The work function is one of the most fundamental surface properties of a material, and understanding and controlling its value is of central importance for manipulating electron flow in applications ranging from high power vacuum electronics to oxide electronics and solar cells. Recent computational studies using Density Functional Theory (DFT) have demonstrated that DFT-calculated work function values for metals tend to agree well (within about 0.3 eV on average) with experimental values. However, a detailed validation of DFT-calculated work functions for oxide materials has not been conducted and is challenging due to the complex dipole structures that can occur on oxide surfaces. In this work, we have focused our investigation on the widely studied perovskite SrTiO_{3} as a case study example. We find that DFT can accurately predict the work function values of clean and reconstructed SrTiO_{3} surfaces vs experiment at about the same level of accuracy as metals when direct comparisons can be made. Furthermore, to aid in understanding the factors governing the work function of oxides, we have performed systematic studies on the influence of common surface features, including surface point defects, doping, adsorbates, reconstructions, and surface steps, on the work function. The relationships between the surface structure and work function for SrTiO_{3} identified here may be qualitatively applicable to other complex oxide materials.

## I. INTRODUCTION

The work function, which is a fundamental electronic property of a material surface, is defined as the energy required to move an electron from the Fermi level to the vacuum level.^{1} The work function is a crucial parameter for the design of materials in numerous applications. For example, for electron emission materials used in high frequency and high power traveling wave tubes,^{2} klystrons,^{3} gyrotrons,^{4} and magnetrons,^{5} a low work function is desired to reach high current density at lower temperature.^{6–8} Materials with low work function are also of central importance for electron emitters used in electron microscopes, electron beam lithography, and electron sources to generate x rays for many types of imaging and medical diagnostic equipment.^{7,9} The work function governs the electronic band offsets between different materials and can control the behavior of electrons across interfaces.^{10} Controlling electronic transport across interfaces is important for engineering electrocatalytic materials,^{11} solid-state Schottky or Ohmic junctions,^{12,13} and efficient charge transport in oxide electronics and solar cells.^{14}

The work function values of pure metals have been well-studied^{15} both experimentally and computationally. Experimentally, the work functions of single- and poly-crystalline metals have been measured and tabulated. For example, the classic work from Michaelson^{15} provides a database of metal work functions collected from many studies and sheds light on the range of work function of a particular metal as a function of surface orientation, showing that for metals, the work functions of different crystalline faces vary by up to about half an eV.^{16} Historically, Density Functional Theory (DFT) has been the computational method of choice to calculate the work function of materials. Recent DFT studies from De Waele *et al.*^{17} and Tran *et al.*^{18} have sought to benchmark the accuracy of DFT methods for calculating the work functions of metals. De Waele *et al.* found that the mean absolute error (MAE) between DFT and experiment for metals was 0.31 eV,^{17} while Tran *et al.* found an MAE value of 0.24 eV.^{18} These comprehensive studies demonstrate that DFT and experimental work functions of metals are broadly in good agreement. In addition, more targeted DFT studies of tungsten with various surface orientations and adsorbate types by Vlahos *et al.*,^{19} Jacobs *et al.*,^{20} and Zhou *et al.*^{21} demonstrated qualitative agreement between calculated DFT work functions and experimentally derived work functions from thermionic emission experiments. As an example of a non-elemental metal, comparisons of the DFT-calculated work functions with experimental values for lanthanum hexaboride (LaB_{6}), a common electron emission material used in electron microscopes and lithography equipment, have also been made and again showed typical errors of about 0.3 eV^{22,23} for different surface orientations.

Contrary to the generally weak dependence of the work function on surface orientation for metals, for oxides, the work function can show much stronger dependence on surface orientation and termination.^{24,25} For example, while the work functions of the two metal orientations may vary by a couple hundred meV,^{15} Jacobs *et al.*^{24} and Zhong and Hansmann^{25} found that the work function difference between the AO- and BO_{2}-terminated (001) perovskite surfaces is predicted to be on the scale of a few eV.^{24,25} This large difference is the result of different perovskite surfaces exhibiting large changes in polarity, which has a direct and large effect on the electronic surface dipole and, thus, the work function. In addition to the effect of various surface terminations and orientations on oxide work functions, surface adsorbates, point defects, interfacial capping layers,^{25} surface reconstructions,^{26–30} and bulk doping or composition modification resulting in shifts to the Fermi level can also change the work function. In particular, the occurrences of adsorbates, surface defects, and reconstructions are environment- and processing-dependent. This dependence between processing and resulting structure can make the one-to-one comparison between simulation and experiment difficult as experimental knowledge of the atomic-level surface structure and corresponding work function for the same surface are not commonly reported in a single study.

There are challenges associated with comparing experimental and DFT-calculated work functions for oxides. As an example, Hong *et al.* reported the work functions of polycrystalline perovskite materials LaMO_{3} (M = Cr, Mn, Fe, Co, Ni) from x-ray photoelectron spectroscopy (XPS) measurements.^{31} While the work functions of these polycrystals followed the same trend in composition as found from DFT, the quantitative values differed from the DFT values on the scale of an eV or more.^{24} The source of such discrepancy is not rigorously known but could be due to the DFT values being reported for clean (001) AO- and BO_{2}-terminated surfaces, whereas the experimental samples likely contained adsorbed surface species as they were calcined in air and may have contained a number of surface orientations, terminations, or reconstructions, as the samples were polycrystalline and the atomic-level surface structure was not characterized. Similar challenges have made quantitative work function comparisons between experiment and DFT on oxides very limited and uncertain. Recently, Chambers and Sushko have combined photoemission experiments and DFT calculations to investigate the work function and surface chemistry of SrO- and TiO_{2}-terminated Nb:SrTiO_{3} (001) surfaces. They found good agreement between DFT-calculated and experimentally measured work function for clean TiO_{2}-terminated Nb:SrTiO_{3} (001).^{32} The authors also found excellent agreement with their calculated work function of 3.6 eV–3.7 eV for SrO-terminated Nb:SrTiO_{3} with 25% Sr vacancies and other surface adsorbates, and their measured work function of 3.6 eV.

The previous works of Jacobs *et al.*^{24} and Zhong and Hansmann^{25} both reported DFT-calculated work functions of SrTiO_{3} (STO). The work of Jacobs *et al.* performed Heyd–Scuseria–Ernzerhof (HSE)-level calculations of clean SrO- and TiO_{2}-terminated pure SrTiO_{3} and reported work functions without shifting the Fermi level to reflect finite-temperature or doping effects. Thus, the reported values for SrTiO_{3} from Jacobs *et al.* are effectively ionization potentials. The work of Zhong and Hansmann used Generalized Gradient Approximation of Perdew–Burke–Ernzerhof (GGA-PBE) calculations and shifted the Fermi level to reside at the conduction band minimum (CBM). This shift to the Fermi level was performed to mimic *n*-type doping of Nb into SrTiO_{3}, so the work functions reported by Zhong and Hansmann most closely reflect the work functions of Nb:SrTiO_{3}, instead of pure SrTiO_{3}. While both of these previous studies made attempts to compare their calculated SrTiO_{3} work functions with selected experimental values, it is not clear from these studies alone what the best practices are for using DFT methods to obtain work functions that agree with the experimental values.

Given the widespread importance of understanding and controlling the work function of oxide materials, in this study, we performed a detailed study of SrTiO_{3} as a first step toward validating DFT-calculated work functions of complex oxides and understanding the scale of the effect of common surface structure features (e.g., steps, defects, and reconstructions) on the work function. We have used SrTiO_{3} as a case study example material because it has been extensively studied, thus making available prior experimental work on both the atomic-level surface structure characterization and corresponding work function^{33,34} to enable one-to-one validation. In addition, the work function of SrTiO_{3} is of interest in its own right, as it is a ubiquitous substrate material for complex oxide thin film growth used for a variety of applications.

## II. METHODS

All calculations were performed with DFT using the *Vienna Ab Initio Simulation Package* (VASP) software using either the Generalized Gradient Approximation of Perdew–Burke–Ernzerhof (GGA-PBE)^{35} functional or the Heyd–Scuseria–Ernzerhof (HSE06)^{36} hybrid functional with 25% exact exchange and an inverse screening length of 0.2 Å^{−1}. The cut-off energy for the plane wave basis set was 500 eV. The convergence criterion of ionic steps is 10^{−3} eV per supercell total energy change. All calculations were performed with spin polarization. To calculate the work function, we used symmetric 9-layer surface slabs separated by at least a 20 Å of vacuum following previous work.^{24} Unreconstructed surface slabs had a cross-sectional area of 2*a*_{0} × 2*a*_{0} [*a*_{0} is the bulk lattice constant, *a*_{0} = 3.909 Å (3.942 Å) for HSE (GGA) methods, and the discrepancy with the experimental value *a*_{0} = 3.905 Å is less than 1%], and a gamma-centered 2 × 2 × 1 *k*-point mesh was used. For some simulations, different surface slab sizes were needed in order to represent a particular surface reconstruction, surface defect concentration, or Nb doping concentration. These simulations used either a 4*a*_{0} × 2*a*_{0} or 4*a*_{0} × 4*a*_{0} cross-sectional area, and the corresponding *k*-point meshes were gamma-centered 1 × 2 × 1 and 1 × 1 × 1, respectively. In all cases, the top three surface layers were relaxed, as well as any defected or adsorbed surface atoms. Tests with more *k*-points and larger slabs suggest that work function errors associated with *k*-point sampling and finite slab thickness and layer relaxation are less than 0.1 eV. The surface vacancies and adsorbates were placed on the surface as far from each other as possible. For Nb-doped STO, Nb dopants were substituted for Ti in the middle of the slab to eliminate possible influence on the surface dipole and to mimic a bulk Nb-doped SrTiO_{3} material. To keep the simulated cell size practical for the HSE-level simulations, the Nb dopant concentrations used in our simulations were *x*_{Nb} = 5 at. % (6.25 at. %) for TiO_{2}-terminated (SrO-terminated) surface, and were thus higher than typical experimental values of *x*_{Nb} = 1 at. %.^{33,37–40} We found that varying Nb concentrations in the range of *x*_{Nb} = 1.3 at. %–12.5 at. % had a small effect on the work function, within about 0.1 eV (see the supplementary material, Sec. S3, for more information). All relaxed structures and VASP simulation files are available via Figshare. In Secs. III A and III B, the HSE functional was used to validate the DFT-calculated work functions. While it is expected that HSE is significantly more accurate than GGA-PBE (and this is consistent with our results here), it is much more computationally expensive. However, we found that GGA-PBE gives *changes* in the work function resulting from different surface structures similar to HSE. Therefore, for the studies of how surface structures alter the work function in Sec. III C, the GGA-PBE functional was used. We note here that the recent work from Chambers and Sushko^{32} using GGA-PBEsol showed work function differences between SrO- and TiO_{2}-Nb:STO quite close to our present HSE results, suggesting that work function differences between GGA-PBEsol and HSE may be similar to GGA-PBE and HSE, although more study is needed to better quantify these differences (see the supplementary material, Sec. S2, for more details).

The work function in this work refers to the energy to move an electron from a material through a specific surface to a position far enough from that surface to no longer interact with the surface but not so far that other surfaces play any role in the energy of the electron. This work function is, therefore, a function of the specific surface being considered and can be expressed as *Φ* = *E*_{vac} − *E*_{F} = *Φ*_{0} + Δ*Φ*_{dipole}. Here, *E*_{vac} is defined as the value of the converged electrostatic potential sufficiently far from the terminating surface of interest such that the restoring force on an emitted electron is negligible. *E*_{F} is the Fermi energy, *Φ*_{0} is the intrinsic work function that would be observed for a surface with no dipole, and Δ*Φ*_{dipole} is the change of the work function resulting from the surface dipole. The relationship between the surface dipole and Δ*Φ*_{dipole} is given by the Helmholtz equation.^{41,42} Since DFT simulations calculate ground state properties (*T* = 0 K), the VASP-calculated Fermi level, *E*_{F,VASP}, is always positioned at the energy of the highest occupied electronic state, which for a semiconductor like STO is the valence band maximum (VBM). For cases explored in this work where this occupancy is not correct for a given comparison or analysis, we perform a correction to shift *E*_{F,VASP} to the appropriate occupied state(s). Specifically, to compare our DFT-calculated work functions of STO with experimental values of pure and *n*-type STO, we shifted *E*_{F,VASP} from the VBM to the middle of the bandgap (to mimic finite temperature behavior) and to the conduction band minimum (CBM), respectively. In other words, to model pure and *n*-type STO from a pure STO calculation, we shift *E*_{F,VASP} by *E*_{G}/2 and *E*_{G}, respectively.

Such corrections require accurate bandgaps, and we have found that the use of HSE correctly reproduces the experimental bandgap of 3.2 eV. In contrast, the use of GGA not only underestimates the magnitude of the bandgap but places the VBM and CBM levels at different positions relative to the vacuum level, thus, affecting the work function values obtained for GGA vs HSE. More information on these calculations can be found in the supplementary material, Sec. S1. Note that we use bandgaps from the slab calculations rather than bulk as we assume that the former best represents the near-surface environment that controls the work function.

## III. RESULTS AND DISCUSSION

### A. Survey of experiments and validation of DFT-calculated work function for ideal surfaces

In this section, we aim to directly compare DFT-calculated work functions with experimental values for clean, undefected, and unreconstructed surfaces (which we call “ideal” surfaces). To aid in this and later comparisons, the work function data and corresponding surface morphologies collected from the literature are shown in Table I. It is worth noting that in some studies, the experimental work functions are reported while the surface structure is not precisely known. In this section, we restrict ourselves to using experimental work from studies where both the work function and surface structure are known, ensuring that the comparison between experimentally measured and DFT-calculated work functions is based on the same surface.

. | Surface orientation . | Experimental WF . | Measurement . | . |
---|---|---|---|---|

Material system . | and termination . | (eV) . | technique . | References . |

STO | (001) SrO | 2.7 | In situ Kelvin probe | 38 |

Nb:STO | (001) SrO | 3.86 | UPS | 43 |

Nb:STO | (001) SrO | 3.6 | UPS | 32 |

STO | (001) TiO_{2} | 4.2 | UPS | 44 |

STO | (001) TiO_{2} | 4.3 | In situ Kelvin probe | 38 |

Nb:STO | (001) TiO_{2} | 4.4 | Retarding potential | 45 |

Nb:STO | (001) TiO_{2} | 4.08 | UPS | 43 |

Nb:STO | (001) TiO_{2} | 3.8 | In situ Kelvin probe | 38 |

Nb:STO | (001) TiO_{2} | 4.0 | In situ Kelvin probe | 37 |

Nb:STO | (001) TiO_{2} | 4.3–4.8 | In situ Kelvin probe | 39 |

Nb:STO | (001) TiO_{2} | 4.7 | UPS | 32 |

STO | (001) unknown | 4.2 | UPS | 46 |

STO | (001) unknown | 3.48 | KPFM | 47 |

STO | (111) unknown | 3.4; 4.0; 5.2 | UPS | 48 |

Nb:STO | (4 × 1) reconstruction | 4.47 | XPS | 40 |

Nb:STO | (2 × 5) reconstruction | 4.05 | XPS | 40 |

Nb:STO | Polycrystalline | 4.10–4.35 | XPEEM | 49 |

Nb:STO | SrO + 14% Sr vacancies | −1.10^{a} | Kelvin force spectroscopy | 33 |

Nb:STO | TiO_{2} + 14% Sr adatoms | −1.88^{a} | Kelvin force spectroscopy | 33 |

STO | ($5$ × $5$) reconstruction | 3.12 | KPFM | 34 |

STO | (2 × 1) | n/a | n/a | 27 |

Nb:STO | (2 × 2) A | n/a | n/a | 28 |

Nb:STO | (2 × 2) C | n/a | n/a | 28 |

STO | c(4 × 2) | n/a | n/a | 26 |

. | Surface orientation . | Experimental WF . | Measurement . | . |
---|---|---|---|---|

Material system . | and termination . | (eV) . | technique . | References . |

STO | (001) SrO | 2.7 | In situ Kelvin probe | 38 |

Nb:STO | (001) SrO | 3.86 | UPS | 43 |

Nb:STO | (001) SrO | 3.6 | UPS | 32 |

STO | (001) TiO_{2} | 4.2 | UPS | 44 |

STO | (001) TiO_{2} | 4.3 | In situ Kelvin probe | 38 |

Nb:STO | (001) TiO_{2} | 4.4 | Retarding potential | 45 |

Nb:STO | (001) TiO_{2} | 4.08 | UPS | 43 |

Nb:STO | (001) TiO_{2} | 3.8 | In situ Kelvin probe | 38 |

Nb:STO | (001) TiO_{2} | 4.0 | In situ Kelvin probe | 37 |

Nb:STO | (001) TiO_{2} | 4.3–4.8 | In situ Kelvin probe | 39 |

Nb:STO | (001) TiO_{2} | 4.7 | UPS | 32 |

STO | (001) unknown | 4.2 | UPS | 46 |

STO | (001) unknown | 3.48 | KPFM | 47 |

STO | (111) unknown | 3.4; 4.0; 5.2 | UPS | 48 |

Nb:STO | (4 × 1) reconstruction | 4.47 | XPS | 40 |

Nb:STO | (2 × 5) reconstruction | 4.05 | XPS | 40 |

Nb:STO | Polycrystalline | 4.10–4.35 | XPEEM | 49 |

Nb:STO | SrO + 14% Sr vacancies | −1.10^{a} | Kelvin force spectroscopy | 33 |

Nb:STO | TiO_{2} + 14% Sr adatoms | −1.88^{a} | Kelvin force spectroscopy | 33 |

STO | ($5$ × $5$) reconstruction | 3.12 | KPFM | 34 |

STO | (2 × 1) | n/a | n/a | 27 |

Nb:STO | (2 × 2) A | n/a | n/a | 28 |

Nb:STO | (2 × 2) C | n/a | n/a | 28 |

STO | c(4 × 2) | n/a | n/a | 26 |

^{a}

These are measured local contact potential differences, not absolute work functions.

The calculated work functions of SrO- and TiO_{2}-terminated (shown in Fig. 1) pure STO from this work are 4.2 eV and 6.0 eV, respectively (i.e., Fermi level is positioned at the VBM). After correcting for thermal effects (see Sec. II), the corrected work functions for SrO- and TiO_{2}-terminated pure STO become 2.6 eV and 4.8 eV, respectively (see Table II). This correction used the DFT-calculated bandgaps of SrO- and TiO_{2}-terminated slabs, which are 3.2 eV and 2.3 eV, respectively. While the SrO-terminated slab bandgap of 3.2 eV matches the calculated (and experimental) bulk STO value, the reduced bandgap for the TiO_{2}-terminated slab is the result of the presence of surface states characterized by undercoordinated O 2*p* orbitals on the surface.^{50} This reduced bandgap for the TiO_{2}-terminated surface is consistent with numerous previous computational studies employing both semilocal and hybrid functionals.^{51–55} However, this reduced bandgap for the TiO_{2}-terminated surface is not consistent with the recent experimental work from Chambers and Sushko,^{32} where the authors found, using XPS, that the bandgap of both SrO- and TiO_{2}-terminated STO is equal to the bulk value. The reason for this discrepancy is not clear, and additional study may be warranted.

. | Experimental value . | DFT-calculated . |
---|---|---|

Material system . | (eV) . | (eV) . |

$\Phi TiO2$ = 3.8, 4.0, 4.3–4.8, 4.7 | $\Phi TiO2$ = 3.8 | |

Nb:STO | $\Delta \Phi $_{14% Sr} = −0.78 | $\Delta \Phi $_{12.5% Sr} = −0.9 |

Φ_{(4 × 1)} = 4.47 | Φ_{(4 × 1)} = 4.5 | |

Φ_{SrO} = 2.7 | Φ_{SrO} = 2.6 | |

Pure STO | $\Phi TiO2$ = 4.3 | $\Phi TiO2$ = 4.8 |

Φ_{($5$ × $5$)} = 3.12 | Φ_{($5$ × $5$)} = 3.2 |

. | Experimental value . | DFT-calculated . |
---|---|---|

Material system . | (eV) . | (eV) . |

$\Phi TiO2$ = 3.8, 4.0, 4.3–4.8, 4.7 | $\Phi TiO2$ = 3.8 | |

Nb:STO | $\Delta \Phi $_{14% Sr} = −0.78 | $\Delta \Phi $_{12.5% Sr} = −0.9 |

Φ_{(4 × 1)} = 4.47 | Φ_{(4 × 1)} = 4.5 | |

Φ_{SrO} = 2.7 | Φ_{SrO} = 2.6 | |

Pure STO | $\Phi TiO2$ = 4.3 | $\Phi TiO2$ = 4.8 |

Φ_{($5$ × $5$)} = 3.12 | Φ_{($5$ × $5$)} = 3.2 |

From Table I, by using *in situ* Kelvin probe measurements, Susaki *et al.* reported the experimental work function values of SrO- and TiO_{2}-terminated pure STO to be 2.7 eV and 4.3 eV,^{38} respectively. The authors used pulsed laser deposition to grow STO thin films with different terminations, followed by annealing in oxygen atmosphere to eliminate oxygen vacancies. Our DFT-calculated SrO termination work function of 2.6 eV matches extremely well with the experimental value, while our DFT-calculated TiO_{2} termination work function of 4.8 eV is half an eV higher than the experimental value, resulting in a mean absolute error of 0.3 eV for pure STO.

For Nb-doped STO (Nb:STO), the calculated work functions of SrO and TiO_{2} terminations are 1.2 eV and 3.8 eV, respectively. Examination of the calculated densities of states (DOS) of Nb:STO showed that Nb degenerately *n*-type dopes STO, causing the Fermi level to reside at the bottom of the conduction band, which is consistent with the experimental behavior of Nb doping in STO (see Fig. 2). Note that these calculated work functions are quantitatively consistent with the impact of Nb being almost entirely to shift the Fermi level and not alter the surface dipoles. Adding the calculated bandgap of 3.2 eV to the calculated SrO-terminated Nb:STO work function of 1.2 eV results in an ionization potential of 4.4 eV, in good agreement with our calculated pure STO ionization potential of 4.2 eV. Similarly for the TiO_{2} termination, adding the bandgap of 2.3 eV to the calculated Nb:STO work function of 3.8 eV results in 6.1 eV, essentially the same value of the pure STO work function of 6.0 eV. Overall, it is clear that doping can have a large effect on the resulting work function, with the case of Nb doping resulting in reduction of the work function from the approximately half-band-gap upshift of the Fermi level. We note here that the presence of dopant species near the surface can also affect the work function by altering the surface dipole; however, we have not treated surface doping here as our Nb is placed in the slab center (see Sec. II). As might be expected from its simple role in shifting the Fermi level, we found there is little dependence of work function on Nb concentration (i.e., about 0.1 eV at most, see the supplementary material, Sec. S3, for more information).

For Nb:STO, the Nb dopant concentrations in experiments are equal to or lower than *x*_{Nb} = 1 at. %, and the measured work functions of TiO_{2}-terminated Nb:STO are 3.8 eV,^{38} 4.0 eV,^{37} 4.3 eV–4.8 eV,^{39} and 4.7 eV,^{32} resulting in a range of 1 eV. It is unclear at this time why the experimental range for this surface is ∼1 eV. Our calculated work function of TiO_{2}-terminated Nb:STO is 3.8 eV (see Table II), residing at the lower bound of these experimental values, and in good agreement with the reported values of 3.8 eV^{38} and 4.0 eV.^{37} We note that Chambers and Sushko obtained 4.7 eV from their GGA calculations, in excellent agreement with their measured values of 4.7 eV.^{32} The large difference between their DFT value of 4.7 eV and our value of 3.8 eV is first due to our use of HSE vs their use of GGA. When we use GGA, we obtain a work function for TiO_{2}-terminated Nb:STO of 4.3 eV, much closer to their value (see the supplementary material, Sec. S2). The remaining differences are likely due to our use of GGA-PBE vs their use of GGA-PBEsol functionals, our use of Nb doping vs their use of extra electron doping, and minor numerical issues in the calculation. Although for this termination, HSE and GGA results both reside in the experimental range (with HSE at the lower bound and GGA at the upper bound), we also checked ideally truncated pure STO and other selected reconstructions and found that HSE is more accurate in general (see the supplementary material, Sec. S1).

To our knowledge, no experimental work function data have been reported for a clean and undefected SrO-terminated Nb:STO. However, Chambers and Sushko identified that the SrO-terminated Nb:STO had 25% surface Sr vacancies by using XPS Ti 2*p* and Sr 3*d* core-level intensities and measured the corresponding work function to be 3.6 eV.^{32} We, therefore, compare our results to this defected SrO-terminated surface. Although we did not perform HSE calculations on this specific surface model, we did explore the influence of surface defects on the work function (see Sec. III C 1 and Fig. 3), which we can use here to make an estimation of the work function using our methods. Introducing 25% Sr vacancies increases the work function by about 1.0 eV for the SrO termination, and adding this correction to the 2.6 eV work function of SrO-terminated pure STO results in a work function of 3.6 eV, in excellent agreement with the work of Chambers and Sushko. This estimation was made using pure STO instead of Nb:STO because, in the surface region under XPS investigation, it is likely that the high concentration of 25% Sr vacancies will deplete the extra electrons introduced by Nb, which are present in a concentration of about 1%.

Overall, in the case of ideal STO and Nb:STO surfaces, our DFT calculated work functions of surfaces agree well with the lower bound of available experimental data, with expected DFT errors vs experiment on the scale of a couple hundred meV. In addition, we can compare the DFT-calculated work functions of this work and previous DFT calculations from Jacobs *et al.* and Zhong and Hansmann for SrO- and TiO_{2}-terminated STO. Zhong and Hansmann used DFT-GGA calculations on STO, and shifted their Fermi level to reside at the CBM, thus mimicking the behavior of Nb:STO. The comparison of their SrO- and TiO_{2}-terminated work functions (1.9 eV and 4.5 eV, respectively) with our DFT-GGA values of 1.9 eV and 4.3 eV shows very good agreement. Jacobs *et al.* used DFT-HSE calculations on STO and did not shift their Fermi levels (i.e., their Fermi levels were at the VBM). The comparison of their SrO- and TiO_{2}-terminated work functions (3.2 eV and 6.3 eV, respectively) with our DFT-HSE values (unshifted) of 4.2 eV and 6.0 eV shows good agreement for the TiO_{2}-terminated surface and poor agreement for the SrO-terminated surface. We believe that the reason for the discrepancy of the SrO-terminated values is because the previous study of Jacobs *et al.* used only a single *k*-point. We, thus, believe that the values reported in the present work are more accurate than those reported in Jacobs *et al.*

Finally, it is worth noting that if we use the experimental value for the TiO_{2}-terminated pure STO work function of 4.3 eV^{38} and assume that Nb reduces the work function by approximately half the bandgap (consistent with our calculations and simple intuitions), then, the expected work functions for TiO_{2}-terminated Nb:STO is 3.1 eV (using our TiO_{2}-terminated slab bandgap) or 2.7 eV (using the experimental or our calculated bulk bandgap). Both of these values are at least 0.7 eV below the lowest measured value for the TiO_{2}-terminated Nb:STO work function. This suggests that the 4.3 eV^{38} work function for TiO_{2}-terminated pure STO may be somewhat low. If we exclude that value from our comparison, the remaining comparisons possible for ideal surfaces including SrO-terminated pure STO and Nb:STO are within 0.1 eV.

### B. Validation of DFT-calculated work function for selected reconstructed surfaces

STO is known to undergo many different surface reconstructions as a result of different sample environments or processing paths.^{26–28,34,40} In this section, we specifically simulated ($5$ × $5$) R26.6° reconstruction on the (001) surface,^{34} (4 × 1) reconstruction on the (011) surface,^{40} and SrO (TiO_{2}) termination with 14% Sr vacancies (adatoms)^{33} at the HSE level in order to compare with the results from the recent studies where both atomic scale surface characterization and work function measurement of such reconstructed surfaces were available.

Wrana *et al.*^{34} annealed STO at 1150 °C in a low oxygen partial pressure for 1 h until the ($5$ × $5$) R26.6° reconstruction was exposed. A work function of 3.12 eV was measured by the Kelvin probe force microscopy (KPFM). There are two alternative explanations for the formation of the ($5$ × $5$) R26.6° reconstruction, both of which occur on the (001) TiO_{2} termination. The first explanation is that an extra Sr-adlayer contributes to the reconstruction,^{30} and the second explanation involves a Ti-adlayer with a composition of either TiO_{2}–Ti_{3/5} or TiO_{2}–Ti_{4/5}.^{29} A recent XPS study revealed that the ($5$ × $5$) R26.6° reconstructed surface is Ti-enriched,^{47} indicating that the latter explanation above is more likely. Therefore, we simulated TiO_{2}–Ti_{3/5} and TiO_{2}–Ti_{4/5} models, whose detailed structures are shown in Fig. 1.

From our DFT simulations, TiO_{2}–Ti_{3/5}- and TiO_{2}–Ti_{4/5}-terminated slabs are both metallic, which agrees with experimental conductivity measurements from Wrana *et al.* The DFT-calculated work functions of TiO_{2}–Ti_{3/5} and TiO_{2}–Ti_{4/5} terminations are 3.5 eV and 3.2 eV, respectively (see Table II). These calculated values are slightly higher than the experimental work function of 3.12 eV and agree to the same level of error as the pure surfaces discussed above. However, in the experiment, there were sparsely distributed Sr adatoms or O vacancies^{34} at a concentration of about 1%, in addition to the known reconstructed surface structure. We can approximate that Sr adatoms (O vacancies) at this concentration would lower the work function by about 0.1 eV (0.05 eV) (see Sec. III C), resulting in DFT errors vs experiment in the range of roughly 0 eV (for TiO_{2}–Ti_{4/5}) to 0.3 eV (for TiO_{2}–Ti_{3/5}), once again indicating good agreement.

Riva *et al.*^{40} obtained a reconstructed (4 × 1) surface on 1 at. % Nb:STO (011) surface and measured the work function to be 4.47 eV from XPS. Such a (4 × 1) surface consists of ten- and six-membered rings of corner-sharing TiO_{2} tetrahedra,^{56} as shown in Fig. 1. Our DFT-calculated work function of this (4 × 1) reconstruction on the (011) surface is 4.7 eV (see Table II). The small discrepancy of about 0.2 eV with the experimental value is again consistent with our previous comparisons. Furthermore, the discrepancy could be at least in part due to the presence of Sr adatoms,^{57} which occupy surface domain boundaries and appear every few nanometers.

Sokolović *et al.* have recently cleaved 1 at. % Nb:STO and obtained pristine (001) surfaces through a novel procedure.^{33} STM/AFM measurements confirmed the presence of a reconstruction consisting of (14 ± 2)% Sr vacancies on the SrO termination and complementary Sr adatoms on the TiO_{2} termination at the same concentration. Using KPFM, the work function of the SrO (TiO_{2}) termination was measured to be 1.10 eV (1.88 eV) lower than that of the reference tip, resulting in a work function difference between the two terminations of $\Delta \Phi TiO2\u2212SrO=\u22121.88\u2009eV\u2212\u22121.10\u2009eV=\u22120.78\u2009eV$. It is interesting to note that typically the TiO_{2} termination of STO has a higher work function; however, here, due to the presence of the reconstruction, the trend is reversed and the TiO_{2} termination work function is 0.8 eV lower than that of SrO termination. This trend reversal is likely due to the Sr adatoms on the TiO_{2} surface functioning as a layer of electropositive species, thus, creating a positive surface dipole resulting in a work function lower than the SrO surface, whose work function is raised due to the Sr vacancies creating a more negative surface dipole.

We simulated both of these surface reconstructions with a slightly reduced concentration of Sr adatoms/Sr vacancies of 12.5%, which was necessary due to supercell size restrictions. From these simulations, the TiO_{2} termination with 12.5% Sr adatoms is metallic and has a work function of 2.0 eV and the SrO termination with 12.5% Sr vacancies is insulating, which is consistent with the STS measurement in the experiments of Sokolović *et al.*^{33} By setting the Fermi level to the middle of the bandgap, the work function is 2.9 eV = 4.5 eV–3.2 eV/2 (where 4.5 eV is the directly calculated work function and 3.2 eV is the bandgap for the SrO-terminated cell with the modeled reconstruction). The DFT-calculated work function difference is, thus, $\Delta \Phi TiO2\u2212SrO=2.0\u2009eV\u22122.9\u2009eV=\u22120.9\u2009eV$, which agrees well with the experimental results (see Table II).

Finally, there are several reconstructed surfaces, which have been structurally resolved from experiments, e.g., (2 × 1),^{27} (2 × 2),^{28} and centered (4 × 2)^{26} (see Fig. 1), but for which no experimental work functions are known. Together with the reconstructions discussed above, we can calculate the work functions of these additional reconstructions to further understand the scale of the effect of reconstructions on the work function. All of the (2 × 1), (2 × 2), and centered (4 × 2) reconstructions considered dramatically increase the work function relative to both pure STO and Nb:STO, typically on the scale of 1 eV–2 eV (see Table III). This increase in the work function is reasonable considering the extra TiO_{x} layer introduced by these reconstructions, which functions as a negative dipole layer. Zhong and Hansmann also reported the work function of (2 × 1) reconstructed Nb:STO, which they found to be 6.18 eV^{25} using the GGA-PBE functional. This value is about 0.5 eV higher than our result of 5.7 eV for Nb:STO, which we use here for comparison as the work of Zhong and Hansmann positioned the Fermi level at the conduction band minimum for SrTiO_{3} (to mimic *n*-type STO). This result of GGA-PBE producing a higher *n*-type work function than HSE is consistent with our findings of GGA-PBE vs HSE work functions of unreconstructed surfaces (see band alignment in Fig. S1 in the supplementary material, Sec. S1), where the use of GGA-PBE shrinks the bandgap, making the CBM further below the vacuum level compared to HSE.

### C. Further understanding the role of surface defects on the work function

#### 1. Role of surface vacancies and adsorbates

In this section, we seek to understand the effect on the work function of common intrinsic surface vacancies and adsorbates observed in the studies referenced in Table I, specifically, O vacancies and O adsorbates on SrO and TiO_{2} terminations, Sr vacancies on SrO terminations, and Sr adatoms on TiO_{2} terminations. Figure 3 shows the DFT-calculated work functions for SrO- [Fig. 3(a)] and TiO_{2}-terminated [Fig. 3(b)] STO as a function of the surface coverage of these defects.

First, we consider O defects. O adsorbates increase the work functions of both terminations at all concentrations likely due to the electronegative O creating a negative surface dipole, which impedes electron emission. Oxygen vacancies might *a priori* be expected to decrease work function, as the removal of a negative charge near the surface might create a positive dipole. However, O vacancies increase the work function of the SrO termination and reduce the work function of the TiO_{2} termination. The physical cause of this opposite and counterintuitive behavior of O vacancies between the SrO- and TiO_{2}-terminated work functions is not rigorously known at this time. However, we speculate that it may be the result of different relaxations in the subsurface layer beneath the defected surface between these two terminations resulting in changes to the surface dipole. This is further evidenced by the fact that the DOS of the defected surfaces showed almost no difference vs undefected surfaces, indicating that it is unlikely that doping or other changes to the electronic structure are responsible for the observed work function changes, although additional focused study beyond the present work would be required to obtain a better understanding of this effect. Zhong and Hansmann also reported the work function of STO with 25% O vacancies. Our results are consistent with Zhong and Hansmann, where they also observed opposite work function changes on SrO- and TiO_{2}-terminations due to the presence of O vacancies. Furthermore, our calculated work functions after introducing 25% O vacancies are 2.28 eV for SrO termination and 3.33 eV for TiO_{2} termination, essentially the same as the results obtained by Zhong and Hansmann, which are 2.26 eV for SrO termination and 3.39 eV for TiO_{2} termination.

Next, we consider Sr defects. Sr vacancies on the SrO termination increase the work function, which is expected as the vacancy is effectively a negative charge and increases the negative dipole at the surface. Sr adatoms on the TiO_{2} termination lower the work function, which is expected as the Sr is positively charged and makes a positive dipole on the surface. For non-interacting surface defects, one would expect a linear dependence of work function on surface coverage. However, in all cases, except for Sr vacancies, the trends with coverage are nonlinear and significantly flatten at higher coverages. This result is not surprising as such trends can involve complex coupling between dipoles through relaxation, direct electronic interaction, and electrostatics (depolarization) (see the supplementary material, Sec. S4); however, more analysis is outside the scope of this work. Overall, it is apparent that surface defects in very high concentrations like those depicted in Fig. 3 have a large effect on the work function on the order of an eV or more. However, for more typically observed O and Sr surface vacancy concentrations of a few percent, the effect on the work function is likely on the scale of a couple tenths of eV.

#### 2. Role of surface steps

Typically, STO thin films and substrates contain surface steps arising from a miscut angle during preparation or termination of thin film growth planes. Here, we seek to understand the scale of effect of (001) surface steps on the work function. From experimental observations, surface steps mostly occur on the TiO_{2} termination along $\u27e8001\u27e9$ directions, with step heights that are integer multiples of the lattice constant.^{34,40} The SrO termination either appears as a terrace on TiO_{2} termination or requires extra-long annealing times to dominate.^{58} Therefore, here, we consider two different types of surface steps on the TiO_{2} termination, denoted as O-edged TiO_{2} and Ti-edged TiO_{2} (see Fig. 4). As with surface defects, the impact of the steps on the work function will depend on their concentration, or equivalently, their separation *L*. For the small cell sizes accessible by DFT, we have to model unphysically small *L* values, and the results for two *L* values as well as a linear extrapolation to infinite *L* value are shown in Fig. 4. The GGA-level calculated work functions for ideal, O-edged, and Ti-edged TiO_{2} [with *L* = 0.39 nm (1/*L* = 2.56 nm^{−1})] are found to be 4.7 eV, 5.2 eV, and 4.3 eV, respectively. These work function changes of 0.5 eV for O-edged TiO_{2} and −0.4 eV Ti-edged TiO_{2} are an extreme upper bound of the scale of the effect of steps due to the large simulated step concentration. The lines in Fig. 4 indicate linear fits to these three data points (for each step type) with an intercept fixed at 4.65 eV, which is the work function of the ideal surface (i.e., no surface steps). The fits have high R^{2} values, suggesting a linear trend in work function with defect concentration. Assuming that the defects are effectively non-interacting, impact the work function only through dipoles, and each contribute the same surface dipole, such linear behavior is expected, since the work function change is linear to the change in the surface dipole. Thus, it is reasonable to assume that the influence of dilute surface step concentrations on a work function will be linear in the defect concentration for low concentrations, and our fit will, therefore, scale accurately to realistic surface step concentrations. Considering a more realistic step separation of *L* = 200 nm,^{34,40,58} a realistic step concentration in experiments is only about 1/500 of that in our simulations. At these physical concentrations, our results show that the surface steps will have a negligible effect on the work function, on the scale of approximately a few meV (see Fig. 4). Even if our linear model is not perfectly correct, it seems unlikely that some deviation from linearity would somehow lead to a significant work function modification for realistic step concentrations.

In addition to the defects and reconstructions of SrTiO_{3} we have discussed above, the formation of interface heterostructures can also influence the work function of perovskites by doping and modification of the surface dipole. For example, the work of Zhong and Hansmann demonstrated that interfacing SrTiO_{3} with SrNbO_{3} results in charge transfer and interactions between the surface dipoles that can result in a work function of the interface material lower than either of the parent materials can obtain in isolation.^{25} While the detailed examination of interfacial effects on the SrTiO_{3} work function is beyond the scope of this study, using interfaces to achieve work function tuning is an interesting area for further research.

## IV. SUMMARY AND CONCLUSIONS

In this work, DFT calculations were used to assess the accuracy of oxide work functions for which both the experimental surface structure and work function are known, using perovskite SrTiO_{3} as a case study example. These results support the fact that DFT-HSE methods can provide accurate work functions for at least one complex oxide system (SrTiO_{3}), and additional detailed comparisons are needed to assess if this agreement extends to other systems. From this work, a number of key facts have emerged, which we believe are of general interest for modeling and understanding the work function of oxide materials: (1) DFT and experimental work functions for SrTiO_{3} are in good agreement, typically within about 0.3 eV, provided that the simulation and experiment are both measuring the same surface. (2) DFT calculations using HSE functionals yield more accurate work function values compared to the GGA-PBE functional. The lower accuracy of GGA-PBE in this case stems from both the underestimation of the bandgap and the incorrect positions of the valence and conduction band edges relative to vacuum. (3) The use of GGA-PBE seems to correctly capture the *changes* in the work function of a given surface with the introduction of common surface defects such as steps and point defects, indicating that the use of GGA-PBE correctly captures the relative surface dipole electrostatics, at least in the case of SrTiO_{3}. In addition, we have assessed the scale of the effect of common bulk and surface modifications on the work function. Nb doping at typical concentrations of 1 at. %, in the absence of any segregation to the surface, primarily elevates the Fermi level of SrTiO_{3} to the CBM and does not significantly impact the surface dipole. Therefore, Nb doping reduces the undoped SrTiO_{3} work function on the scale of half the bandgap. The presence of surface steps has a negligible influence on the work function at typical concentrations, on the scale of just a few meV. Surface reconstructions and adsorbates can increase or reduce the work function on the scale of an eV or more. For surface vacancies, typical concentrations of up to a few percent likely result in work function changes of a couple tenths of eV. The direction and magnitude of the work function change follows known chemical intuition of surface species electronegativities and resulting surface dipoles, except for O deficient SrO termination, where subsurface relaxation alters the surface dipole. Overall, as a first step of the validation of the DFT calculated work functions for complex oxides, this work suggests that the DFT-calculated work functions of SrTiO_{3} are accurate vs experiment to within a couple tenths of eV, codifies the scale of effect of common surface defects on the work function, and sheds light on the quantitative work function values and physics of SrTiO_{3}, an essential oxide material with numerous technological applications.

## SUPPLEMENTARY MATERIAL

See the supplementary material for more information on the changes in the work function using GGA vs HSE, analysis of GGA and HSE band alignment, analysis of effect of Nb doping concentration on the work function, and fits of work functions of defected surfaces to the depolarization curve.

## DATA AVAILABILITY

The data that support the findings of this study are openly available in Figshare at http://doi.org/10.6084/m9.figshare.12403226.^{59}

## ACKNOWLEDGMENTS

This work was funded by the Defense Advanced Research Projects Agency (DARPA) through the Innovative Vacuum Electronic Science and Technology (INVEST) program. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant No. ACI-1548562. This research was also performed using the computer resources and assistance of the UW-Madison Center for High Throughput Computing (CHTC) in the Department of Computer Sciences.

## REFERENCES

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