + U method [39], meta-GGA functionals [40], or the Tran–Blaha-modified Becke–Johnson (TB-mBJ) exchange potential approximations [41]. We note, however, that since the band gap of this material is large enough, bipolar transport is suppressed, and we have considered the conduction bands and valence bands separately in the transport calculations. With regards to the accuracy of the band structure parameters, previous work has compared the band structure using GGA and TB-mBJ corrections for SnO

_{2}[35] and found very similar overall behavior of the band structures, except for the value of band gap. Thus, we take that GGA is reliable enough to describe the overall behavior of the band structures to be used in our transport calculations.

#### 3.1. Effective Mass Extraction Method

**k**points and bands in the first Brillouin zone, ${f}_{{E}_{(\mathbf{k},n)}}$ is the Fermi–Dirac distribution, and $d{V}_{\mathbf{k}}$ is the volume element in

**k**space, which usually depends only on the mesh.

#### 3.2. Scattering Rates

_{2}[28], respectively. $\rho $ is the mass density. ${v}_{\mathrm{s}}$ is the sound velocity of the material, where 4.3 km/s is used [47]. $g\left(E\right)$ is the density of states for the initial state.

_{2}, respectively. Compared to the electron–phonon scattering rates, the ionized impurity scattering rates for electrons at high impurity concentrations, e.g., at 10${}^{20}$ cm

^{−3}, are comparable to the POP scattering rates. However, for holes, even at a high impurity concentration ($1.26\times {10}^{20}$ cm

^{−3}), the ionized impurity scattering rates are still lower than the POP scattering rates. Thus, the POP will always dominate the scattering for holes in SnO

_{2}at room temperature.

#### 3.3. Mobility Calculations

^{2}V

^{−1}s

^{−1}and ${\mu}_{\mathrm{h}}$ = 7.6 cm

^{2}V

^{−1}s

^{−1}.

^{2}V

^{−1}s

^{−1}(lower compared to experiments) and ${\mu}_{\mathrm{h}}$ = 10.8 cm

^{2}V

^{−1}s

^{−1}(much higher compared to experiments) [28]. Considering the whole range of carrier concentrations, our predicted mobilities, including electron–phonon and ionized impurity scattering, are comparable to the mobilities from single crystals and are higher than those from thin films, as expected (see Figure 4) [9,10,12,13,14,15,52,53]. This can be attributed to significant carrier scattering from the grain boundaries and dislocations induced by lattice mismatch between the film and substrates such as corundum Al

_{2}O

_{3}and rutile TiO

_{2}[54,55]. On the other hand, SnO

_{2}single crystals are found to have higher mobility than the epitaxial thin films [9,10,11]. Using very thick self-buffer layers [16], SnO

_{2}epitaxial thin films on TiO

_{2}(001) substrates are also found to have high electrons mobilities, which agrees very well with our calculated mobility with both electron–phonon and ionized impurity scattering.

## 4. Conclusions

_{2}. We consider acoustic deformation potential, polar optical phonon, and ionized impurity scattering processes. Both electron and hole mobilities are found to be predominantly limited by the polar optical phonon scattering at room temperature. The calculated effective masses of electrons and holes are directly related to the difference in mobilities observed between n-type and p-type SnO

_{2}. The mobilities, as a function of the carrier concentration, show an upper limit of ${\mu}_{\mathrm{e}}$ = 265 cm

^{2}V

^{−1}s

^{−1}and ${\mu}_{\mathrm{h}}$ = 7.6 cm

^{2}V

^{−1}s

^{−1}, which agrees well with previous experimental values, at least for the n-type SnO

_{2}.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Total energy versus

**k**meshes along the x and y directions, where the

**k**mesh along the z direction is set to 10. (

**b**) Total energy versus cutoff energy.

**Figure 2.**(

**a**) Lattice structure for SnO

_{2}. (

**b**) Band structure for SnO

_{2}along high-symmetry lines. (

**c**) Fermi surface at energy E = 0.2 eV above the conduction band minimum (CBM). (

**d**) Fermi surface at energy E = 0.2 eV below the valance band maximum (VBM).

**Figure 3.**Scattering rates arising from polar optical phonon (POP) and acoustic deformation potential (ADP) scattering processes in SnO

_{2}for (

**a**) electrons and (

**b**) holes at 300 K. Ionized impurity scattering rates in SnO

_{2}for (

**c**) electrons and (

**d**) holes at different impurity concentrations. The conduction band minimum is set to zero eV in (

**a**,

**c**), while the valence band maximum is set to zero eV in (

**b**,

**d**).

**Figure 4.**The calculated mobility versus carrier concentration for (

**a**) electrons and (

**b**) holes in SnO

_{2}. Phonon-limited (solid lines) and phonon plus ionized impurity scattering (dotted lines) are shown. Experimental measurements from single crystals (s.c., green triangles) and thin films (black triangles) are also indicated. References for the data in (

**a**): single crystals (refs. [9,10]) and thin films (refs. [12,13,14,15,16]) for electrons. References for the data in (

**b**): films (refs. [52,53]) for holes.

**Figure 5.**Calculated (

**a**) transport distribution functions, (

**b**) band velocities, (

**c**) density of states, and (

**d**) carrier concentrations for electrons and holes in SnO

_{2}. In (

**a**), the transport distribution functions are averaged from ${\mathsf{\Xi}}_{xx}$, ${\mathsf{\Xi}}_{yy}$, and ${\mathsf{\Xi}}_{zz}$. The inset of (

**b**) shows the different $|\mathbf{k}-{\mathbf{k}}^{\prime}{|}^{2}$ for the heavy band and light band.

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**MDPI and ACS Style**

Li, Z.; Graziosi, P.; Neophytou, N.
Electron and Hole Mobility of SnO_{2} from Full-Band Electron–Phonon and Ionized Impurity Scattering Computations. *Crystals* **2022**, *12*, 1591.
https://doi.org/10.3390/cryst12111591

**AMA Style**

Li Z, Graziosi P, Neophytou N.
Electron and Hole Mobility of SnO_{2} from Full-Band Electron–Phonon and Ionized Impurity Scattering Computations. *Crystals*. 2022; 12(11):1591.
https://doi.org/10.3390/cryst12111591

**Chicago/Turabian Style**

Li, Zhen, Patrizio Graziosi, and Neophytos Neophytou.
2022. "Electron and Hole Mobility of SnO_{2} from Full-Band Electron–Phonon and Ionized Impurity Scattering Computations" *Crystals* 12, no. 11: 1591.
https://doi.org/10.3390/cryst12111591