Ramanathan Sh. (Ed.) Thin Film Metal-Oxides: Fundamentals and Applications in Electronics and Energy
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200 Y. Hikita and H.Y. Hwang
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| Current Density | (A/cm
2
)
| Current Density | (A/cm
2
)
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
Voltage (V) Voltage (V)
300 K
20 K
300 K
20 K
40 K step
40 K step
ab
Fig. 5.27 Temperature dependence of the I V characteristics in (a)SrRuO
3
=NbWSrTiO
3
and
(b)SrRu
0:95
Mn
0:05
O
3
=NbWSrTiO
3
junctions. Measurements taken between 300 and 20 K in 40 K
steps (From [114], by permission)
Mn-doped
SrRuO
3
Nb:SrTiO
3
q
F
SB
E
loc
SrRu4
+
O
3
SrMn4
+
O
3
e
g
e
g
¯
e
g
t
2
g
t
2
g
¯
t
2
g
O 2p
O 2p
E
F
q
V
Fig. 5.28 A schematic diagram of the density of states for SrRuO
3
and SrMnO
3
, with an interface
band diagram illustrating the localized state E
loc
on the metal side of the interface (From [114], by
permission)
the SrRuO
3
side of the interface. From these and other spectroscopic experiments,
it could be concluded that the NDR observed at the SrRu
0:95
Mn
0:05
O
3
=NbWSrTiO
3
interface is caused by tunneling into the localized Mn e
g
states in SrRu
0:95
Mn
0:05
O
3
.
The change in the junction characteristics induced by impurity doping is unexpect-
edly large considering the small differences of the in-plane bulk transport properties.
The poor electrostatic screening generally present in metallic perovskites and the
close similarity in the chemical bonding between metallic and semiconducting per-
ovskites can result in robust interface states exhibiting the character of the impurity
element. Further studies may lead to designing other resonant tunneling structures
by simple doping of a metallic host, or as a prototypical structure for spectroscopic
investigation of screening, localization, and metal–insulator transitions in strongly
correlated electron systems.
5 Complex Oxide Schottky Junctions 201
5.4.4 Resistive Switching at Complex Oxide Schottky Junctions
For completeness, we note that another application attracting recent attention is the
reversible resistance switching observed at complex oxide Schottky junctions. This
is a subset of a broader class of structures where the resistance of the device is
reversibly switchable by an externally applied voltage, drawing interest for non-
volatile memory applications [117,118]. In many cases, the resistance switching is
a result of changes in the storage media, such as a crystalline to amorphous phase
transition, electromigration, or creating and breaking conducting filaments. In some
rectifying heterojunctions, however, the active region appears to be the Schottky
junction itself, and can be attributed to changes in the SBH. For perovskites, these
effects have been observed using Pr
0:7
Ca
0:3
MnO
3
[119]andCrW SrZrO
3
[120].
Some current aspects of this problem are discussed in Chap. 4 of this book, as well
as a recent review [121].
5.5 Summary
In this chapter, we have described some of the recent developments in the emerging
field of perovskite Schottky junctions. The importance of the non-linear permittivity
of SrTiO
3
, although long recognized, leads to a highly counterintuitive temperature
evolution of the barrier profile in Schottky junctions using NbWSrTiO
3
, giving un-
usual features such as a temperature dependent polarity reversal. The capability
to fabricate heterostructures with atomic precision, developed in recent years, has
given new abilities to engineer interfaces, such as the design of resonant states, or
tuning the barrier height by controlling the termination layer at the interface. This
latter feature may have broad applicability in the renewed interest in energy appli-
cations using oxides, such as photovoltaics, photocatalysis and fuel cells, where the
design of band alignments is crucial in the context of many stringent materials re-
quirements. Possible direct applications could include engineering band alignments
to maximize the efficiency of water splitting at NiO=NaTaO
3
oxide interfaces [122]
or improving the charge transfer rate in dye-sensitized solar cells [123].
While we are using the language and framework of conventional semiconductor
devices, many of the most interesting scientific issues in complex oxide Schottky
junctions are precisely in areas where this viewpoint fails. When the Schottky metal
is a heteroepitaxial manganite, the strong spin-charge coupling of the manganite
is manifest in large junction magnetoresistance and magnetocapacitance, implying
magnetic control of the energy band diagram. This has no direct semiconductor
analogue, but the analogy is helpful to frame the many open questions when using
strongly correlated electron materials in such a device. What is the nature of band
bending for materials where a rigid single particle band description is known to be
inapplicable? What is the semiconductor equivalent of the dopant density in such
cases? What is the relevant permittivity on the short length scales relevant for in-
terfaces? Schottky junctions are an ideal platform for probing these issues, which
202 Y. Hikita and H.Y. Hwang
are central to the further development of novel devices with new functionalities us-
ing complex oxides. The nature of phase transitions, the continuous photocarrier
injection across them, and the search for new interface states are all active areas of
exploration using complex oxide Schottky junctions, which promises much more
to come.
References
1. Goodenough JB (1971) Prog Solid Stat Chem 5:145
2. Tsuda N, Nasu K, Fujimori A, Siratori K (2000) Electronic conduction in oxides, 2nd rev.
Springer, Berlin
3. Imada M, Fujimori A, Tokura Y (1998) Rev Mod Phys 70:1039
4. Galasso FS (1970) Structure and properties of inorganic solids. Pergamon, Oxford
5. Chrisey DB, Hubler GK (1994) Pulsed laser deposition of thin films. Wiley, New York
6. Kawasaki M, Takahashi K, Maeda T, Tsuchiya R, Shinohara M, Ishiyama O, Yonezawa T,
Yoshimoto M, Koinuma H (1994) Science 266:1540
7. Koster G, Kropman BL, Rijnders GJHM, Blank DHA, Rogalla H (1998) Appl Phys Lett
73:2920
8. Mukunoki Y, Nakagawa N, Susaki T, Hwang HY (2005) Appl Phys Lett 86:171908
9. Ohtomo A, Muller DA, Grazul JL, Hwang HY (2002) Nature 419:378
10. Ohtomo A, Hwang HY (2004) Nature 427:423
11. Okamoto S, Millis AJ (2004) Nature 428:630
12. Nakagawa N, Hwang HY, Muller DA (2006) Nature Mater 5:204
13. Hamann DR, Muller DA, Hwang HY (2006) Phys Rev B 73:195403
14. Huijben M, Rijnders G, Blank DHA, Bals S, van Aert S, Verbeeck J, van Tendeloo G,
Brinkman A, Hilgenkamp H (2006) Nature Mater 5:556
15. Yunoki S, Moreo A, Dagotto E, Okamoto S, Kancharla SS, Fujimori A (2007) Phys Rev B
76:064532
16. Smadici S, Abbamonte P, Bhattacharya A, Zhai X, Jiang B, Rusydi A, Eckstein JN, Bader
SD, Zuo J-M (2007) Phys Rev Lett 99:196404
17. Chakhalian J, Freeland JW, Habermeier H-U, Cristiani G, Khaliullin G, van Veenendaal M,
Keimer B (2007) Science 318:1114
18. Hotta Y, Susaki T, Hwang HY (2007) Phys Rev Lett 99:236805
19. Cen C, Thiel S, Hammerl G, Schneider CW, Andersen KE, Hellberg CS, Mannhart J, Levy J
(2008) Nature Mater 7:298
20. Okamoto S, Maier ThA (2008) Phys Rev Lett 101:156401
21. Gozar A, Logvenov G, Fitting Kourkoutis L, Bollinger AT, Giannuzzi LA, Muller DA,
Bozovic I (2008) Nature 455:782
22. Pentcheva R, Pickett W (2008) Phys Rev B 78:205106
23. Caviglia AD, Gariglio S, Reyren N, Jaccard D, Schneider T, Gabay M, Thiel S, Hammerl G,
Mannhart J, Triscone J-M (2008) Nature 456:624
24. Popovic ZS, Satpathy S, Martin RM (2008) Phys Rev Lett 101:256801
25. Levy A, Falck JP, Kastner MA, Gallagher WJ, Gupta A, Kleinsasser AW (1991) J Appl Phys
69:4439
26. Mannhart J, Schlom DG, Bednorz JG, Muller KA (1991) Phys Rev Lett 67:2099
27. Xi XX, Doughty C, Walkenhorst A, Kwon C, Li Q, Venkatesan T (1992) Phys Rev Lett
68:1240
28. Atalla MM, Soshea RW (1963) Solid-State Electron 6:245
29. Suzuki H, Yamamoto T, Suzuki S, Iyori M, Takahashi K, Usuki T, Yoshisato Y, Nakano S
(1993) Jpn J Appl Phys 32:783
30. Toda F, Abe H, Ogiwara M (1993) IEEE Electron Device Lett 14:100
5 Complex Oxide Schottky Junctions 203
31. Yamamoto T, Suzuki S, Suzuki H, Kawaugchi K, Takahashi K, Yoshisato Y (1997) Jpn J Appl
Phys 36:L390
32. Yamamoto T, Suzuki S, Kawaguchi K, Takahashi K (1998) Jpn J Appl Phys 37:4737
33. Tamura H, Yoshida A, Gotoh K, Hasuo S, Van Duzer T (1991) IEEE Trans Magn 27:2594
34. Yoshida A, Tamura H, Gotoh K, Takauchi H, Hasuo S (1991) J Appl Phys 70:4976
35. Blom PWM, Wolf RM, Cillessen JFM, Krijn MPCM (1994) Phys Rev Lett 73:2107
36. Watanabe Y (1998) Phys Rev B 57:R5563; (1999) Phys Rev B 59:11257
37. Watanabe Y, Okano M (2001) Appl Phys Lett 78:1906
38. Sze SM (1981) Physics of semiconductor devices, 2nd edn. Wiley, New York
39. Brillson LJ (1982) Surf Sci Rep 2:123
40. Rhoderick EH, Williams RH (1988) Metal–semiconductor contacts, 2nd edn. Clarendon,
Oxford
41. Tung R (2001) Mater Sci Eng R 35:1
42. Kurtin S, McGill TC, Mead CA (1969) Phys Rev Lett 22:1433
43. Bardeen J (1947) Phys Rev 71:717
44. Heine V (1965) Phys Rev 138:A1689
45. Spicer WE, Chye PW, Skeath PR, Su CY, Lindau I (1979) J Vac Sci Technol 16:1422
46. Hasegawa H, Ohno H (1986) J Vac Sci Technol B 4:1068
47. Tung R (2000) Phys Rev Lett 84:6078
48. Padovani FA, Stratton R (1966) Solid-State Electron 9:695
49. Fonash SJ (1983) J Appl Phys 54:1966
50. Margaritondo G, Rowe JE, Christman SB (1976) Phys Rev B 14:5396
51. Hecht MH (1990) Phys Rev B 41:7918
52. Fowler RH (1931) Phys Rev 38:45
53. Crowell CR, Sze SM, Spitzer WG (1964) Appl Phys Lett 4:91
54. Sze SM, Moll JL, Sugano T (1964) Solid-State Electron 7:509
55. Dalal VL (1971) J App Phys 42:2274
56. Noland JA (1954) Phys Rev 94:724
57. Tufte ON, Chapman PW (1967) Phys Rev 155:796
58. Frederikse HPR, Hosler WR (1967) Phys Rev 161:822
59. Lee C, Yahia J, Brebner JL (1971) Phys Rev B 3:2525
60. M¨uller KA, Burkard H (1979) Phys Rev B 19:3593
61. Sawaguchi E, Kikuchi A, Kodera Y (1962) J Phys Soc Jpn 17:1666
62. Saifi MA, Cross LE (1970) Phys Rev B 2:677
63. Carnes JE, Goodman AM (1967) J Appl Phys 38:3091
64. Hasegawa H, Nishino T (1991) J Appl Phys 69:1501
65. Naito M, Sato H (1994) Physica C 229:1
66. Yoshimoto M, Maeda T, Shimozono K, Koinuma H, Shinohara M, Ishiyama O, Ohtani F
(1994) Appl Phys Lett 65:3197
67. Shimizu T, Okushi H (1999) J Appl Phys 85:7244
68. Inoue A, Izumisawa K, Uwe H (2001) Jpn J Appl Phys 40:3153
69. Barrett JH (1952) Phys Rev 86:118
70. Susaki T, Kozuka Y, Tateyama Y, Hwang HY (2007) Phys Rev B 76:155110
71. Eom CB, Cava RJ, Fleming RM, Phillips JM, van Dover RB, Marshall JH, Hsu JWP,
Krajewski JJ, Peck WF, Jr. (1992) Science 258:1766
72. Hikita Y, Kozuka Y, Susaki T, Takagi H, Hwang HY (2007) Appl Phys Lett 90:143507
73. Fujii T, Kawasaki M, Sawa A, Kawazoe Y, Akoh H, Tokura Y (2007) Phys Rev B 75:165101
74. Goodman AM, Perkins DM (1964) J Appl Phys 35:3351
75.
Stoecky B (1980) Appl Phys Lett 36:384
76. Sroubek Z (1970) Phys Rev B 2:3170
77. Fang X, Kobayashi T (1999) Appl Phys A 69:S587
78. Robertson J, Chen CW (1999) Appl Phys Lett 74:1168
79. Minohara M, Ohkubo I, Kumigashira H, Oshima M (2007) Appl Phys Lett 90:132123
80. Katsufuji T, Tokura Y (1994) Phys Rev B 50:2704
81. Miyasaka S, Okuda T, Tokura Y (2000) Phys Rev Lett 85:5388
204 Y. Hikita and H.Y. Hwang
82. Urushibara A, Moritomo Y, Arima T, Asamitsu A, Kido G, Tokura Y (1995) Phys Rev B
51:14103
83. Sawa A, Yamamoto A, Yamada H, Fujii T, Kawasaki M, Matsuno J, Tokura Y (2007) Appl
Phys Lett 90:252102
84. A. Fujimori, A. Ino, J. Matsuno, T. Yoshida, K. Tanaka, T. Mizokawa (2002) J Electron Spec-
trosc. Relat. Phenom. 124:127
85. Matsuno J, Fujimori A, Takeda Y, Takano M (2002) Europhys Lett 59:252
86. Wadati H, Kobayashi D, Kumigashira H, Okazaki K, Mizokawa T, Fujimori A, Horiba K,
Oshima M, Hamada N, Lippmaa M, Kawasaki M, Koinuma H (2005) Phys Rev B 71:035108
87. Hikita Y, Nishikawa M, Yajima T, Hwang HY (2009) Phys Rev B 79:073101
88. Kozuka Y, Susaki T, Hwang HY (2006) Appl Phys Lett 88:142111
89. Lee WC, MacDonald AH (2006) Phys Rev B 74:075106
90. Ruegg A, Pilgram S, Sigrist M (2007) Phys Rev B 75:195117
91. Tokura Y (ed) (2000) Colossal magnetoresistive oxides Gordon and Breach, New York
92. Haghiri-Gosnet A-M, Renard J-P (2003) J Phys D: Appl Phys 36:R127
93. Sugiura M, Uragou K, Noda M, Tachiki M, Kobayashi T (1999) Jpn J Appl Phys 38:2675
94. Tanaka H, Zhang J, Kawai T (2001) Phys Rev Lett 88:027204
95. Sun JR, Xiong CM, Zhao TY, Zhang SY, Chen YF, Shen BG (2004) Appl Phys Lett 84:1528
96. Wang DJ, Xie YW, Xiong CM, Shen BG, Sun JR (2006) Europhys Lett 73:401
97. Sheng ZG, Song WH, Sun YP, Sun JR, Shen BG (2005) Appl Phys Lett 87:032501
98. Jin KJ, Lu HB, Zhou QL, Zhao K, Cheng BL, Chen ZH, Zhou YL, Yang GZ (2005) Phys
Rev B 71:184428
99. Matsuno J, Sawa A, Kawasaki M, Tokura Y (2008) Appl Phys Lett 92:122104
100. Postma FM, Ramaneti R, Banerjee T, Gokcan H, Haq E, Blank DHA, Jansen R, Lodder CJ
(2004) J Appl Phys 95:7324
101. Sawa A, Fujii T, Kawasaki M, Tokura Y (2005) Appl Phys Lett 86:112508
102. Nakagawa N, Asai M, Mukunoki Y, Susaki T, Hwang HY (2005) Appl Phys Lett 86:082504
103. Susaki T, Nakagawa N, Hwang HY (2007) Phys Rev B 75:104409
104. Wang DJ, Sun JR, Lu WM, Xie YW, Liang S, Shen BG (2007) J Phys D: Appl Phys 40:5075
105. Wang DJ, Sun JR, Xie YW, Lu WM, Liang S, Zhao TY, Shen BG (2007) Appl Phys Lett
91:062503
106. Chen YF, Ziese M, Esquinazi P (2007) J Appl Phys 101:123906
107. Ahn CH, Bhattacharya A, Di Ventra M, Eckstein JN, Frisbie CD, Gershenson ME,
Goldman AM, Inoue IH, Mannhart J, Millis AJ, Morpurgo AF, Natelson D, Triscone J-M
(2006) Rev Mod Phys 78:1185
108. Katsu H, Tanaka H, Kawai T (2001) J Appl Phys 90:4578
109. Muraoka Y, Hiroi Z (2003) J Phys Soc Jpn 72:781
110. Muraoka Y, Yamauchi T, Muramatsu T, Yamaura J, Hiroi Z (2004) J Magn Magn Mater
272–276:448
111. Muraoka Y, Muramatsu T, Hiroi Z (2005) Thin Solid Films 486:82
112. Eguchi R, Tsuda S, Kiss T, Chainani A, Muraoka Y, Hiroi Z, Shin S (2007) Phys Rev B
75:073102
113. Muramatsu T, Muraoka Y, Yamauchi T, Yamaura J, Hiroi Z (2004) J Magn Magn Mater
272–276:e787
114. Hikita Y, Fitting Kourkoutis L, Susaki T, Muller DA, Takagi H, Hwang HY (2008) Phys Rev
B 77:205330
115. Cao G, Chikara S, Lin XN, Elhami E, Durairaj V, Shlottmann P (2005) Phys Rev B 71:035104
116. Tsu R, Esaki L (1973) Appl Phys Lett 22:562
117. Dearnaley G, Steoneham AM, Morgan DV (1970) Rep Prog Phys 33:1129
118. Pgnia H, Sotnik N (1988) Phys Status Solidi a108:11
119. Liu SQ, Wu NJ, Ignatiev A (2000) Appl Phys Lett 76:2749
120. Beck A, Bednorz JG, Gerber Ch, Rossel C, Widmer D (2000) Appl Phys Lett 77:139
121. Sawa A (2008) Mater Today 11:28
122. Kato H, Kudo A (2003) Catal Today 78:561
123. Diamant Y, Chappel S, Chen SG, Melamed O, Zaban A (2004) Coord Chem Rev 248:1271
Chapter 6
Theory of Ferroelectricity and Size Effects
in Thin Films
Umesh V. Waghmare
Abstract Ferroelectrics form an important class of materials with giant elec-
tromechanical and dielectric response properties, which are crucial in technologies
ranging from micro-electromechanical systems (MEMS) to computer memory de-
vices. Ferroelectrics exhibit a spontaneous electric polarization (electric dipole
moment per unit volume) that can be switched with large enough external fields
to its symmetry equivalent states, in particular to the one with reversed direction.
Fundamentally, the strong coupling of spontaneous polarization with external stress
and electric fields is linked with a structural (or ferroelectric) phase transition ex-
hibited by ferroelectrics as a function of temperature, which depends sensitively on
applied stress and electric fields. The coupling between the polarization and stress
facilitates their use as sensors and actuators, and the dielectric coupling and switch-
ability of polarization facilitate their use in memory devices. With a constant trend
of miniaturization of devices, it is essential to understand how these properties of
bulk ferroelectrics evolve to the ones at nano-scale.
Measurement of properties of a nano-structure of a material and its potential
use in a device depend very much on its chemical, electrical and mechanical en-
vironment. Fundamentally, this is because (a) a sizeable fraction of atoms in a
nano-structure belongs to its surface or interface with environment, and (b) the
length-scale(s) associated with its interface depend on the nature of its environment.
This becomes even more important in the context of a ferroelectric due to intrinsi-
cally strong coupling of its polarization with strain, electric field and chemistry. For
example, shape of a nano-structure and the metallic versus insulating nature of its
surrounding material determine the electro-static boundary conditions. The lattice
mismatch between a ferroelectric nano-structure and its surrounding material deter-
mine the mechanical boundary conditions, and chemical bonding or charge transfer
between them determine the changes in the local dipole moments arising from local
chemistry. Thus, an accurate description of properties of a ferroelectric at nano-scale
requires realistic treatment of the electrostatic, mechanical and chemical boundary
conditions.
U.V. Waghmare (
)
Theoretical Sciences Unit, J Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore
560 064, India
e-mail: waghmare@jncasr.ac.in
S. Ramanathan (ed.), Thin Film Metal-Oxides: Fundamentals and Applications
in Electronics and Energy, DOI 10.1007/978-1-4419-0664-9
6,
c
Springer Science+Business Media, LLC 2010
205
206 U.V. Waghmare
6.1 Introduction
Ferroelectrics form an important class of materials with giant electromechanical and
dielectric response properties,which are crucial in technologies ranging from micro-
electromechanical systems (MEMS) to computer memory devices. Ferroelectrics
exhibit a spontaneous electric polarization (electric dipole moment per unit volume)
that can be switched with large enough external fields to its symmetry equivalent
states, in particular to the one with reversed direction. Fundamentally, the strong
coupling of spontaneous polarization with external stress and electric fields is linked
with a structural (or ferroelectric) phase transition exhibited by ferroelectrics as
a function of temperature, which depends sensitively on applied stress and elec-
tric fields. The coupling between the polarization and stress facilitates their use
as sensors and actuators, and the dielectric coupling and switchability of polariza-
tion facilitate their use in memory devices. With a constant trend of miniaturization
of devices, it is essential to understand how these properties of bulk ferroelectrics
evolve to the ones at nano-scale.
Measurement of properties of a nano-structure of a material and its potential
use in a device depend very much on its chemical, electrical and mechanical en-
vironment. Fundamentally, this is because (a) a sizeable fraction of atoms in a
nano-structure belongs to its surface or interface with environment, and (b) the
length-scale(s) associated with its interface depend on the nature of its environment.
This becomes even more important in the context of a ferroelectric due to intrinsi-
cally strong coupling of its polarization with strain, electric field and chemistry. For
example, shape of a nano-structure and the metallic versus insulating nature of its
surrounding material determine the electro-static boundary conditions. The lattice
mismatch between a ferroelectric nano-structure and its surrounding material deter-
mine the mechanical boundary conditions, and chemical bonding or charge transfer
between them determine the changes in the local dipole moments arising from local
chemistry. Thus, an accurate description of properties of a ferroelectric at nano-scale
requires realistic treatment of the electrostatic, mechanical and chemical boundary
conditions.
A wide spectrum of materials, such as polymers, H-bonded phosphates, oxides,
chalcogenides exhibit ferroelectricity, one of the simplest ones being GeTe. Among
these, ABO
3
oxides occurring in the perovskite structure form a large and impor-
tant class of ferroelectrics with properties attractive to technological applications.
This is largely due to an important feature of the perovskite structure: linear chains
made of corner-shared .BO
6
/
n
octahedra occur in all the three directions. Typi-
cally A atom is a simple alkali, alkaline earth or simple divalent or trivalent metal,
B is a transition metal in the ionic state with null occupation of its d -orbital –
the so-called d
0
-ness that facilitates ferroelectricity. Some examples of perovskite
ferroelectrics are KNbO
3
; BaTiO
3
; PbTiO
3
and BiAlO
3
. These materials are ferro-
electric well above room temperature, and become paraelectric above a ferroelectric
transition temperature. Diversity in the properties of such oxides is quite remarkable,
and involves interesting fundamental physical and chemical concepts: for example,
KNbO
3
is a high-temperature ferroelectric, while KTaO
3
remains paraelectric at
6 Theory of Ferroelectricity and Size Effects in Thin Films 207
all temperatures! Properties of these perovskite oxide ferroelectrics can be tuned
further by forming their solid solutions, growing ordered superlattices, etc.
To capture the diversity in ferroelectrics as well as universality in their transition
behavior, particularly at nano-scale, a material-specific atomistic theoretical frame-
work is highly desirable. First-principles calculations based on density functional
theory (DFT) provide a parameter-free, fairly realistic and accurate basis for this:
they provide an approximate solution to quantum mechanical many-body ground
state of electrons with a material-specific Hamiltonian defined by atomic numbers
and positions, and its energy as a function of atomic positions control the motion
of nuclei in a material. Since the transition metal in ferroelectrics is in the d
0
-state,
effects of electron correlations on the nature of electronic ground state of ferro-
electrics are rather weak, such first-principles calculations can be quite useful in
their studies [1]. While this energy surface can be used in statistical mechanical
analysis of the behavior of a material at finite temperature, such simulations directly
based on DFT calculations are prohibitively expensive at present for systems con-
taining more than a few hundred atoms. Construction of a model Hamiltonian [2]
that would reproduce the material-specific transition behavior while readily permit-
ting large-scale simulations has proven to be a systematic and effective approach to
such analysis.
Landau-Devonshire theories of the free energy landscape near a ferroelectric
transition provide a phenomenological approach for continuum mechanical analysis
of behavior of ferroelectrics and have been quite useful in both (a) fundamental anal-
ysis of various phases in complex phase diagrams and formation of micro-structures,
and (b) simulation of devices based on ferroelectrics [3]. Such a description is ideal
for phenomena occurring at long length-scales, but is limited in its ability at atomic
or sub-nanometer scales. Parameters in the Landau free energy functions have so
far been obtained by fitting to experimental material-specific information. It is very
much desirable to bridge the gap between first-principlesand Landau-like theories at
nonzero temperature, and have a consistent description at all length-scales entirely
based on information available from first-principles.
Ferroelectricity in perovskite oxides arises mostly from phonons. Soft modes
[4–6], phonon modes with frequencies typically less than 100 cm
1
, have long been
known to dominate the material behavior near ferroelectric and structural phase
transitions: (a) soft modes are responsible for a large response of a material to ex-
ternal perturbations, and (b) a large entropy associated with soft modes leads to
greater thermodynamic stability of a given structure of a material. Important mi-
croscopic insights into properties of ferroelectric oxides in the bulk form have been
obtained through first-principles calculations [7]. We highlight a few of these here:
(a) hybridization between the d -states of transition metal and p-states of oxygen
are crucial to the stability of ferroelectric phase [8], (b) the pd-hybridization is also
known for anomalously large Born dynamical charges of ions [9,10], which can be
considered as indicators of off-centering of a cation from its high symmetry posi-
tion [11, 12], (c) the Born dynamical charges, which are the measure of coupling
of atomic displacements with electric field, are largest for the softest mode as re-
flected in its giant LO–TO phonon frequency splitting [9], (d) ferroelectric phase
208 U.V. Waghmare
transitions are first-order transitions driven by fluctuations arising from coupling of
polarization with strain [13], (e) polarization rotation through monoclinic phases is
largely responsible for large piezoelectric [14] and soft switching properties [15]
of ferroelectrics. Most of these findings are essential to understanding ferroelectric-
ity at nano-scale, but need to be used carefully along with appropriate mechanical,
electrostatic and chemical boundary conditions.
Definition of an intrinsic property of a nano-structure that can be meaningfully
compared with that of its bulk form is rather tricky. For example, the bulk modulus
of a cluster of Si compared with that of its crystal. While the bulk modulus of a
zero-dimensional object, such as a cluster, has the dimension of energy, the bulk
modulus of a three-dimensional infinite crystal has the dimension of energy per
unit volume. Thus, a meaningful comparison of the two necessary to estimate size
dependence of properties requires a definition of the volume of a cluster or the
corresponding length-scale. Of course, above a certain large enough size (greater
than this length-scale), the size-dependence of various properties exhibit scaling
that can be deduced from continuum analysis. In the reported size-dependence of
properties of nano-structures (like the ones which we will discuss here) smaller than
this “critical size”, one needs to keep in mind this subtleness and that the estimates
of properties of such nano-structures depend on the length-scale relevant to their
measurement or calculations, that is, on the surrounding environment.
It would be interesting to determine how the soft modes, Born dynamical
charges, and other properties of ferroelectric oxides evolve to nano-scale. In fact,
the boundary conditions, which have a great impact on properties of ferroelectrics
at nano-scale, can be exploited to tune or enhance some of their properties. For
example, epitaxial growth of ferroelectric films can be used to change mechanical
boundary conditions and tune their properties. It is indeed very hard, yet possible
these days, to grow and characterize ferroelectric nano-structures and determine
their properties. First-principles computational methodologies play important roles
in this endeavor: (a) they provide fundamental insights by accessing controlled
microscopic information, (b) they complement experimental efforts by helping to
“see” what is being seen in characterization, and (c) they can predict properties and
act as guideline in design of ferroelectric nano-structures.
In this chapter, we restrict to the theoretical analysis of ferroelectricity in thin
films of perovskite oxides. We first review briefly the principles of first-principles
density functional theory–based calculations, modeling and simulations with a focus
on ferroelectric oxides and films. Then, we examine the phenomenological Landau-
like theories relevant to ferroelectrics and their thin films. We then present what and
how interesting aspects of nano-thin ferroelectric films are uncovered through (a) di-
rect use of first-principles techniques in simulations of free-standing films and their
interfaces, (b) determination of temperature-strain phase diagrams using a model
Hamiltonian, and (c) polarization switching in films using first-principles model and
phenomenology. We clarify that our goal here is not to review all the work in the
vast field of ferroelectric films, but rather to give a pedagogic presentation through
use of select case studies and author’s experience in the field.
6 Theory of Ferroelectricity and Size Effects in Thin Films 209
6.2 Methodology
6.2.1 First-Principles Theoretical Methodology
In the analysis of most problems in materials that involve only the structure and
phonons, Born-Oppenheimer approximation can be used to (a) decouple solutions
of electronic and nuclear motion, and (b) obtain an inter-atomic potential that in-
cludes the effects of electrons and controls the motion of nuclei or ions. The latter
is obtained by assuming that electrons remain in their ground state as a function
of nuclear positions and is based on the fact that electrons are much lighter than
nuclei and typically move much faster. This approximation is quite harmless for
insulators such as ferroelectric oxides, where the large band-gap makes electronic
excitations are very unlikely. The effective inter-atomic potential is thus given by a
“total energy” as a function of atomic numbers Z
I
and positions R
I
:
E
tot
.Z
I
;R
I
/ D E
G
.Z
I
;R
I
/ C E
C
.Z
I
;R
I
/;
where E
G
is the many-particle ground state energy of electrons and E
C
is the elec-
trostatic interaction energy of nuclei. While the latter is a simple expression that
can be computed using O.N
2
/ operations, N being the system size, determination
of the former involves solution of an eigenvalue problem of a differential equation
with functions involving a vary large number of variables and is very hard. Numer-
ically, exact solution of the former can be obtained using Quantum Monte Carlo
methods, but their applicability to real and complex materials is rather limited on
the present-day computers.
A practical, albeit approximate, approach to determination of E
G
isbasedonthe
density functional theory [16]. Here, the many-electron quantum ground state en-
ergy is shown to depend uniquely on ground state electron density, and the problem
of many interacting electrons is mapped onto an effective single or non-interacting
electron problem with density n.r/ D
P
iD1;N
e
j
i
.r/j
2
;
i
.r/s being single-particle
wave functions. In this picture, the effective potential felt by each electron in-
cludes its electrostatic interaction with nuclei and other electrons, and many body
exchange–correlation energy in an approximate form:
E
G
D T
s
Œ
f
i
.r/
g
C E
ne
ŒZ
I
; n.r/ C E
ee
Œn.r/ C E
xc
Œn.r/;
where T
s
is the kinetic energy of non-interacting electrons, E
n-e
;E
e-e
are the
electrostatic interaction energy between nuclei and electrons, and among electrons,
respectively, E
xc
is the exchange–correlation energy and the total energy is min-
imum for the ground state density and wavefunctions. The exchange–correlation
energy is most commonly evaluated in a local density approximation in which it
is (a) assumed to depend on the local electron density, and (b) assumed to be the
same as that of homogeneous electron gas of the same density. We note that such an
approximate treatment of many electron exchange correlation energy limits DFT’s