Non-equilibrium state defines physical properties of materials in many technologies, including architectural, metallic, and semiconducting amorphous glasses. In contrast, crystalline electronic and energy materials, such as transparent conductive oxides (TCO), are conventionally thought to be in equilibrium. Here, we demonstrate that high electrical conductivity of crystalline Ga-doped ZnO TCO thin films occurs by virtue of metastable state of their defects. These results imply that such defect metastability may be important in other functional oxides. This finding emphasizes the need to understand and control non-equilibrium states of materials, in particular, their metastable defects, for the design of novel functional materials.

The description and control of matter away from equilibrium is one of the grand challenges in basic energy sciences.1 Some of the most important physical phenomena in nature occur away from equilibrium, ranging in scale from ion exchange in the cells of living organisms to expansion of our universe. In applied science and technology, non-equilibrium phenomena most often manifest themselves in amorphous materials, including silica glass in architecture, hydrogenated amorphous silicon in solar cells, bulk metallic glasses in engineering, chalcogenide glasses in electronics and photonics, amorphous oxide semiconductors in flat panel displays, etc. In the near future, metastable materials will become even more important to our society due to the growing demand for sustainable fabrication processes with low energy budgets, in particular, with regards to advanced generation, storage, transportation, and conversion of alternative and renewable energy.

One class of increasingly important materials for electronic and energy applications is transparent conductive oxides (TCOs), for example, crystalline In2O3:Sn, SnO2:F, ZnO:Al and amorphous In-Zn-O, Ga-In-Zn-O, Zn-In-Sn-O.2 The crystalline TCOs are materials that combine the counter-indicated properties of good optical transparency and high electrical conductivity, which is highly desirable in photovoltaics (PV), flat panel displays, light emitting diodes (LEDs), transparent electronics, and solar fuels, also known as photoelectrochemical (PEC) water splitting. Despite heavy use of the crystalline TCOs in current applied science and technology, understanding the physical origin of simultaneous high optical transparency and high electrical conductivity in these materials is still a challenging scientific question. For example, doped In2O3 has been in practical use as a TCO for more than half a century, but the origins of its good optical transparency (optically forbidden band gap transition3) and high conductivity (surface electron accumulation layer4 and surface defects5) in pure In2O3 thin films have been uncovered only in the past several years. As another example of particular relevance to this study, ZnO has numerous technological applications,6 yet quantitative physical understanding of doping and defect mechanisms in this material is still the subject of very active research.7–10 

The mechanisms of extrinsic doping and intrinsic defect compensation in crystalline oxides, including both traditional electrically insulating ceramic oxides (such as Al2O3) and TCOs (such as ZnO), have been traditionally described by Brouwer (Kröger-Vink) diagrams.11 During the past two decades, it became possible to predict the defect formation enthalpy ΔHD of the dopants and defects, and hence the Brouwer diagrams, from first principles, by means of total-energy calculations.7–10 However, these theoretical approaches inherently invoke equilibrium concepts to relate the formation enthalpies of dopants and defects to their actual concentrations, as well as the resulting concentration of free charge carriers. Contrasting this paradigm is the fact that many thin film deposition techniques used to fabricate TCOs, such as physical or chemical vapor deposition processes and solution growth are known to be non-equilibrium methods.12,13 Thus, the purpose of the present work is to establish the connection between defect theory and thin-film growth of oxides, using the example of the Ga:ZnO TCO material.

We start by discussing the theoretical predictions for electron concentrations in gallium doped zinc oxide. Since Ga acts as a shallow donor in ZnO,14 the electron concentration equals the net doping density cnet = c(GaZn)–2c(VZn), where c(GaZn) is the concentration of electron-producing Ga donors, and c(VZn) is the concentration of compensating doubly charged acceptor-like Zn vacancies7,15 in the wurtzite ZnO phase. The theoretical predictions for cnet are based on supercell calculations of formation enthalpies ΔHD of GaZn and VZn defects, as well as their defect pairs (GaZn + VZn) and (2GaZn + VZn).16 These calculations include recent advances in determining appropriate reference energies for the electronic and atomic chemical potentials,17,18 in addition to usual corrections19 applied in the previous study.7 For the subsequent thermodynamic simulations of cnet from a numerical solution of the charge neutrality equation19 as a function of temperature (T) and oxygen partial pressure (pO2), the Ga concentration in ZnO is constrained to 4% of the cation sites (1.7 × 1021 cm−3). This “constrained equilibrium” approach is more realistic for modeling TCOs than complete equilibrium model, where Ga concentration is limited by precipitation of the Ga2ZnO4 spinel, which would occur only at very high temperature over long times.20,21 For more theoretical details and for comparison of the present results with those published earlier see supplemental material.22 

The predicted net doping density cnet as a function of T and pO2 for ZnO with a fixed concentration of 4 cat. % Ga is shown in Fig. 1(a) As expected for a fixed Ga concentration, a higher pO2 leads to a lower cnet due to increase in the c(VZn). In contrast, the increase of cnet with increasing T may seem counter-intuitive, since a simple proportionality with the Boltzmann factor would suggest the opposite behavior (the concentration of compensating zinc vacancies should increase as T increases). The two reasons for the predicted negative temperature dependence of cnet (Fig. 1(a)) are the decreasing band gap of ZnO23 and the decreasing chemical potential of oxygen.24 Both of these factors lead to increasing formation energy of compensating VZn acceptor defects with temperature and therefore to a decrease in c(VZn) and an increase of cnet. Such “negative temperature dependence” is not uncommon and has been discussed in detail for n-type GaAs.25 

FIG. 1.

Dependence of electrical properties of ZnO:Ga on temperature and oxygen partial pressure from (a) first principles theory, and (b) combinatorial experiments. The dashed square in (a) indicates the pO2-Ts range covered in (b). The dashed arrows in both (a) and (b) indicate the direction of increasing conductivity or net doping. The colored background in (b) indicates different regions of equilibration (see supplemental material)22 

FIG. 1.

Dependence of electrical properties of ZnO:Ga on temperature and oxygen partial pressure from (a) first principles theory, and (b) combinatorial experiments. The dashed square in (a) indicates the pO2-Ts range covered in (b). The dashed arrows in both (a) and (b) indicate the direction of increasing conductivity or net doping. The colored background in (b) indicates different regions of equilibration (see supplemental material)22 

Close modal

Experimental studies of room-temperature conductivity of gallium zinc oxide thin films prepared using pulsed laser deposition (PLD) from an 8% Ga doped ZnO target as function of substrate temperature Ts and pO2 in the chamber during the growth were performed using a thin film high-throughput combinatorial approach.26 The approach consists of three steps: (a) deposition of thin film (“sample libraries”) with a spatial gradient during the growth in the composition, the substrate temperature Ts (as in the present case) or both;27 (b) subsequent room-temperature characterization (“mapping”) of the libraries as a function of position and hence as a function of the composition and/or Ts during the growth;28 (c) semi-automated analysis and visualization of the resulting experimental data. More details about combinatorial experiments are provided in the supplemental material.22 

The results of the combinatorial experiments (Fig. 1(b)) indicate that conductivity measured at room temperature increases with decreasing pO2 and Ts during the growth. This measured pO2-Ts conductivity trend (Fig. 1(b)) is consistent with the individual Ts and pO2 trends reported in literature,12,29 but it contrasts with the calculated pO2-T trend of net doping density (Fig. 1(a)). The origin of this difference lies in a convolution of thermodynamic and kinetic processes that take place during the thin film growth. This effect, along with the three distinct pO2-Ts regions of Ga-doped ZnO thin films properties, depending on the degree of equilibration, are described in more detail in supplemental material.22 

Overall, the results shown in Fig. 1(b) lead to the conclusion that higher electrical conductivity should be achievable at low pO2 as well as lower Ts and Ga concentration than shown in Fig. 1(b) Indeed, subsequent synthesis of gallium zinc oxide thin films with 2%– 4% Ga doping at Ts = 250 °C and pO2 = 10−8 atm led to conductivity of σ = 3000 S/cm with electron concentration of n = 8 × 1020 cm−3 as determined by Hall effect, comparable to the best gallium zinc oxide TCO films reported in the literature.8,9,29,30 This experimental result is a factor of 100 000 larger than the theoretically predicted net doping density (cnet = 1015–1016 cm−3). This discrepancy indicates the defects and dopants in gallium zinc oxide thin films are so far from equilibrium that even the results of the constrained-equilibrium theoretical model (Fig. 1(a)) cannot describe their electrical transport properties. Such large difference can be explained only by non-equilibrium defect concentrations (affects electron concentration) rather than metastable distribution of defects or their mutual interaction (affects electron mobility) in the gallium zinc oxide thin films.

In order to reconcile the experimental results (Fig. 1(b)) with the theoretical predictions (Fig. 1(a)), we measured conductivity of the Ga doped ZnO thin films in situ as a function of T and pO2 during the measurement.21 In addition, we performed a series of complementary room-temperature ex-situ Hall effect measurements on the analogous samples. For both sets of experiments, more information is given in the supplemental material.22 

As shown in Fig. 2(a), the room temperature electron concentration of ZnO:Ga thin films measured ex-situ decreases modestly from ∼1021 cm−3 to ∼1020 cm−3 after heating from 250 °C to 450 °C in pO2 = 10−8 atm. In contrast, upon heating from ambient temperature to T = 500 °C in pO2 = 1 atm, the in-situ conductivity measured at temperature drops from >103 S/cm to <10−1 S/cm, but then recovers to >1 S/cm at 800 °C. This non-monotonic behavior is a strong indication of an initial non-equilibrium state of the samples, transitioning into the thermodynamic equilibrium state above 500 °C in pO2 = 1 atm. It is likely that this non-monotonic trend would not be observed for the samples measured ex-situ at room temperature, since regardless of the annealing temperature the defect concentrations would be frozen at the same temperature during cool-down procedure. At measurement temperatures above 500 °C, the conductivity trend agrees well with the theoretical prediction for the electron mobility of 1 cm2/V s, as seen in Fig. 2(a). This mobility value was obtained by extrapolation to pO2 = 1 atm of the ex-situ pO2 dependent Hall effect measurement results on the samples equilibrated in the pO2 = 10−8–10−4 atm range at 450 °C (Fig. 2(b)), and it is quite typical for doped ZnO.31 

As shown in Fig. 2(b), the experimental and theoretical electrical properties in ZnO:Ga thin films heated in pO2 = 1 atm to T = 600 °C and measured in-situ agree well both in terms of the slope and the magnitude of the pO2 dependence for the extrapolated 1 cm2/V s mobility. These in-situ results were reversible with respect to pO2, indicating the thermodynamic equilibrium. In contrast, for the samples that were measured ex-situ after cooling, the carrier concentration as a function of pO2 shows hysteretic behavior indicative of the initial non-equilibrium state of the samples. In the first branch of the hysteresis (going from 10−8 atm to 10−4 atm in pO2), the slope of the pO2 dependence equals about −1/2, which is steeper than the limit of −1/4 expected from the Brouwer diagram and observed in numerical simulations (Fig. 2(b)) for a VZn compensation mechanism. In the second branch of hysteresis (going from 10−4 atm to 10−8 atm in pO2), the −1/4 slope is consistent with the theoretical predictions, but the magnitude of electron concentration remains a factor of 2–5 higher than expected. This difference could be attributed to incomplete equilibration of the samples even after 36 h in pO2 = 10−4 atm at T = 450 °C, so it is likely that at least some hysteresis would persist upon further cycling of the samples.

In summary, the experimental in-situ temperature dependence (Fig. 2(a)) and oxygen partial pressure dependence (Fig. 2(b)) of electrical conductivity of the gallium zinc oxide thin films equilibrated by heating to 500 °C–600 °C in 1 atm of oxygen fully support the theoretical model of gallium-on-zinc (GaZn) substitutional donor-like defects partially compensated by acceptor-like zinc vacancy (VZn) defects (Fig. 1(a)) in this material. However, these equilibrated thin films have much lower conductivities (0.1–1 S/cm) than as-grown gallium zinc oxide TCO samples (>3000 S/cm). For these non-equilibrium ZnO:Ga samples, the conductivity dependence on temperature and oxygen partial pressure during both deposition (Fig. 1(b)) and post-deposition annealing (Fig. 2) strongly deviate from the expected Brouwer diagram and from numerical defect theory results (Fig. 1(a) and Fig. 2). These results demonstrate that high electrical conductivity of the typical gallium zinc oxide thin films used for TCO and other practical applications occurs only by virtue of their highly non-equilibrium metastable state, resulting from growth at relatively low temperature and low oxygen partial pressure.

FIG. 2.

Electrical properties of ZnO:Ga thin films determined from in-situ (red symbols) and ex-situ (blue symbols) experiments and theory (blue lines) as a function of (a) temperature and (b) oxygen partial pressure for different levels of Ga doping and different substrates (triangles—2%, SiO2; squares—4%, SiO2; circles—4%, Al2O3). The scales in both of the conductivity and the carrier density in (a) and (b) have been aligned to reflect an electron mobility of 1 cm2/V s.

FIG. 2.

Electrical properties of ZnO:Ga thin films determined from in-situ (red symbols) and ex-situ (blue symbols) experiments and theory (blue lines) as a function of (a) temperature and (b) oxygen partial pressure for different levels of Ga doping and different substrates (triangles—2%, SiO2; squares—4%, SiO2; circles—4%, Al2O3). The scales in both of the conductivity and the carrier density in (a) and (b) have been aligned to reflect an electron mobility of 1 cm2/V s.

Close modal

The results of our work suggest that non-equilibrium phenomena can be used intentionally to design metastable materials with new functionalities that are enabled by non-equilibrium processing. The vision of non-equilibrium properties by design calls for closely coupled development of new experimental and theoretical techniques that can describe, predict and control non-equilibrium phenomena in oxides and other functional metastable materials.

This research was supported by the U.S. Department of Energy under Contract No. DE-AC36-08GO28308 to the National Renewable Energy Laboratory (NREL). The theoretical calculations and the thin film experiments were supported by the Office of Energy Efficiency and Renewable Energy, Solar Energy Technology Program. The in-situ van der Pauw measurements were supported by the Office of Science, Basic Energy Science Program, as a part of the Energy Frontier Research Center “Center for Inverse Design”. Useful discussions with P. F. Ndione, A. Adler, and J. D. Perkins are gratefully acknowledged.

1.
G. R.
Fleming
and
M. A.
Ratner
,
Phys. Today
61
(7),
28
(
2008
).
2.
Handbook of Transparent Conductors
, edited by
D. S.
Ginley
,
H.
Hosono
, and
D. C.
Paine
(
Springer
,
Heidelberg
,
2011
).
3.
A.
Walsh
,
J. L. F.
Da Silva
,
S.-H.
Wei
,
C.
Körber
,
A.
Klein
,
L. F. J.
Piper
,
A.
DeMasi
,
K. E.
Smith
,
G.
Panaccione
,
P.
Torelli
,
D. J.
Payne
,
A.
Bourlange
, and
R. G.
Egdell
,
Phys. Rev. Lett.
100
,
167402
(
2008
).
4.
P. D. C.
King
,
T. D.
Veal
,
F.
Fuchs
,
Ch. Y.
Wang
,
D. J.
Payne
,
A.
Bourlange
,
H.
Zhang
,
G. R.
Bell
,
V.
Cimalla
,
O.
Ambacher
,
R. G.
Egdell
,
F.
Bechstedt
, and
C. F.
McConville
,
Phys. Rev. B
79
,
205211
(
2009
).
5.
S.
Lany
,
A.
Zakutayev
,
T. O.
Mason
,
J. F.
Wager
,
K. R.
Poeppelmeier
,
J. D.
Perkins
,
J. J.
Berry
,
D. S.
Ginley
, and
A.
Zunger
,
Phys. Rev. Lett.
108
,
016802
(
2012
).
6.
Ü.
Özgür
,
Ya. I.
Alivov
,
C.
Liu
,
A.
Teke
,
M. A.
Reshchikov
,
S.
Doğan
,
V.
Avrutin
,
S.-J.
Cho
, and
H.
Morkoç
,
J. Appl. Phys.
98
,
041301
(
2005
).
7.
S.
Lany
and
A.
Zunger
,
Phys. Rev. Lett.
98
,
045501
(
2007
).
8.
D. C.
Look
,
K. D.
Leedy
,
L.
Vines
,
B. G.
Svensson
,
A.
Zubiaga
,
F.
Tuomisto
,
D. R.
Doutt
, and
L. J.
Brillson
,
Phys. Rev. B
84
,
115202
(
2011
).
9.
D. O.
Demchenko
,
B.
Earles
,
H. Y.
Liu
,
V.
Avrutin
,
N.
Izyumskaya
,
Ü.
Özgür
, and
H.
Morkoç
,
Phys. Rev. B
84
,
075201
(
2011
).
10.
C. G.
van de Walle
and
J.
Neugebauer
,
J. Appl. Phys.
95
,
3851
(
2004
).
11.
F. A.
Kröger
and
H. J.
Vink
,
Physica
20
,
950
(
1954
).
12.
H.
Ryokena
,
I.
Sakaguchia
,
N.
Ohashia
,
T.
Sekiguchia
,
S.
Hishitaa
, and
H.
Haneda
,
J. Mater. Res.
20
,
2866
(
2005
).
13.
A.
Tsukazaki
,
A.
Ohtomo
,
T.
Onuma
,
M.
Ohtani
,
T.
Makino
,
M.
Sumiya
,
K.
Ohtani
,
S. F.
Chichibu
,
S.
Fuke
,
Y.
Segawa
,
H.
Ohno
,
H.
Koinuma
, and
M.
Kawasaki
,
Nature Mater.
4
,
42
(
2005
).
14.
B. K.
Meyer
,
J.
Sann
,
D. M.
Hofmann
,
C.
Neumann
, and
A.
Zeuner
,
Semicond. Sci. Technol.
20
,
S62
(
2005
).
15.
L. J.
Brillson
,
Y.
Dong
,
F.
Tuomisto
,
B. G.
Svensson
,
A. Yu.
Kuznetsov
,
D.
Doutt
,
H. L.
Mosbacker
,
G.
Cantwell
,
J.
Zhang
,
J. J.
Song
,
Z.-Q.
Fang
, and
D. C.
Look
,
Physica Status Solidi C
9
,
1566
(
2012
).
16.
K.
Biswas
and
S.
Lany
,
Phys. Rev. B
80
,
115206
(
2009
).
17.
L. Y.
Lim
,
S.
Lany
,
Y. J.
Chang
,
E.
Rotenberg
,
A.
Zunger
, and
M. F.
Toney
,
Phys. Rev. B
86
,
235113
(
2012
).
18.
19.
S.
Lany
and
A.
Zunger
,
Phys. Rev. B
78
,
235104
(
2008
).
20.
R.
Wang
and
A. W.
Sleight
,
Chem. Mater.
8
,
433
(
1996
).
21.
G. B.
González
,
T. O.
Mason
,
J. P.
Quintana
,
O.
Warschkow
,
D. E.
Ellis
,
J.-H.
Hwang
,
J. P.
Hodges
, and
J. D.
Jorgensen
,
J. Appl. Phys.
96
,
3912
(
2004
).
22.
See supplementary material at http://dx.doi.org/10.1063/1.4841355 for more information on theoretical and experimental details, for comparison of the present theoretical results with those published earlier, and for more discussion of combinatorial experiments.
23.
F. J.
Manjón
,
M.
Mollar
,
M. A.
Hernández-Fenollosa
,
B.
Marí
,
R.
Lauck
, and
M.
Cardona
,
Solid State Commun.
128
,
35
(
2003
).
24.
J.
Osorio-Guillén
,
S.
Lany
,
S. V.
Barabash
, and
A.
Zunger
,
Phys. Rev. Lett.
96
,
107203
(
2006
).
25.
T. Y.
Tan
,
H. M.
You
, and
U. M.
Gösele
,
Appl. Phys. A
56
,
249
(
1993
).
26.
A.
Zakutayev
,
F. J.
Luciano
,
V. P.
Bollinger
,
A. K.
Sigdel
,
P. F.
Ndione
,
J. D.
Perkins
,
J. J.
Berry
,
P. A.
Parilla
, and
D. S.
Ginley
,
Rev. Sci. Instrum.
84
,
053905
(
2013
).
27.
A.
Zakutayev
,
T. R.
Paudel
,
P. F.
Ndione
,
J. D.
Perkins
,
S.
Lany
,
A.
Zunger
, and
D. S.
Ginley
,
Phys. Rev. B
85
,
085204
(
2012
).
28.
A.
Zakutayev
,
J. D.
Perkins
,
P. A.
Parilla
,
N. E.
Widjonarko
,
A. K.
Sigdel
,
J. J.
Berry
, and
D. S.
Ginley
,
MRS Commun.
1
,
23
(
2011
).
29.
R. C.
Scott
,
K. D.
Leedy
,
B.
Bayraktaroglu
,
D. C.
Look
, and
Y.-H.
Zhang
,
J. Cryst. Growth
324
,
110
(
2011
).
30.
S.-M.
Park
,
T.
Ikegami
, and
K.
Ebihara
,
Thin Solid Films
513
,
90
(
2006
).
31.
K.
Ellmer
,
A.
Klein
, and
B.
Rech
,
Transparent Conductive Zinc Oxide: Basics and Applications in Thin Film Solar Cells
(
Springer
,
Heidelberg
,
2008
).

Supplementary Material