Coexistence of trapped and free excess electrons in ${\mathrm{SrTiO}}_{3}$

Coexistence of trapped and free excess electrons in ${SrTiO}_{3}$

Xianfeng Hao, Zhiming Wang, Michael Schmid, Ulrike Diebold, and Cesare Franchini

Phys. Rev. B 91, 085204 – Published 12 February 2015

Abstract

The question whether excess electrons in ${SrTiO}_{3}$ form free or trapped carriers is a crucial aspect for the electronic properties of this important material. This fundamental ambiguity prevents a consistent interpretation of the puzzling experimental situation, where results support one or the other scenario depending on the type of experiment that is conducted. Using density functional theory with an on-site Coulomb interaction U, we show that excess electrons form small polarons if the density of electronic carriers is higher than $\approx 10^{20} {cm}^{- 3}$ . Below this value, the electrons stay delocalized or become large polarons. For oxygen-deficient ${SrTiO}_{3}$ , small polarons confined to ${Ti}^{3 +}$ sites are immobile at low temperature but can be thermally activated into a conductive state, which explains the metal-insulator transition observed experimentally.

Received 8 August 2014
Revised 26 January 2015

DOI:https://doi.org/10.1103/PhysRevB.91.085204

Authors & Affiliations

Xianfeng Hao^1,2,3, Zhiming Wang¹, Michael Schmid¹, Ulrike Diebold¹, and Cesare Franchini^2,*

¹Institute of Applied Physics, Vienna University of Technology, Vienna, Austria
²University of Vienna, Faculty of Physics and Center for Computational Material Science, Vienna, Austria
³Key Laboratory of Applied Chemistry, Yanshan University, Qinhuangdao 066004, People's Republic of China

^*cesare.franchini@univie.ac.at

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 91, Iss. 8 — 15 February 2015

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Nb-doped ${SrTiO}_{3}$ . (a) Isosurfaces of the spin density (dark blue; $0.045 e$ /Å) for an $N = 3$ supercell. The green (big), light-blue/turquoise (medium size), and red (small) balls represent Sr, Ti/Nb, and O atoms, respectively. (b) Electronic density of states $(N = 3)$ ; the (blue) peak in the band gap corresponds to the polaron state formed at a $Ti^{3 +}$ site. (c) Variation of the polaron formation energy as a function of carrier density for the cubic ( $N = 3, 6.03 \times 10^{20} {cm}^{- 3}$ , 135 atoms; $N = 4, 2.54 \times 10^{20} {cm}^{- 3}$ , 320 atoms; $N = 5, 1.30 \times 10^{20} {cm}^{- 3}$ , 625 atoms; $N = 6, 0.75 \times 10^{20} {cm}^{- 3}$ , 1080 atoms) and low-T AFD phases ( $2.54 \times 10^{20} {cm}^{- 3}$ , 320 atoms; $1.50 \times 10^{20} {cm}^{- 3}$ , 540 atoms; $0.75 \times 10^{20} {cm}^{- 3}$ , 1080 atoms) for the optimized (Theor.) and experimental (Expt.) volume. The AFD Expt. curves are obtained by shifting down the AFD Theor. curve by the calculated energy shift at $N = 4, 90$ meV. For the 540 atoms AFD phase we adopted a supercell based on the $a_{A F D} \sqrt{2} \times a_{A F D} \sqrt{2} \times c_{A F D}$ . The data are aligned with respect to the electron density (1 Nb per SC). Interpolating the polaron formation energy for $N \to \infty$ as a function of $1 / N^{3}$ [42] confirms the delocalization solution in the dilute limit. The dotted line serves as a guide to the eye.
Reuse & Permissions
Figure 2
Local structural distortions of the Nb-induced polaron shown in Fig. 1 within the $x y$ and $x z$ planes. The numbers indicate the expansion (in Å) with respect to the ideal cubic Ti-O bond length.
Reuse & Permissions
Figure 3
Isosurfaces of the spin density ( $0.045 e / Å^{2}$ ) for one $V_{O}$ (circle) in an $N = 5$ SC. The insets show the density of states near the Fermi level (set at the CBM). The filled peaks indicate the polaron states. In the P1 polaron configuration, considered to be the ground state in previous computational studies [25, 26, 27, 28], one electron is self-trapped near the $V_{O}$ , while the other one is delocalized. For each polaronic configuration (P1–P4) the relative energy with respect to the P1 phase is given (in meV). The most stable polaron solution is P3. The two small polarons (high spin density) are labeled Pol1 and Pol2. A three dimensional view of the four polaron configurations is given in Fig. 4.
Reuse & Permissions
Figure 4
Three dimensional view of the four polaron configurations shown in Fig. 3 and corresponding spin density isosurfaces. Small (red) and large (blue) spheres represent oxygen and titanium atoms, respectively. Strontium atoms are not shown. The $V_{O}$ is indicated by a (black) filled circle. (a) P1: one electron is trapped near the $V_{O}$ in a $d_{z^{2}}$ orbital, while the second one is delocalized; (b) P2: both electrons are trapped near the $V_{O}$ in $d_{x z}$ orbitals; (c) P3: one electron is trapped near the $V_{O}$ in a $d_{x z}$ orbital, while the second one is located in a ${Ti}^{3 +}$ site $\approx 8$ Å away with a $d_{y z}$ configuration (shown in the insets). (d) Both electrons are about 8 Å away from the $V_{O}$ , and adopt $d_{x z}$ and $d_{x y}$ configurations.
Reuse & Permissions
Figure 5
MD in ${SrTiO}_{3 - δ}, δ = 0.00267 (N = 5)$ . Mobility of the two polarons Pol1 and Pol2 starting from the P3 configuration (see Fig. 3: Pol1 near $V_{O}$ ; Pol2 about 8 Å apart), given with respect to the distance $d_{P o l 1 / P o l 2 - V_{O}}$ between the polaron and $V_{O}$ for two simulation temperatures, 700 K (top) and 1000 K (bottom). One polaron usually remains in the vicinity of $V_{O}$ , whereas the other one frequently hops to neighboring sites.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics

Coexistence of trapped and free excess electrons in ${SrTiO}_{3}$

Xianfeng Hao, Zhiming Wang, Michael Schmid, Ulrike Diebold, and Cesare Franchini

Phys. Rev. B 91, 085204 – Published 12 February 2015

Abstract

Authors & Affiliations

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

Physical Review B

covering condensed matter and materials physics

Coexistence of trapped and free excess electrons in SrTiO3

Xianfeng Hao, Zhiming Wang, Michael Schmid, Ulrike Diebold, and Cesare Franchini

Phys. Rev. B 91, 085204 – Published 12 February 2015

Abstract

Authors & Affiliations

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

Authorization Required

Other Options

Download & Share

Images

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Log In

Search

Article Lookup

Paste a citation or DOI

Enter a citation

Coexistence of trapped and free excess electrons in ${SrTiO}_{3}$