Transition Metal Oxides
A. J. Millis
Department of Physics
Columbia University
NSF DMR 1006282
DOE BES ER 046160 and CMCSN program
Copyright A. J. Millis 2013
Columbia University
Outline
1. Introduction: History
2. Overview: Phenomena
3. Basic Chemistry and Electronic Structure
4. Theory--General
5. Dynamical Mean Field Theory and Oxides
6. Applications and Prospects
Copyright A. J. Millis 2013
Columbia University
Transition metal ``oxides’’
Periodic Table of the Elements
GROUP 1
IA
1
1
1.00794
H
!"#$%&'()$*+,
12$*#3
2
1^gX
2.2
Hydrogen
0.0899
-259.14
=:?'37
13.5984
-252.87
-
3
2
6.941
Li
2
1^gX
0.98
Lithium
0.535
180.54
=$?'152
5.3917
1342
BCC
4
1
1_
1.57
1.848
1287
=$?'112
9.3227
2470
HCP
11
Na
Period
3
2
1^gX
0.82
Rubidium
5
X
12
B\+C'XA
`2
24.3050
Mg
1_
1
1_
1.00
Calcium
Alkali Metals
Noble Gas
Alkaline Earth
Metals
Halogens
!*A#3)"+'Q+,#
!"#$%&'G8AA'R5%"
!:#/86,#'I#5A"85"
J8A+'#L'(8"),83'9#/8,%"0$A
J#3"E$855'&#5A"85"
<3+&",#5'G8AA
Poor Metals
3
IIIB
21
44.955910
Sc
Metalloids
4
IVB
2
@bgX
1.36
Scandium
22
5
VB
47.867
Ti
23
3
UX
1.54
Titanium
e
k
m+
$+c X
,_
+F
e
U
D
2 S hc X
6
VIB
50.9415
V
24
Cr
4
UbgX
1.63
Vanadium
7
1b
1.66
Chromium
25
2.8179403x10 m
1.602176x10-19 J
1.602176x10-19 C
96 485.3399 C/mol
0.0072973525
3.7417749x10-16 W m2
Mn
1+&#56'T86%8"%#5'I#5A"85"
1K++6'#L'9%/0"'%5'8'F8&))$
1K++6'#L'A#)56'%5'8%,'8"'1V>
1"8568,6'>,+AA),+
8
VIII
54.938049
6
1agX
1.55
Manganese
p
h
m +gm K
R’
R ’c
R ’hc
ch/k
c
-15
7
VIIB
51.9961
G
R
4,8:%"8"%#5'I#5A"85"
G#38,'48A'I#5A"85"
G#38,'F#3)$+'=D6+83'48A?
>D
>385&S'I#5A"85"
>,#"#57<3+&",#5'G8AA'T8"%#
T26*+,/'I#5A"85"
-273.15 °C
1.660539x10-27 kg
23
-1
6.022142x10 mol
2.718281828
-23
1.380650x10 J/K
9.10938215x10-31 kg
0.5110 MeV
m)
<3+&",#5'T86%)A'=I38AA%&83?
<3+&",#5'F#3"
<3+$+5",2'I08,/+
U8,8682'I#5A"85"
L%5+7A",)&"),+'&#5A"85"
U%,A"'T86%8"%#5'I#5A"85"
Non Metals
Rare Earth Metals
BWXC
1#),&+O'K02A%&AP5%A"P/#:
Common Constants
Solid Synthetic
Transition Metals
1
1.31
Magnesium
Ca
†
Liquid
Categories
Sr
1
1_
0.95
Strontium
Y
2
D3/2
1.22
Yttrium
1.532
2.63
4.472
4.1771
5.6949
6.2173
39.31
688
777
1382
1526
3345
=$?'248
BCC
=$?'215
FCC
=$?'180
HCP
Bh,C'aA ^
Bh,C'aA X
Bh,C'W6 ^'aAX
`1
`2
`3
132.90545 56
137.327
55
2
1^gX
0.79
Cesium
Ba
1
1_
0.89
Barium
1.879
3.51
3.8939
5.2117
28.44
671
727
1870
=$?'265
BCC
=$?'222
BCC
Bi+C'eA ^
Bi+C'eA X
`1
`2
(223) 88
(226)
87
Fr
7
1^gX
13
IIIA
9
VIII
26
55.845
Fe
Iron
5
@W
1.83
27
10
VIII
58.933200
Co
6.67428x10-11 m3 kg-1 s-2
8.314472 J mol-1 K-1
3
0.02241410 m /mol
3.14159265358979
-34
6.626069x10 J s
1836.15267247
10 973 732 m-1
3.289842x1015 Hz
13.6057 eV
0.01438769 m K
299 792 458 m/s
343.2 m/s
101 325 Pa
4
UfgX
1.88
Cobalt
28
11
IB
58.6934
Ni
3
UW
1.91
Nickel
63.546
Cu
2
1^gX
1.90
Copper
10.811
B
8.2980
4000
rhom.
2.26
3550
=:?'77
11.2603
4027
hex
1.251
-210.1
=:?'75
>H^gX
1.61
Aluminum
2.7
5.9858
YWXZ 660.32
2519
=$?'143
FCC
B(+C'bA X'bK^
`3
65.409 31
69.723
30
Zn
Zinc
1
1_
1.65
Ga
2
>H^gX
1.81
Gallium
2
1^gX
0.7
Francium
-
4.0727
BT5C'cA ^
`1
Ra
Nb
Mo
6
D1/2
1.60
Niobium
7
S3
2.16
Molybdenum
Tc
6
S5/2
1.9
Technetium
Ru
5
F5
2.20
Ruthenium
Rh
4
F9/2
2.28
Rhodium
Pd
1
S0
2.20
Palladium
Ag
2
S1/2
1.93
Silver
Cd
1
S0
1.69
Cadmium
In
2
P°1/2
1.78
Indium
6.511
8.57
10.28
11.5
12.37
12.45
12.023
10.49
8.65
7.31
6.6339
6.7589
7.0924
7.28
7.3605
7.4589
8.3369
7.5762
8.9938
5.7864
1855
4409
2477
4744
2623
4639
2157
4265
2334
4150
1964
3695
1554.9
2963
961.78
2162
321.07
767
156.6
2072
=$?'160
HCP
=$?'146
BCC
=$?'139
BCC
=$?'136
HCP
=$?'134
HCP
=$?'134
FCC
=$?'137
FCC
=$?'144
FCC
=$?'151
§hex
=$?'167
§tetra.
Bh,C'W6 X'aAX
Bh,C'W6 W'aA^
Bh,C'W6 a'aA^
Bh,C'W6 a'aAX
Bh,C'W6 c'aA^
Bh,C'W6 ]'aA^
Bh,C'W6 ^_
Bh,C'W6 ^_'aA^
Bh,C'W6 ^_'aAX
Bh,C'W6 ^_'aAX'aK^
+4
`3[a
`X[b[W[a[6
`W[7
`X[b[4[e[]
`X[3[W
`2[W
`1
`2
`3
178.49 73
180.9479 74
183.84 75
186.207 76
190.23 77
192.217 78
195.078 79
196.96655 80
200.59 81
204.3833
72
Hf
3
UX
1.3
Ta
W
4
UbgX
1.5
Tantalum
985"085%6+'1+,%+A 13.31
5
@_
2.36
Tungsten
Re
6
1agX
1.9
Rhenium
Os
Ir
5
@W
2.2
Osmium
4
UfgX
2.2
Iridium
Pt
3
@b
2.28
Platinum
Au
2
1^gX
Gold
2.54
Hg
1
1_
2
Mercury
Tl
2
>H^gX
1.62
Thallium
16.65
19.25
21.02
22.61
22.65
21.09
19.3
13.534 10.4375
11.85
6.8251
7.5496
7.8640
7.8335
8.4382
8.9670
8.9588
9.2255
6.1082
2233
4603
3017
5458
3422
5555
3186
5596
3033
5012
2466
4428
1768.3
3825
1064.18
2856
-38.83
356.73
304
1473
=$?'159
HCP
=$?'146
BCC
=$?'139
BCC
=$?'137
HCP
=$?'135
HCP
=$?'136
FCC
=$?'139
FCC
=$?'144
FCC
=$?'151 §rhom. =$?'170
HCP
Bi+C'WL ^W'a6X'eAX
Bi+C'WL ^W'a6b'eAX
Bi+C'WL ^W'a6W'eAX
Bi+C'WL ^W'a6a'eAX
Bi+C'WL ^W'a6e'eAX
Bi+C'WL ^W'a6c'eAX
Bi+C'WL ^W'a6f'eA^
Bi+C'WL ^W'a6^_'eA^
Bi+C'WL ^W'a6^_'eAX
B\/C'eK^
+4
`5
`X[b[W[a[6
`X[W[e[7[7^
`X[b[4[e[]
`X[b[4[e
`X[4
`^[3
`^[2
`1[b
(261) 105
(262) 106
(266) 107
(264) 108
(277) 109
(268) 110
(281) 111
(272) 112
(285) 113
104
Rf
1
1_
Radium
3
F2
1.33
Zirconium
Hafnium
0.9
5
5.2784
700
1737
BCC
BT5C'cA X
`2
Zr
!&"%5%6+
1+,%+A
3
UX'd
Rutherfordium
Db
Sg
Dubnium
Seaborgium
Bh
Bohrium
Hs
Hassium
Mt
Meitnerium
Ds
Rg
Darmstadtium Roentgenium
Cn
Copernicium
Uut
Ununtrium
>_
Carbon
^
2
3
X
1
1_
14
Si
15
>_
Silicon
2.33
8.1517
1414
2900
=:?'111
cubic
B(+C'bA X'bKX
`X[4[7W
72.64
32
Ge
3
>_
2.01
Germanium
-
5.323
7.8994
938.3
2820
=:?'122
§cubic
^_
X
X
B!,C'b6 'WA 'WK
`X[4
118.710
50
Sn
3
P0
1.96
Tin
7.31
7.3439
231.93
2602
=:?'141
§tetra.
Bh,C'W6 ^_'aAX'aKX
`X[4
207.2
82
Pb
3
>_
Lead
2.33
1.429
-218.3
=:?'73
13.6181
-182.9
-
1.696
-219.6
=:?'71
4
1HbgX
2.18
4
S°3/2
2.05
Antimony
6.697
8.6084
630.63
1587
=:?'138
§rhom.
Bh,C'W6 ^_'aAX'aKb
`3[a[73
208.98038
83
Bi
16
1HbgX
2.19
5.727
9.7886
817
614
=:?'119
rhom.
^_
X
b
B!,C'b6 'WA 'WK
`3[a[73
121.760
51
Sb
X
4
Arsenic
4
1HbgX
2.02
Bismuth
>X
Oxygen
b
1.823
10.4867
44.2
280.5
=:?'106
§
B(+C'bA X'bKb
`b[W[5[7b
74.92160
33
3
S
3
>X
2.58
Sulfur
1.96
10.3600
115.21
444.72
=:?'102
FCO
B(+C'bA X'bKW
`X[W[6[7X
78.96
34
Se
3
>X
2.55
Selenium
4.819
9.7524
221
685
=:?'116
§hex
^_
X
W
B!,C'b6 'WA 'WK
`X[4[e[7X
127.60
52
Te
2
>HbgX
3.16
Br
2
>HbgX
2.96
Bromine
1
1_
-
Neon
a
3.214
12.9676
-101.5
-34.04
=:?'99
B(+C'bA X'bKa
`1[b[a[c[71
79.904
35
21.5645
-246.08
X
18
e
B\+C'XA 'XK
_
39.948
Ar
1
1_
-
Argon
1.784
15.7596
-189.3
-185.8
=:?'97
B(+C'bA X'bKe
_
83.798
36
Kr
I
2
P°3/2
2.66
Iodine
1_
3
Xe
1
S0
2.60
Xenon
6.24
4.94
5.9
9.0096
10.4513
12.1298
449.51
988
113.7
184.3
-111.8
-108
=:?'135
hex
=:?'133
BCO
=:?'130
Bh,C'W6 ^_'aAX'aKW
Bh,C'W6 ^_'aAX'aKa
Bh,C'W6 ^_'aAX'aKe
`X[4[e[7X
`1[a[c[71
_
(209) 85
(210) 86
(222)
84
Po
3
>X
2.0
Polonium
Ununquadium Ununpentium
Ununhexium
Uuh
At
2
>HbgX
2.2
Astatine
302
B\/C'eKa
`1[b[a[c[71
117
Rn
1
1_
-
Radon
9.73
10.7485
-71
-61.7
=:?'145
B\/C'eKe
_
Ununseptium
Uuo
Ununoctium
6.0 ?
BT5C'aL ^W'e6X'cAX'd
+4
>+,%#6%&'V8*3+'#L'"0+'<3+$+5"A
@+A%/5'*2'F+,"+NWXP&#$
l'X_^^'F+,"+NWX'99IP'!33',%/0"A',+A+,:+6P
Lanthanides
57
Actinides
References:
-(%A"P/#:[';.#3L,8$P&#$'=G8"0+$8"%&?[
ITI'\856*##S'#L'I0+$%A",2'856'>02A%&A
]^A"'<6%"%#5['X___7X__^['856'#"0+,A
1 (or more) from
group VIA (O,
or S, Se)
118
Uus
Notes:
7'@+5A%"2')5%"A'8,+'/g&$ b'L#,'A#3%6A'856'/g9'#,'S/g&$ b'8"'
_H'I+3A%)A'L#,'/8A+A
7'!"#$%&'.+%/0"'*8A+6'#5' ^XI
7'='?'%56%&8"+'$8AA'5)$*+,'#L'$#A"'A"8*3+'%A#"#K+
7'I#$$#5'MN%68"%#5'1"8"+A'%5'*#36
7'<3+&",#5'I#5L%/P'*8A+6'#5'DR>!I'/)%6+3%5+A
7'j'%56%&8"+A'&,2A"83'A",)&"),+'%A')5)A)83'#,'$82',+k)%,+'
+NK3858"%#5
7'=$?'G+"833%&',86%)A['=:?'I#:83+5"',86%)A
1 (or more) from
transition metal
group
1
Krypton
3.12
3.75
11.8138
13.9996
-7.3
59
-157.36 -153.22
=:?'114
BCO
=:?'110
^_
X
a
^_
X
e
B!,C'b6 'WA 'WK
B!,C'b6 'WA 'WK
`1[a[71
_
126.90447 54
131.293
53
3
P2
2.10
9.196
8.414
254
962
§cubic
B\/C'eKW
`X[4
(292)
116
Uup
0.9
-248.59
=:?'69
Chlorine
Tellurium
11.34
9.78
7.4167
7.2855
327.46
1749
271.3
1564
=$?'175
FCC
=:?'146
§rhom.
B\/C'eKX
B\/C'eKb
`2[W
`3[a
(289) 115
114
Uuq
17.4228
-188.12
-
B\+C'XA 'XK
71
35.453
Cl
20.1797
Ne
>HbgX
X
17
10
3.98
2
Fluorine
W
B\+C'XA 'XK
72
32.065
24.5874
-268.93
^AX
_
18.9984032
14.5341
-195.79
-
Phosphorus
As
9
F
B\+C'XA 'XK
`X[3[W[a[7X[73
30.97361
P
15.9994
3.44
X
3
1.90
8
O
1HbgX
0.1785
=:?'32
17
VIIA
3.04
4
Nitrogen
X
B\+C'XA 'XK
`X[4[7W
28.0855
16
VIA
14.0067
N
B\+C'XA 'XK
`3
26.981538
Al
7
2.55
X
13
12.0107
C
>H^gX
2.46
2075
=:?'82
6
15
VA
2.04
2
Boron
12
IIB
29
14
IVA
0.856
1.55
2.985
4.507
6.11
7.14
7.47
7.874
8.9
8.908
8.92
7.14
5.904
4.3407
6.1132
6.5615
6.8281
6.7462
6.7665
7.4340
7.9024
7.8810
7.6398
7.7264
9.3942
5.9993
63.38
759
842
1484
1541
2830
1668
3287
1910
3407
1907
2671
1246
2061
1538
2861
1495
2927
1455
2913
1084.62
2927
419.53
907
29.76
2204
=$?'227
BCC
=$?'197
FCC
=$?'162
HCP
=$?'147
HCP
=$?'134
BCC
=$?'128
BCC
=$?'127 §cubic =$?'126
BCC
=$?'125
HCP
=$?'124
FCC
=$?'128
FCC
=$?'134
§hex
=$?'135
§BCO
^
X
^
X
X
X
b
X
a
^
a
X
e
X
c
X
]
X
^_
^
^_
X
^_
X
^
B!,C'WA
B!,C'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA
B!,C'b6 'WA 'WK
`1
`2
`3
`X[b[4
`X[3[W[5
`X[3[e
`2[b[W[e[c
`X[3
`2[b
`X[3
`^[2
`2
`3
85.4678 38
87.62 39
88.90585 40
91.224 41
92.90638 42
95.94 43
(98) 44
101.07 45
102.90550 46
106.42 47
107.8682 48
112.411 49
114.818
37
Cs
6
2
0.82
Potassium
Gas
Beryllium
0.968
1.738
5.1391
7.6462
97.72
883
650
1090
=$?'186
BCC
=$?'160
HCP
B(+C'bA ^
B(+C'bA X
`1
`2
39.0983 20
40.078
19
Rb
5
1^gX
Sodium
K
4
2
0.93
Phase at STP
4.002602
He
Helium
Free Downloads at Vertex42.com
9.012182
Be
^
B\+C'XA
`1
22.989770
2
1.00794 !"#$%&'.+%/0"
2
1^gX -4,#)5671"8"+'9+:+3
2.2
;<3+&",#5+/8"%:%"2'=>8)3%5/?
H
(8$+
Hydrogen
;@+5A%"2'B(#"+C 0.0899 13.5984 -D#5%E8"%#5'<5+,/2'=+F?
;G+3"%5/'>#%5"'=HI? -259.14 -252.87 ;J#%3%5/'>#%5"'=HI?
FCC I,2A"83'1",)&"),+'B(#"+C
!"#$%&',86%)A'=K$?B(#"+C =:?'37
^A^
<3+&",#5'I#5L%/),8"%#5
`^[7^
>#AA%*3+'MN%68"%#5'1"8"+A'B(#"+C
2
IIA
^A^
`1[7^
1
18
VIIIA
-
138.9055
La
2
@bgX
1.10
Lanthanum
58
140.116
Ce
59
1
G°4
1.12
Cerium
Pr
140.90765
4
I°9/2
1.13
60
144.24
Nd
5
I4
1.14
Praseodymium Neodymium
61
Pm
(145)
6
H°5/2
-
Promethium
62
150.36
Sm
7
F0
1.17
Samarium
63
151.964
Eu
8
S°7/2
-
Europium
64
157.25
Gd
9
D°2
1.20
Gadolinium
65
158.92534
Tb
6
H°15/2
-
Terbium
66
Dy
162.500
5
I8
1.22
Dysprosium
67
164.93032
Ho
4
I°15/2
1.23
Holmium
68
167.259
Er
3
H6
1.24
Erbium
69
168.93421
Tm
2
F°7/2
1.25
Thulium
70
173.04
Yb
1
S0
-
Ytterbium
71
174.967
Lu
2
@bgX
1.27
Lutetium
7.353
5.244
7.901
8.219
8.551
8.795
9.066
9.321
6.57
9.841
6.146
6.689
6.64
7.01
7.264
5.582
5.6437
5.6704
6.1498
5.8638
5.9389
6.0215
6.1077
6.1843
6.2542
5.4259
5.5769
5.5387
5.473
5.5250
920
3464
798
3360
931
3290
1021
3100
1100
3000
1072
1803
822
1527
1313
3250
1356
3230
1412
2567
1474
2700
1497
2868
1545
1950
819
1196
1663
3402
=$?'187
§hex
=$?'182
FCC
=$?'182
§hex
=$?'181
§hex
=$?'183
HCP
=$?'180
§hex
=$?'180
BCC
=$?'180
HCP
=$?'177
HCP
=$?'178
HCP
=$?'176
HCP
=$?'176
HCP
=$?'176
HCP
=$?'176
FCC
=$?'174
HCP
^
X
^
^
X
b
X
W
X
a
X
e
X
c
X
c
^
X
f
X
^_
X
^^
X
^X
X
^b
X
^W
X
^W
Bi+C'a6 'eA
Bi+C'WL 'a6 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'a6 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'eA
Bi+C'WL 'a6 ^'eAX
+3
+3[W
+3[W
+3
+3
`X[3
`X[3
+3
+3[W
+3
+3
+3
`X[3
`X[3
+3
(227) 90
232.0381 91
231.0359 92
238.0289 93
(237) 94
(244) 95
(243) 96
(247) 97
(247) 98
(251) 99
(252) 100
(257) 101
(258) 102
(259) 103
(262)
89
Ac
2
@bgX
1.1
Actinium
10.07
5.17
1050
3200
FCC
BT5C'e6 ^'cAX
+3
Th
3
UX
1.3
Thorium
11.724
6.3067
1750
4820
=$?'179
FCC
BT5C'e6 X'cAX
+4
Copyright A. J. Millis 2013
Pa
4
h^^gX
1.5
Protactinium
15.37
5.89
1572
4000
=$?'163
§tetra
BT5C'aL X'e6 ^'cAX
`W[5
U
5
L°6
1.38
Uranium
19.05
6.1941
1135
3927
=$?'156
BCP
BT5C'aL b'e6 ^'cAX
`b[W[a[6
Np
6
L11/2
1.36
Neptunium
20.45
6.2657
644
4000
=$?'155
SO
BT5C'aL W'e6 ^'cAX
`b[W[5[e
Pu
7
F0
1.28
Plutonium
19.816
6.0260
640
3230
=$?'159 §mono.
BT5C'aL e'cAX
`b[4[a[e
Am
8
S°7/2
1.3
Americium
5.9738
1176
2011
=$?'173
HCP
BT5C'aL c'cAX
+3[W[a[e
Cm
9
D°2
1.3
Curium
13.51
5.9914
1345
3110
=$?'174
HCP
BT5C'aL c'e6 'cAX
+3
Bk
6
H°15/2
1.3
Berkelium
14.78
6.1979
1050
=$?'170
hex
BT5C'aL f'cAX
+3[W
Cf
5
I8
1.3
Californium
15.1
6.2817
900
hex
BT5C'aL ^_'cAX
+3
Es
4
I°15/2
1.3
Einsteinium
6.42
860
BT5C'aL ^^'cAX
+3
Fm
3
H6
1.3
Fermium
6.50
1527
BT5C'aL ^X'cAX
+3
Md
2
F°7/2
1.3
Mendelevium
6.58
827
BT5C'aL ^b'cAX
`X[3
No
1
S0
1.3
Nobelium
6.65
827
BT5C'aL ^W'cAX
`X[3
Lr
2
P°1/2 ?
-
Lawrencium
4.9 ?
1627
BT5C'aL ^W'cAX'cK'd
+3
Plus other atoms
(playing mainly
structural role)
Columbia University
Some jargon
Transition metal series: fill d-shell
`Early’ (left side)
`Late’ (right side)
3d
4d
5d
Copyright A. J. Millis 2013
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Long history in condensed matter physics
Frederich Wohler 1823
WO3: insulator (dull black)
add sodium
NaxWO3:
shiny metal
Copyright A. J. Millis 2013
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In modern terms: doping-driven metal
insulator transition
NaxTayWO3: # added electrons=x-y
Dubson and Holcomb,
Phys. Rev. B32 1955 (1985)
Copyright A. J. Millis 2013
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More history
1897: J.J. Thompson discovers electron.
Natural inference: transport and optical properties
largely determined by electrons
1900: Drude--classical model of electronic conduction
Classical mechanics: mean velocity ~ T1/2
idea: conduction should cease as T->0
Copyright A. J. Millis 2013
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More history
Classical mechanics: mean velocity ~ T1/2
idea: conduction should cease as T->0
This idea motivated KamerlinghOnnes to investigate properties of
metals at very low T
Copyright A. J. Millis 2013
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More history
Classical mechanics: mean velocity ~ T1/2
idea: conduction should cease as T->0
This idea motivated KamerlinghOnnes to investigate properties of
metals at very low T
Copyright A. J. Millis 2013
=>superconductivity!!
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More history
Classical mechanics: mean velocity ~ T1/2
idea: conduction should cease as T->0
This idea motivated KamerlinghOnnes to investigate properties of
metals at very low T
=>superconductivity!!
Importance of theory (even if wrong or incomplete) in motivating experiment
Copyright A. J. Millis 2013
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More history
1920s: Sommerfeld, Peierls, Bloch--quantum theory
Fermi statistics=> non-zero electron velocity even at T=0
Neglect electron-electron interactions =>basic equation:
� ∇2
i
H=−
+ Vlattice (ri )
2m
i
eigenfunctions: ψnk (r) = eik·r unk (r)
modulated plane wave, extended throughout solid; propagates
at nonzero group velocity=>theory of metallic conduction
Copyright A. J. Millis 2013
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1931: Enter A. H. Wilson
The Master and Fellows of Trinity
College Cambridge; taken from
http://www.computerhistory.org/
semiconductor/timeline/1931The-Theory.html
Note: first sharp statement
metal and insulator distinct phases
Copyright A. J. Millis 2013
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Key role of energy gap=>
sharp distinction, metal vs insulator
Mattheis, PRB5 290 (1972)
� ∇2
i
H=−
+ Vlattice (ri )
2m
i
µ in bands:
metal
µ in gap:
insulator
In this theory, insulator
possible only if even #
electrons per cell
Copyright A. J. Millis 2013
FeO, CoO and NiO:
predicted to be metals
Columbia University
NiO is a very good insulator
Discussion of the paper by de Boer and
Verwey, reported by N. F. Mott with the
help of some notes of R. Peierls 1937
Yamaka and Sawamoto PR 112 1861 1958
Peierls: Importance of electron-electron interactions in partially filled
transition metal d-shell. Fundamental modification of usual ``band’’
picture. Need for real-space view of the physics.
Copyright A. J. Millis 2013
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Electronic configuration of atoms complicated
basic concept: ‘Hubbard U’
Number of electrons
on ion nd
configuration: label by nd, other
(L,S) quantum numbers
Charging energy U:
U = E[d
nd +1
U
E[nd ] = nd (nd − 1)
2
] + E[d
nd −1
] − 2E[d
nd
]
In transition metal oxides: Udd big because d wave
function small (esp in ‘3d’ series)
Copyright A. J. Millis 2013
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Consequence of ‘Charging energy’:
Mott “blocking” effect
Copyright A. J. Millis 2013
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Consequence of ‘Charging energy’:
Mott “blocking” effect
Copyright A. J. Millis 2013
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Consequence of ‘Charging energy’:
Mott “blocking” effect
Copyright A. J. Millis 2013
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Consequence of ‘Charging energy’:
Mott “blocking” effect
2dn=>dn-1dn+1 E~U
process suppressed
Copyright A. J. Millis 2013
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Consequence of ‘Charging energy’:
Mott “blocking” effect
2dn=>dn-1dn+1 E~U
process suppressed
Copyright A. J. Millis 2013
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Consequence of ‘Charging energy’:
Mott “blocking” effect
2dn=>dn-1dn+1 E~U
process suppressed
Copyright A. J. Millis 2013
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Consequence of ‘Charging energy’:
Mott “blocking” effect
2dn=>dn-1dn+1 E~U
process suppressed
Copyright A. J. Millis 2013
dndn-1=>dn-1dn
process allowed
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Consequence of ‘Charging energy’:
Mott “blocking” effect
2dn=>dn-1dn+1 E~U
process suppressed
dndn-1=>dn-1dn
process allowed
If U large enough, only holes (or doubly occupied
sites) can move=> if integer number of electrons
per site: Mott insulator
Copyright A. J. Millis 2013
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Something like this actually occurs
LaTiO3
http://dmft.rutgers.edu/
LDA/lmto/lmto_run.htm
Copyright A. J. Millis 2013
LaTiO3:
•# electrons/unit cell is odd
•material is insulating!
•YTiO3 even more insulating
Okimoto et al. PRB51 9581 (1995)
Columbia University
`Mott’ insulator:
fundamental paradigm for oxides
But be careful: in many cases ``Mott’’
is not the controlling physics
Copyright A. J. Millis 2013
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Basic properties of transition metals and
their oxides
Copyright A. J. Millis 2013
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Transition metal ion:
many accessible valence states
*Partly filled d-shell =>spin and
orbital degrees of freedom active
*Multiple valence states=>many
electronic configurations, many
bonding modalities easily available
*energies of different transition metal
valences compatible with energies of
different oxygen charge states
=>tunability of materials properties
Copyright A. J. Millis 2013
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Application: Catalysis
(ex: photon-induced water splitting)
sciencedirect.com
spigse.chem.es.osaka-u.ac.jp
Works, in essence, because of multiple d-valence: it is
easy to transfer electrons back and forth between free
oxygen and transition metal ions.
Goal: manipulate band-gap to match solar spectrum
Copyright A. J. Millis 2013
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Application: Respiration etc in biochemistry
From the following article
Versatility of biological non-heme Fe(II)
centers in oxygen activation reactions
Elena G Kovaleva & John D Lipscomb
Nature Chemical Biology 4, 186 - 193
(2008)
Works, in essence, because of multiple d-valence: it is
easy to transfer electrons back and forth between
free oxygen and Fe complex.
Copyright A. J. Millis 2013
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Application: Batteries
http://www.electronicslab.com/articles/
Li_Ion_reconstruct/
Works, in essence, because of multiple d-valence: it is
easy to transfer electrons into and out of CoO2 complex
Copyright A. J. Millis 2013
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Application: ``Memristor’’
Nature 453, 80-83 (2008)
Electrical observations of filamentary conductions for the resistive memory
switching in NiO films D. C. Kim, S. Seo, S. E. Ahn, D.-S. Suh, M. J. Lee et al.
Appl. Phys. Lett. 88, 202102 (2006)
In oxides works, in
essence, because of
multiple d-valence: it is
easy to transfer
electrons into and out of
NiO complex
Copyright A. J. Millis 2013
Columbia University
Many structures possible
Monoxide (e.g NiO): rocksalt structure
http://www.webelements.com/
compounds/nickel/
nickel_oxide.html
Copyright A. J. Millis 2013
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Many structures possible
ABO3 Perovskite (e.g LaNiO3):
A and B sites: 2 interpenetrating
cubic lattices
A-site. Typically
electrically inert
B-site: in our case,
transition metal
dont forget oxygen!
Fundamental motif: BO6 octahedron
Copyright A. J. Millis 2013
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Many structures possible
A2BO4 Perovskite (e.g La2CuO4):
Layered structure
Octahedra as in perovskite but
now organized two
dimensionally, with weak
interplane coupling
Copyright A. J. Millis 2013
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Many structures possible
Ruddlesden-Popper series An+1BnO3n+1
(e.g Sr2RuO4->Sr3Ru2O7->Sr4Ru3O10->SrRuO3)
Interpolates between 2
and 3 dimensional
coordination
Copyright A. J. Millis 2013
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Many structures possible
Double perovskite AA’BB’O6
Sr2FeMoO6
Copyright A. J. Millis 2013
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Many structures possible
Other structures: spinel (2 inequivalent sites: one rocksalt
and one tetrahedrally bonded_
tf.uni-kiel.de
Copyright A. J. Millis 2013
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Many structures possible
Corundum structure: V2O3; hematite....
Here, only
transition
metal ions
are shown
http://www.irm.umn.edu/hg2m/hg2m_b/hg2m_b.html
Copyright A. J. Millis 2013
Columbia University
Many structures possible
Many other structures: for example quasi one
dimensional arrays of transition metal ions (e.g. CuGeO3
below),
Basic point: multiple
configurations of transition
metal=>multiple structures
possible.
Copyright A. J. Millis 2013
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New development:
artificially constructed superlattices
LaTiO3/SrTiO3
LaMnO3/SrTiO3
(a): La (b) Ti
(c) Mn (d) composite
A. Ohtomo, D. A. Muller, J. L. Grazul
and H. Hwang, Nature 419 378 (2002)
Copyright A. J. Millis 2013
D. Muller Nat. Mat. 9 263 (2009)
Columbia University
New window onto basic questions
Photoemission (many-body
density of states--here
symmetrized) for n layers of
SrVO3 (paramagnetic metal)
embedded in SrTiO3 (insulator)
As SrVO3 becomes thinner,
gap opens
Copyright A. J. Millis 2013
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Strain control of metal-insulator transition in LaNiO3
Son et al Appl. Phys. Lett. 96 062114 (2010)
LAO: 1.3% compressive strain
LSAT: 0.8% tensile strain
DSO: 2.5% tensile strain
Critical thickness for metalinsulator transition depends on
magnitude and sign of strain
Copyright A. J. Millis 2013
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More exotic: ``design in new properties’’
e.g. Chaloupka/Khalliulin: ?New high-Tc superconductor?
Idea:
Bulk LaNiO3 Ni [d]7
(1 electron in two
degenerate eg bands).
Phys. Rev. Lett. 100 016404 (2008)
Copyright A. J. Millis 2013
In correctly chosen
structure, split eg
bands, get 1 electron in
1 band--”like” copperoxide high-Tc
Columbia University
Structure =>interesting behavior
Spins on geometrically
frustrated lattices
Copyright A. J. Millis 2013
Columbia University
Structure =>interesting behavior
Anisotropic Transport
Copyright A. J. Millis 2013
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Structure =>interesting behavior
Sr2RuO4: p-wave superconductor
Sr3Ru2O7: almost ferromagnetic (100x
susceptibility enhancement).
SrRuO3: Ferromagnetic (Tc ~150K).
Power-law conductivity. CaRuO3: no
fm, but power law
Copyright A. J. Millis 2013
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Important property of transition oxides:
partially filled d-shells
Hund’s first rule (maximize total spin)
=>magnetism
Copyright A. J. Millis 2013
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Magnetism and full spin polarization at temperatures
above room T
Nature 395, 677-680
(15 October 1998)
Copyright A. J. Millis 2013
Columbia University
Hunds second rule:
maximize orbital angular momentum
In solid state context: occupy a subset of d-orbitals
<=> ``orbital order’’=>anisotropic transport
Ex: Manganese perovskites band filling x
controls orbital ordering=>conductivity
Copyright A. J. Millis 2013
Nagaosa/Tokura Science 288 462
Columbia University
These effects can be induced by strain
Ex: CMR films-substrate-induced
strain=>(putative)
orbital ordering=>
conductivity.
Mechanism: ‘orbital
ordering’ triggered by
strain
Konishi et al , J. Phys Soc. Jpn 68 3790
Copyright A. J. Millis 2013
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Proximity (in energy) of several spin and
charge configurations=>
interesting and potentially useful
transport properties
Copyright A. J. Millis 2013
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Thermoelectric effects
Lots of waste heat in the
world:
Thermal conductivity
∆V
S=
∆T
A.Maignan, APs tutorial 2010
Copyright A. J. Millis 2013
=>from temperature
difference, can get
voltage=>current
=>recapture some
energy
Columbia University
Thermoelectric figure of merit
A.Maignan, APs tutorial 2010
Insulators Semicond.
Semimetals
Metals
S T
ZT =
ρ (κel + κlattice )
2
Want: high thermal
conductivity, low electrical
resistance, low thermal
resistance
Carrier density (simple band picture)
Copyright A. J. Millis 2013
Columbia University
(Over)simplification of physics
F = Energy − T × Entropy
Temperature gradient drives
carriers by coupling to entropy
Insulators Semicond.
Semimetals
Metals
Insulator: no carriers
Metal: entropy per carrier is
very small
Carrier density (simple band picture)
Transition metal oxide: partially filled d-shell=>high density of nearly
localized carriers, high spin=>high entropy/carrier
Copyright A. J. Millis 2013
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A.Maignan, APs tutorial 2010
Copyright A. J. Millis 2013
Columbia University
More extreme transport anomaly: ``Mott’’ transition
V2O3--and other oxides--can be switched from metallic to insulating
behavior by modest changes in temperature and pressure.
Note! Transitions
are first order
McWhan et al,PRL 27 941 (1971)
Copyright A. J. Millis 2013
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Device Application: bolometer
VO2: thermally driven metal-insulator transition
Tune device to critical
temperature. Small
amount of absorbed
heat=>large change in
resistance
Wu et al, Nanoletters 6 2313 (2006)
Copyright A. J. Millis 2013
Columbia University
VO2 proposed for actual use
Copyright A. J. Millis 2013
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Hysteresis is a problem
hard to ‘reset’ after
initial temperature pulse
Gurvitch et al http://www.ee.sunysb.edu/~serge/234.pdf
Copyright A. J. Millis 2013
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Partially filled d-shell:
``emergent’’ electronic properties
Copyright A. J. Millis 2013
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Emergent electronic properties
``Colossal’’
Magnetoresistance
Copyright A. J. Millis 2013
p-wave superconductivity
(and many other phases)
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Colossal Magnetoresistance: Phase competition
High T phase: ``insulating’’ (resistivity large
and increasing as T is decreased)
Low T phase metallic and ferromagnetic
Transition appears more or less continuous,
and for many years was thought to be second
order.
BUT......
Copyright A. J. Millis 2013
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Manganese perovskites
Average high-T structure
cubic perovskite
Many low-T phases
``charge/orbital order’’
(2d slice shown).
Note 4 unit cell period
Copyright A. J. Millis 2013
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Transition is first order
High T phase has period 4 ordering
Adams et al PRL 85 2553 and 3954 2000
Copyright A. J. Millis 2013
Columbia University
Transition is first order
Low T phase: ferromagnetic.
Well defined spin stiffness T phase has period 4
ordering
Copyright A. J. Millis 2013
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Transition is first order
Spin stiffness jumps discontinuously
as does 1/4,0,0 intensity
Copyright A. J. Millis 2013
Morals:
--Unusual high T behavior <=>short ranged
order
--First order transitions
(may be hard to see)
Columbia University
Emergent electronic properties
High Tc Superconductivity: in copper oxide
and iron arsenide superconductors
SCIENCE VOL 329 13 AUGUST 2010 825
J.Zhao, et.al, Nature Mater. 7, 953(2008)
Copyright A. J. Millis 2013
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Copper-oxide high Tc:
again competing density-wave phases
Wu et al Nature 477 7363 2011
Open question: how much of the physics of oxide
superconductivity arises from competing phases
Copyright A. J. Millis 2013
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summary:
profusion of emergent electronic phases
Manganite
Ruthenate
FeAs
Cuprate
Copyright A. J. Millis 2013
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Key Features of Transition Metal Oxides
•Multiple d configurations (both valence and spin/orbital
configuration at fixed valence)
•``Easy’’ charge exchange with other ions (esp O)
•Multiple bonding modalities <=> many possible structures
•Unusual transport: high carrier density but low coherence
scales
•Multiple electronic phases (some, unexpected)
All of these properties stem from electron-electron
interactions in the partly filled transition metal d-shell
The question: relate crystal structure,
chemistry to material properties
Copyright A. J. Millis 2013
Columbia University
Periodic Table to Properties:
Stereotypical Theorist’s View
Copyright A. J. Millis 2013
Columbia University
Periodic Table to Properties:
Stereotypical Theorist’s View
PeriodicTable
Periodic table from J H Wood; quoted in Z Fisk 2010 KITP talk
Copyright A. J. Millis 2013
Columbia University
Periodic Table to Properties:
Stereotypical Theorist’s View
PeriodicTable
Ignore details: write
Quantum Field Theory
=>
κ µνλ
� Aµ ∂ν Aλ + ...
2
Periodic table from J H Wood; quoted in Z Fisk 2010 KITP talk
Copyright A. J. Millis 2013
Columbia University
Periodic Table to Properties:
Stereotypical Theorist’s View
PeriodicTable
Ignore details: write
Quantum Field Theory
=>
κ µνλ
� Aµ ∂ν Aλ + ...
2
These lectures--where we are on attempts to do better:
Periodic table from J H Wood; quoted in Z Fisk 2010 KITP talk
Copyright A. J. Millis 2013
Columbia University
Periodic Table to Properties:
Stereotypical Theorist’s View
PeriodicTable
Periodic Table of the Elements
GROUP 1
IA
1
!"#$%&'()$*+,
12$*#3
2
1^gX
2.2
Hydrogen
0.0899
-259.14
=:?'37
13.5984
-252.87
-
2
IIA
^A^
`1[7^
3
Li
Period
2
1^gX
0.82
Potassium
13
IIIA
5
Gas
Mg
1
1_
1.31
1
1_
1.00
Calcium
Liquid
!*A#3)"+'Q+,#
!"#$%&'G8AA'R5%"
!:#/86,#'I#5A"85"
J8A+'#L'(8"),83'9#/8,%"0$A
J#3"E$855'&#5A"85"
<3+&",#5'G8AA
Noble Gas
Alkaline Earth
Metals
Halogens
Transition Metals
Non Metals
Poor Metals
3
IIIB
21
Metalloids
4
IVB
44.955910
Sc
2
@bgX
1.36
Scandium
22
5
VB
47.867
Ti
23
3
UX
1.54
Titanium
m)
e
k
m+
$+c X
,_
+F
e
U
D
2 S hc X
<3+&",#5'T86%)A'=I38AA%&83?
<3+&",#5'F#3"
<3+$+5",2'I08,/+
U8,8682'I#5A"85"
L%5+7A",)&"),+'&#5A"85"
U%,A"'T86%8"%#5'I#5A"85"
Rare Earth Metals
BWXC
1#),&+O'K02A%&AP5%A"P/#:
Common Constants
Solid Synthetic
Alkali Metals
Magnesium
Ca
†
Categories
2
1^gX
0.82
Rubidium
Sr
1
1_
0.95
Strontium
Y
2
D3/2
1.22
Yttrium
1.532
2.63
4.472
4.1771
5.6949
6.2173
39.31
688
777
1382
1526
3345
=$?'248
BCC
=$?'215
FCC
=$?'180
HCP
Bh,C'aA ^
Bh,C'aA X
Bh,C'W6 ^'aAX
`1
`2
`3
132.90545 56
137.327
55
2
1^gX
0.79
Cesium
Ba
1
1_
0.89
Barium
1.879
3.51
3.8939
5.2117
28.44
671
727
1870
=$?'265
BCC
=$?'222
BCC
Bi+C'eA ^
Bi+C'eA X
`1
`2
(223) 88
(226)
87
Fr
7
Phase at STP
1_
1.57
Beryllium
6
VIB
50.9415
V
4
UbgX
1.63
Vanadium
24
7
1b
1.66
Chromium
25
4,8:%"8"%#5'I#5A"85"
G#38,'48A'I#5A"85"
G#38,'F#3)$+'=D6+83'48A?
>D
>385&S'I#5A"85"
>,#"#57<3+&",#5'G8AA'T8"%#
T26*+,/'I#5A"85"
2.8179403x10-15 m
1.602176x10-19 J
1.602176x10-19 C
96 485.3399 C/mol
0.0072973525
3.7417749x10-16 W m2
1+&#56'T86%8"%#5'I#5A"85"
1K++6'#L'9%/0"'%5'8'F8&))$
1K++6'#L'A#)56'%5'8%,'8"'1V>
1"8568,6'>,+AA),+
8
VIII
54.938049
Mn
6
1agX
1.55
Manganese
G
R
-273.15 °C
1.660539x10-27 kg
23
-1
6.022142x10 mol
2.718281828
1.380650x10-23 J/K
9.10938215x10-31 kg
0.5110 MeV
7
VIIB
51.9961
Cr
9
VIII
26
55.845
Fe
5
Iron
@W
1.83
27
-11
p
h
m +gm K
R’
R ’c
R ’hc
ch/k
c
4
UfgX
1.88
Cobalt
28
-1 -2
11
IB
58.6934
Ni
3
6.67428x10 m kg s
8.314472 J mol-1 K -1
3
0.02241410 m /mol
3.14159265358979
6.626069x10-34 J s
1836.15267247
-1
10 973 732 m
15
3.289842x10 Hz
13.6057 eV
0.01438769 m K
299 792 458 m/s
343.2 m/s
101 325 Pa
10
VIII
58.933200
Co
2
1^gX
0.7
Francium
-
4.0727
BT5C'cA ^
`1
Ra
985"085%6+'1+,%+A
3
UW
1.91
Nickel
63.546
Cu
2
1^gX
1.90
Copper
10.811
B
2
>H^gX
2.04
Boron
2.46
8.2980
2075
4000
=:?'82
rhom.
B\+C'XA X'XK^
`3
26.981538
13
Al
2
>H^gX
1.61
Aluminum
2.7
5.9858
660.32
2519
=$?'143
FCC
B(+C'bA X'bK^
`3
65.409 31
69.723
YWXZ
12
IIB
29
14
IVA
30
Zn
1
Zinc
1_
1.65
Ga
2
>H^gX
1.81
Gallium
1_
!&"%5%6+
1+,%+A
Notes:
7'@+5A%"2')5%"A'8,+'/g&$ b'L#,'A#3%6A'856'/g9'#,'S/g&$ b'8"'
_H'I+3A%)A'L#,'/8A+A
7'!"#$%&'.+%/0"'*8A+6'#5' ^XI
7'='?'%56%&8"+'$8AA'5)$*+,'#L'$#A"'A"8*3+'%A#"#K+
7'I#$$#5'MN%68"%#5'1"8"+A'%5'*#36
7'<3+&",#5'I#5L%/P'*8A+6'#5'DR>!I'/)%6+3%5+A
7'j'%56%&8"+A'&,2A"83'A",)&"),+'%A')5)A)83'#,'$82',+k)%,+'
+NK3858"%#5
7'=$?'G+"833%&',86%)A['=:?'I#:83+5"',86%)A
References:
-(%A"P/#:[';.#3L,8$P&#$'=G8"0+$8"%&?[
ITI'\856*##S'#L'I0+$%A",2'856'>02A%&A
]^A"'<6%"%#5['X___7X__^['856'#"0+,A
>+,%#6%&'V8*3+'#L'"0+'<3+$+5"A
@+A%/5'*2'F+,"+NWXP&#$
l'X_^^'F+,"+NWX'99IP'!33',%/0"A',+A+,:+6P
3
F2
1.33
Zirconium
Nb
6
D1/2
1.60
Niobium
Mo
7
S3
2.16
Molybdenum
Tc
6
S5/2
1.9
Technetium
Ru
5
F5
2.20
Ruthenium
Rh
4
F9/2
2.28
Rhodium
Pd
1
S0
2.20
Palladium
Ag
2
S1/2
1.93
Silver
Cd
1
S0
1.69
Cadmium
In
2
P°1/2
1.78
Indium
6.511
8.57
10.28
11.5
12.37
12.45
12.023
10.49
8.65
7.31
6.6339
6.7589
7.0924
7.28
7.3605
7.4589
8.3369
7.5762
8.9938
5.7864
1855
4409
2477
4744
2623
4639
2157
4265
2334
4150
1964
3695
1554.9
2963
961.78
2162
321.07
767
156.6
2072
=$?'160
HCP
=$?'146
BCC
=$?'139
BCC
=$?'136
HCP
=$?'134
HCP
=$?'134
FCC
=$?'137
FCC
=$?'144
FCC
=$?'151
§hex
=$?'167
§tetra.
Bh,C'W6 X'aAX
Bh,C'W6 W'aA^
Bh,C'W6 a'aA^
Bh,C'W6 a'aAX
Bh,C'W6 c'aA^
Bh,C'W6 ]'aA^
Bh,C'W6 ^_
Bh,C'W6 ^_'aA^
Bh,C'W6 ^_'aAX
Bh,C'W6 ^_'aAX'aK^
+4
`3[a
`X[b[W[a[6
`W[7
`X[b[4[e[]
`X[3[W
`2[W
`1
`2
`3
178.49 73
180.9479 74
183.84 75
186.207 76
190.23 77
192.217 78
195.078 79
196.96655 80
200.59 81
204.3833
72
Hf
3
UX
1.3
Hafnium
Ta
4
UbgX
1.5
Tantalum
W
5
@_
2.36
Tungsten
Re
6
1agX
1.9
Rhenium
Os
Ir
5
@W
2.2
Osmium
4
UfgX
2.2
Iridium
Pt
3
@b
2.28
Platinum
Au
2
1^gX
Gold
2.54
Hg
1
1_
2
Mercury
Tl
2
>H^gX
1.62
Thallium
13.31
16.65
19.25
21.02
22.61
22.65
21.09
19.3
13.534 10.4375
11.85
6.8251
7.5496
7.8640
7.8335
8.4382
8.9670
8.9588
9.2255
6.1082
2233
4603
3017
5458
3422
5555
3186
5596
3033
5012
2466
4428
1768.3
3825
1064.18
2856
-38.83
356.73
304
1473
=$?'159
HCP
=$?'146
BCC
=$?'139
BCC
=$?'137
HCP
=$?'135
HCP
=$?'136
FCC
=$?'139
FCC
=$?'144
FCC
=$?'151 §rhom. =$?'170
HCP
Bi+C'WL ^W'a6X'eAX
Bi+C'WL ^W'a6b'eAX
Bi+C'WL ^W'a6W'eAX
Bi+C'WL ^W'a6a'eAX
Bi+C'WL ^W'a6e'eAX
Bi+C'WL ^W'a6c'eAX
Bi+C'WL ^W'a6f'eA^
Bi+C'WL ^W'a6^_'eA^
Bi+C'WL ^W'a6^_'eAX
B\/C'eK^
+4
`5
`X[b[W[a[6
`X[W[e[7[7^
`X[b[4[e[]
`X[b[4[e
`X[4
`^[3
`^[2
`1[b
(261) 105
(262) 106
(266) 107
(264) 108
(277) 109
(268) 110
(281) 111
(272) 112
(285) 113
104
Rf
1
0.9
Radium
5
5.2784
700
1737
BCC
BT5C'cA X
`2
Zr
3
UX'd
Rutherfordium
Db
Dubnium
Sg
Seaborgium
Bh
Bohrium
Hs
Hassium
Mt
Meitnerium
4.002602
He
1
1_
-
Helium
Free Downloads at Vertex42.com
0.856
1.55
2.985
4.507
6.11
7.14
7.47
7.874
8.9
8.908
8.92
7.14
5.904
4.3407
6.1132
6.5615
6.8281
6.7462
6.7665
7.4340
7.9024
7.8810
7.6398
7.7264
9.3942
5.9993
63.38
759
842
1484
1541
2830
1668
3287
1910
3407
1907
2671
1246
2061
1538
2861
1495
2927
1455
2913
1084.62
2927
419.53
907
29.76
2204
=$?'227
BCC
=$?'197
FCC
=$?'162
HCP
=$?'147
HCP
=$?'134
BCC
=$?'128
BCC
=$?'127 §cubic =$?'126
BCC
=$?'125
HCP
=$?'124
FCC
=$?'128
FCC
=$?'134
§hex
=$?'135
§BCO
B!,C'WA ^
B!,C'WA X
B!,C'b6 ^'WAX
B!,C'b6 X'WAX
B!,C'b6 b'WAX
B!,C'b6 a'WA^
B!,C'b6 a'WAX
B!,C'b6 e'WAX
B!,C'b6 c'WAX
B!,C'b6 ]'WAX
B!,C'b6 ^_'WA^
B!,C'b6 ^_'WAX
B!,C'b6 ^_'WAX'WK^
`1
`2
`3
`X[b[4
`X[3[W[5
`X[3[e
`2[b[W[e[c
`X[3
`2[b
`X[3
`^[2
`2
`3
85.4678 38
87.62 39
88.90585 40
91.224 41
92.90638 42
95.94 43
(98) 44
101.07 45
102.90550 46
106.42 47
107.8682 48
112.411 49
114.818
37
Cs
6
Be
1
0.968
1.738
5.1391
7.6462
97.72
883
650
1090
=$?'186
BCC
=$?'160
HCP
B(+C'bA ^
B(+C'bA X
`1
`2
39.0983 20
40.078
19
Rb
5
1^gX
Sodium
K
4
2
0.93
18
VIIIA
2
1.00794 -!"#$%&'.+%/0"
2
1^gX -4,#)5671"8"+'9+:+3
2.2
;<3+&",#5+/8"%:%"2'=>8)3%5/?
H
9.012182
0.535
1.848
5.3917
9.3227
180.54
1342
1287
2470
=$?'152
BCC
=$?'112
HCP
B\+C'XA ^
B\+C'XA X
`1
`2
22.989770 12
24.3050
11
Na
3
1^gX
0.98
Lithium
4
Ds
Rg
Darmstadtium Roentgenium
Cn
Copernicium
Uut
Ununtrium
6
15
VA
12.0107
C
3
>_
2.55
Carbon
2.26
11.2603
3550
4027
=:?'77
hex
B\+C'XA X'XKX
`X[4[7W
28.0855
14
Si
7
14.0067
N
3
>_
2.01
Germanium
5.323
7.8994
938.3
2820
=:?'122
§cubic
B!,C'b6 ^_'WAX'WKX
`X[4
118.710
50
Sn
3
P0
1.96
Tin
7.31
7.3439
231.93
2602
=:?'141
§tetra.
Bh,C'W6 ^_'aAX'aKX
`X[4
207.2
82
Pb
3
>_
Lead
2.33
1HbgX
3.04
Nitrogen
P
>_
Ge
4
1.251
14.5341
-210.1
-195.79
=:?'75
B\+C'XA X'XKb
`X[3[W[a[7X[73
30.97361
15
3
1.90
Silicon
2.33
8.1517
1414
2900
=:?'111
cubic
B(+C'bA X'bKX
`X[4[7W
72.64
32
16
VIA
4
1HbgX
2.18
Arsenic
4
S°3/2
2.05
Antimony
6.697
8.6084
630.63
1587
=:?'138
§rhom.
Bh,C'W6 ^_'aAX'aKb
`3[a[73
208.98038
83
Bi
4
1HbgX
2.02
Bismuth
S
3
>X
2.58
Sulfur
1.96
10.3600
115.21
444.72
=:?'102
FCO
B(+C'bA X'bKW
`X[W[6[7X
78.96
34
Se
3
>X
2.55
Selenium
4.819
9.7524
221
685
=:?'116
§hex
B!,C'b6 ^_'WAX'WKW
`X[4[e[7X
127.60
52
Te
>HbgX
3.98
Fluorine
10
20.1797
Ne
1
1_
-
Neon
1.696
0.9
17.4228
21.5645
-219.6
-188.12 -248.59 -246.08
=:?'71
=:?'69
B\+C'XA X'XKa
B\+C'XA X'XKe
71
_
35.453 18
39.948
17
Cl
2
>HbgX
3.16
Chlorine
3.214
12.9676
-101.5
-34.04
=:?'99
B(+C'bA X'bKa
`1[b[a[c[71
79.904
35
Br
2
>HbgX
2.96
Bromine
Ar
1
1_
-
Argon
1.784
15.7596
-189.3
-185.8
=:?'97
B(+C'bA X'bKe
_
83.798
36
Kr
I
3
P2
2.10
Po
3
>X
2.0
Polonium
Ununquadium Ununpentium
Ununhexium
Uuh
1
1_
3
Krypton
2
P°3/2
2.66
Iodine
Xe
1
S0
2.60
Xenon
At
2
>HbgX
2.2
Astatine
302
B\/C'eKa
`1[b[a[c[71
117
Rn
1
1_
-
Radon
9.73
10.7485
-71
-61.7
=:?'145
B\/C'eKe
_
118
Uus
Ununseptium
Uuo
Ununoctium
6.0 ?
57
138.9055
La
2
@bgX
1.10
Lanthanum
58
140.116
Ce
1
G°4
1.12
Cerium
59
Pr
140.90765
4
I°9/2
1.13
60
144.24
Nd
5
I4
1.14
Praseodymium Neodymium
61
Pm
(145)
6
H°5/2
-
Promethium
62
150.36
Sm
7
F0
1.17
Samarium
63
151.964
Eu
8
S°7/2
-
Europium
64
157.25
Gd
9
D°2
1.20
Gadolinium
65
158.92534
Tb
6
H°15/2
-
Terbium
66
Dy
162.500
5
I8
1.22
Dysprosium
67
164.93032
Ho
4
I°15/2
1.23
Holmium
68
167.259
Er
3
H6
1.24
Erbium
69
168.93421
Tm
2
F°7/2
1.25
Thulium
70
173.04
Yb
1
S0
-
Ytterbium
71
174.967
Lu
2
@bgX
1.27
Lutetium
7.353
5.244
7.901
8.219
8.551
8.795
9.066
9.321
6.57
9.841
6.146
6.689
6.64
7.01
7.264
5.582
5.6437
5.6704
6.1498
5.8638
5.9389
6.0215
6.1077
6.1843
6.2542
5.4259
5.5769
5.5387
5.473
5.5250
920
3464
798
3360
931
3290
1021
3100
1100
3000
1072
1803
822
1527
1313
3250
1356
3230
1412
2567
1474
2700
1497
2868
1545
1950
819
1196
1663
3402
=$?'187
§hex
=$?'182
FCC
=$?'182
§hex
=$?'181
§hex
=$?'183
HCP
=$?'180
§hex
=$?'180
BCC
=$?'180
HCP
=$?'177
HCP
=$?'178
HCP
=$?'176
HCP
=$?'176
HCP
=$?'176
HCP
=$?'176
FCC
=$?'174
HCP
Bi+C'a6 ^'eAX
Bi+C'WL ^'a6 ^'eAX
Bi+C'WL b'eAX
Bi+C'WL W'eAX
Bi+C'WL a'eAX
Bi+C'WL e'eAX
Bi+C'WL c'eAX
Bi+C'WL c'a6 ^'eAX
Bi+C'WL f'eAX
Bi+C'WL ^_'eAX
Bi+C'WL ^^'eAX
Bi+C'WL ^X'eAX
Bi+C'WL ^b'eAX
Bi+C'WL ^W'eAX
Bi+C'WL ^W'a6 ^'eAX
+3
+3[W
+3[W
+3
+3
`X[3
`X[3
+3
+3[W
+3
+3
+3
`X[3
`X[3
+3
(227) 90
232.0381 91
231.0359 92
238.0289 93
(237) 94
(244) 95
(243) 96
(247) 97
(247) 98
(251) 99
(252) 100
(257) 101
(258) 102
(259) 103
(262)
89
Ac
2
@bgX
1.1
Actinium
10.07
5.17
1050
3200
FCC
BT5C'e6 ^'cAX
+3
Th
3
UX
1.3
Thorium
11.724
6.3067
1750
4820
=$?'179
FCC
BT5C'e6 X'cAX
+4
Pa
4
h^^gX
1.5
Protactinium
15.37
5.89
1572
4000
=$?'163
§tetra
BT5C'aL X'e6 ^'cAX
`W[5
U
5
L°6
1.38
Uranium
19.05
6.1941
1135
3927
=$?'156
BCP
BT5C'aL b'e6 ^'cAX
`b[W[a[6
Np
6
L11/2
1.36
Neptunium
20.45
6.2657
644
4000
=$?'155
SO
BT5C'aL W'e6 ^'cAX
`b[W[5[e
Pu
7
F0
1.28
Plutonium
19.816
6.0260
640
3230
=$?'159 §mono.
BT5C'aL e'cAX
`b[4[a[e
Am
8
S°7/2
1.3
Americium
5.9738
1176
2011
=$?'173
HCP
BT5C'aL c'cAX
+3[W[a[e
Cm
9
D°2
1.3
Curium
13.51
5.9914
1345
3110
=$?'174
HCP
BT5C'aL c'e6 'cAX
+3
Bk
6
H°15/2
1.3
Berkelium
14.78
6.1979
1050
=$?'170
hex
BT5C'aL f'cAX
+3[W
Cf
5
I8
1.3
Californium
15.1
6.2817
900
hex
BT5C'aL ^_'cAX
+3
Es
4
I°15/2
1.3
Einsteinium
6.42
860
BT5C'aL ^^'cAX
+3
Fm
3
H6
1.3
Fermium
6.50
1527
BT5C'aL ^X'cAX
+3
Md
2
F°7/2
1.3
Mendelevium
6.58
827
BT5C'aL ^b'cAX
`X[3
Richer Description of
Properties
3.12
3.75
11.8138
13.9996
-7.3
59
-157.36 -153.22
=:?'114
BCO
=:?'110
B!,C'b6 ^_'WAX'WKa
B!,C'b6 ^_'WAX'WKe
`1[a[71
_
126.90447 54
131.293
53
Tellurium
9.196
8.414
254
962
§cubic
B\/C'eKW
`X[4
(292)
116
Uup
2
6.24
4.94
5.9
9.0096
10.4513
12.1298
449.51
988
113.7
184.3
-111.8
-108
=:?'135
hex
=:?'133
BCO
=:?'130
Bh,C'W6 ^_'aAX'aKW
Bh,C'W6 ^_'aAX'aKa
Bh,C'W6 ^_'aAX'aKe
`X[4[e[7X
`1[a[c[71
_
(209) 85
(210) 86
(222)
84
11.34
9.78
7.4167
7.2855
327.46
1749
271.3
1564
=$?'175
FCC
=:?'146
§rhom.
B\/C'eKX
B\/C'eKb
`2[W
`3[a
(289) 115
114
Uuq
F
>X
24.5874
-268.93
^AX
_
18.9984032
3.44
Oxygen
1HbgX
5.727
9.7886
817
614
=:?'119
rhom.
B!,C'b6 ^_'WAX'WKb
`3[a[73
121.760
51
Sb
O
9
3
1.429
13.6181
-218.3
-182.9
=:?'73
B\+C'XA X'XKW
72
32.065
16
2.19
1.823
10.4867
44.2
280.5
=:?'106
§
B(+C'bA X'bKb
`b[W[5[7b
74.92160
33
17
VIIA
15.9994
4
Phosphorus
As
8
0.1785
=:?'32
BT5C'aL ^W'e6X'cAX'd
+4
Lanthanides
2
6.941
2
1
(8$+
Hydrogen
;@+5A%"2'B(#"+C 0.0899 13.5984 -D#5%E8"%#5'<5+,/2'=+F?
;G+3"%5/'>#%5"'=HI? -259.14 -252.87 ;J#%3%5/'>#%5"'=HI?
FCC I,2A"83'1",)&"),+'B(#"+C
!"#$%&',86%)A'=K$?B(#"+C =:?'37
^A^
<3+&",#5'I#5L%/),8"%#5
`^[7^
>#AA%*3+'MN%68"%#5'1"8"+A'B(#"+C
Actinides
1
1.00794
H
No
1
S0
1.3
Nobelium
6.65
827
BT5C'aL ^W'cAX
`X[3
Lr
2
P°1/2 ?
-
Lawrencium
4.9 ?
1627
BT5C'aL ^W'cAX'cK'd
+3
Understand and control
what situations give which
properties
These lectures--where we are on attempts to do better:
Periodic table from J H Wood; quoted in Z Fisk 2010 KITP talk
Copyright A. J. Millis 2013
Columbia University
Basic chemistry and physics
•Crystal type and nominal valence
•Crystal structure and level ordering
•Energetics
•Interactions
•Charge transfer
•Hybridization
•Classification and Summary
Copyright A. J. Millis 2013
Columbia University
Crystal type and nominal valence
Basic idea
--Oxygen: very electronegative.
In vacuum: 1s22s22p4
In solid: oxygen ``wants`` to fill up p-shell
(O2-: 1s22s22p6)
--Transition metal: less electronegative
In vacuum: [Ar]dn4s2
In solid: ``wants`` to lose 4s electrons, change # d =>[Ar]dn+m
By controlling number of oxygen per transition metal,
and number and electronegativity of other elements, can
control filling of d-shell
Copyright A. J. Millis 2013
Columbia University
Crystal type and nominal valence
Monoxide: rocksalt structure
Simple picture: removal energy of Ni
4s low; electron affinity of O 2p high
=>
Ni: [Ar]3d8 O: [He]2s22p6
http://www.webelements.com/
compounds/nickel/
nickel_oxide.html
Complication: Ni-O hybridization
Ni: [Ar]3d8+x
O: [He]2s22p6-x
??how big is x??
Copyright A. J. Millis 2013
Columbia University
Crystal type and nominal valence
ABO3 perovskite, cubic phase
A-site. Typically
electrically inert;
choice of A ion
controls valence of
B site
B-site: in our case,
transition metal with
partly filled d-shell
dont forget oxygen!
http://en.wikipedia.org/wiki/File:Perovskite.jpg
Copyright A. J. Millis 2013
Columbia University
Crystal type and nominal valence
ABO3: Presence of A-site ion stabilizes different valence
Simple picture: removal energies of La
5d, 6s V 4s, 3d low; electron affinity of
O2p high
=>LaVO3
La3+ [Xe] V3+: [Ar]3d2 O2-: [He]2s22p6
Complication: V-O hybridization
V: [Ar]3d2+3x
Copyright A. J. Millis 2013
??how big is x??
O: [He]2s22p6-x
Columbia University
Crystal type and nominal valence
Vary A-site ion: change valence
Simple picture: removal energies of La
5d, 6s V 4s, 3d low; electron affinity of
Simple picture: removal energies
O2p high
of La 5d, 6s V 4s, 3d low;
=>LaVO3
electron affinity of O2p high
La3+ [Xe] V3+: [Ar]3d2 O2-: [He]2s22p6
=>SrVO3
Complication: V-O hybridization
V4+: [Ar]3d1+3x O2-: [He]2s22p6-x
V: [Ar]3d2+3x
Copyright A. J. Millis 2013
??how big is x??
O: [He]2s22p6-x
Columbia University
Crystal structure and level ordering
Basic idea
--d-levels degenerate in free space.
--Solid state: local environment breaks the symmetry
By controlling the local (point) symmetry of the
transition metal ion, can control which orbitals are
occupied.
Copyright A. J. Millis 2013
Columbia University
Symmetry breaking: Group Theory
Simplest example: nearly cubic environment
Free space
(no spin orb)
Cubic
O(3)
Oh
group:
representation
L=2
eg
t2g
x2-y2; 3z2-r2
xy; xz; yz
picture
http://winter.group.shef.ac.uk/
orbitron/AOs/3d/index.html
Copyright A. J. Millis 2013
Columbia University
Electrostatic contribution to crystal field splitting small
Simple estimate: point charge model:
O charge -2 at distance RO~2A; ``size’’ of d orbital Rd~0.5A
-2
-2
V(�r) =
i=1...6
-2
4e2
� i|
|�r − R
Level splitting: expand for small r. First
deviation from spherical symmetry is
quadrupole term
-2
Vquad
Copyright A. J. Millis 2013
�
4e2
∼
RO
�
Rd
RO
�4
∼ 0.1eV
Columbia University
Dominant contribution: hybridization to ligand (oxygen)
tpd
+
-
+
dx2 −y2
-
+
pσ
εd
εp
p-d anti
bonding
p-d
bonding
In almost all transition metal oxides, the near-fermi
surface states are p-d ANTIBONDING states.
Question: how much d and how much p character
Copyright A. J. Millis 2013
Columbia University
d-p overlap stronger for eg than t2g
-
+
in solid
(cubic symm)
pσ
dxy
Copyright A. J. Millis 2013
free ion
eg − pσ
antibonding
t2g − pπ
antibonding
t2g − pπ
bonding
eg − pσ
bonding
-
-
+
-
dx2 −y2
+
-
-
+
+
+
pπ
Columbia University
Lower than cubic symmetry
Rondinelli and Spaldin, Adv. Mater. 23 3363 (2011)
(Recall! what is depicted are the p-d
antibonding bands)
Copyright A. J. Millis 2013
Columbia University
Jahn-Teller (cubic-tetragonal) distortion
Define distortions
Copyright A. J. Millis 2013
Columbia University
Consequence of structural distortion:
change of hybridization
Jahn-Teller type: change Tm-O bond length
Change transition metaloxygen distance:
hybridization changes.
=>shifts bondingantibonding splitting
Copyright A. J. Millis 2013
Columbia University
Alternative: GdFeO3 distortion
Rotate octahedra about two axes
Induced by choice of A-site ion.
If dAA<2dBO then B-O-B bond is under compressive strain.
B-O bond relatively rigid =>B-O-B bond buckles.
Angle can be as small as 140o
Change in bandwidth: 30%
Copyright A. J. Millis 2013
Columbia University
Consequence of structural distortion:
change of hybridization
dxy
-
+
-
+
pπ
+
+
-
+
-
tpd
-
GdFeO3 type
dxy
Move oxygen out of alignment: overlap decreases
Copyright A. J. Millis 2013
Columbia University
How big are these effects?
Cubic perovskite structure:
eg-t2g splitting ~2eV
Copyright A. J. Millis 2013
Columbia University
Cartoon
p
O
p
θ
Tm
1
tpd → tpd cos θ
Copyright A. J. Millis 2013
Tm
2
Bandwidth ∼ cos2 θ
Columbia University
Density of states of t2g antibonding band as function of
amplitude of GdFeO3 distortion: ``LaVO3’’ (H. T. Dang)
Bandwidth varies from ~2.5eV to ~1.6eV
Copyright A. J. Millis 2013
Columbia University
Jahn-Teller (cubic-tetragonal) distortion
Analyse band structure in distorted case
λ : 2 − 3eV/Å
Copyright A. J. Millis 2013
Ederer and AJM, PRB 76 155105 (2007)
Popovic and Satpathy, PRL 84 1603 (2000)
Yin et al PRL 96 116405 (2006)
Columbia University
GdFeO3 rotation (trigonal) distortion
Analyse band structure in distorted case
◦
θ = 12
Local basis rotated WRT mean
crystal structure
θ = 12 Level splitting : 0.15eV
◦
θ = 20 Level splitting : 0.25eV
◦
Pavarini et al PRL 92 176403 (2004)
Copyright A. J. Millis 2013
Columbia University
free-space d-orbital: 5-fold degenerate
lattice effects lift degeneracy
Rondinelli and Spaldin, Adv. Mater. 23 3363 (2011)
Jahn-Teller (0.1/AA: 0.2eV) GdFeO3 Rotations (15o: 0.2eV)
Splittings from structural effects by themselves
are typically small compared to bandwidths.
Overall band-width renormalization important.
?Enhancement of splitting by manybody effects: orbital ordering?
Copyright A. J. Millis 2013
Columbia University