lecture 12 magnetic band structures and density of stateskasinath/pqm/lecture... · -9 -6 -3 0 3 6...
TRANSCRIPT
Lecture 12
Magnetic band structures and density of states
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
bcc Fe
space group = 229 (Im3m) one wyckoff position
Non magnetic bcc - Fe
-9 -6 -3 0 3 6 9Energy (eV)
0
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2
3
4
5
6
7
Densi
ty o
f S
tate
s
bcc FeNonmagnetic
Non magnetic bcc - Fe
space group = 229 (Im3m) one wyckoff position
Non magnetic bcc - Fe
space group = 229 (Im3m) one wyckoff position
Non magnetic à No. of spin-up electrons = No. of spin-down electrons
-9 -6 -3 0 3 6 9Energy (eV)
0
1
2
3
4
5
6
7
Densi
ty o
f S
tate
s
bcc FeNonmagnetic
spin-up + spin-down electrons
-9 -6 -3 0 3 6 9Energy (eV)
0
1
2
3
4
5
6
7
De
nsi
ty o
f S
tate
s
bcc FeNonmagnetic
-9 -6 -3 0 3 6 9Energy (eV)
0
1
2
3
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5
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7
De
nsi
ty o
f S
tate
s
-9 -6 -3 0 3 6 9Energy (eV)
0
1
2
3
4
5
6
7
De
nsi
ty o
f S
tate
s
divide by 2 spin-up
spin-down
-9 -6 -3 0 3 6 9Energy (eV)
0
1
2
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6
7
De
nsi
ty o
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tate
s
bcc FeNonmagnetic
-9 -6 -3 0 3 6 9Energy (eV)
0
1
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5
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7
De
nsi
ty o
f S
tate
s
-9-6-30369Energy (eV)
0
1
2
3
4
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7
De
nsity o
f Sta
tes
divide by 2 spin-up
spin-down
-9 -6 -3 0 3 6 9Energy (eV)
0
1
2
3
4
5
6
7
De
nsi
ty o
f S
tate
s
bcc FeNonmagnetic
-9 -6 -3 0 3 6 9Energy (eV)
-4
-2
0
2
4
Densi
ty o
f S
tate
s
spin-down
spin-up
Magnetic moment = No. of spin-up electrons – No. of spin-down electrons
A ferromagnet has a spontaneous magnetic moment — a magnetic moment even in zero applied magnetic field. The existence of a spontaneous moment suggests that electron spins and magnetic moments are arranged in a regular manner.
Ferromagnetic bcc-Fe
space group = ? No. of wyckoff positions = ?
The arrows correspond to magnetic moments, not electron spins ! There are electrons with “both” spins at each site. In other words, the arrows represent the “net” direction from “summing up” all the spins at each site
Ferromagnetic bcc-Fe
space group = ? No. of wyckoff positions = ?
All space group operations for the non-magnetic space group are still satisfied even upon the introduction of ferromagnetic order !
Ferromagnetic bcc-Fe
space group = 229 (Im3m) No. of wyckoff positions = 1
Ferromagnetic bcc-Fe
space group = 229 (Im3m) No. of wyckoff positions = 1
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
Ferromagnetic bcc−Fespin−upspin−downspin−upspin−down
Ferromagnetic bcc-Fe
space group = 229 (Im3m) No. of wyckoff positions = 1
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
Ferromagnetic bcc−Fespin−downspin−down
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
Ferromagnetic bcc−Fespin−upspin−up
Ferromagnetic bcc-Fe
-9 -6 -3 0 3 6 9Energy (eV)
-4
-2
0
2
4
Desn
ity o
f st
ate
s
total
Ferromagnetic bcc-Fe
What is the sign of the net magnetic moment ? How will it compare to atomic value ?
Fe: 26 electrons = 1s2 2s2 2p6 3s2 3p6 4s2 3d6
l = -2 -1 0 +1 +2
Atomic value = 4 μB Ferromagnetic bcc-Fe = 2.21 μB Why the discrepancy ?
-9 -6 -3 0 3 6 9Energy (eV)
-4
-2
0
2
4
Desn
ity o
f st
ate
stotalFe 3d
-9 -6 -3 0 3 6 9Energy (eV)
-4
-2
0
2
4
Desn
ity o
f st
ate
stotalFe 4sFe 3d
-9 -6 -3 0 3 6 9Energy (eV)
-0.2
-0.1
0
0.1
0.2D
esn
ity o
f st
ate
sFe 4s
Fe: 26 electrons = 1s2 2s2 2p6 3s2 3p6 4s2 3d6
l = -2 -1 0 +1 +2
Atomic value = 4 μB Ferromagnetic bcc-Fe = 2.21 μB Why the discrepancy ?
The bond formation in the bcc lattice, creates bonding and anti-bonding 4s-states, thereby reducing the net magnetic moment
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
Ferromagnetic bcc−Fespin−downspin−down
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
Ferromagnetic bcc−Fespin−upspin−up
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
bcc Fe
Ferromagnet No. of non-magnetic bands
= No. of spin-up/spin- down
Other magnetic orders
Other magnetic orders – simple cubic
Other magnetic orders – body centered cubic
Other magnetic orders – face centered cubic
Nonmagnetic bcc-Mn
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
Non magnetic bcc−Mn (delta − Mn)
-8 -6 -4 -2 0 2 4 6 8 10Energy (eV)
0
2
4
6
8
Densi
ty o
f st
ate
s
total
Non magnetic bcc-Mn (delta-Mn)
space group = 229 (Im3m) one wyckoff position
Antiferromagnetic bcc-Mn (delta-Mn)
space group = ? No. of wyckoff positions = ?
Antiferromagnetic bcc-Mn (delta-Mn)
space group = ? No. of wyckoff positions = ?
No. of symmtery operations for the non-magnetic crystal = 48 Unit cell of AFM structure = 2*non-magnetic
cell No. of symmetry operations for the AFM crystal = 8
Antiferromagnetic bcc-Mn (delta-Mn)
space group = Cmmm (65) No. of wyckoff positions = 2
How do the band structure and density of states of AFM-Mn differ from FM-Fe ? Is there any similarity ?
Antiferromagnetic bcc-Mn (delta-Mn)
-8 -6 -4 -2 0 2 4 6 8Energy (eV)
-4
-2
0
2
4
De
nsi
ty o
f st
ate
s
total
Antiferromagnetic bcc-Mn
−5.0
0.0
5.0
¡F
K K U Y Z T C P
Ener
gy ¡
n(k) [
eV]
Antiferromagnetic bcc−Mn
spin−upspin−up
−5.0
0.0
5.0
¡F
K K U Y Z T C P
Ener
gy ¡
n(k) [
eV]
Antiferromagnetic bcc−Mn
spin−downspin−down
Ferromagnetic bcc-Fe
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
Ferromagnetic bcc−Fespin−downspin−down
−5.0
0.0
5.0
10.0
¡F
K H N K P N P
Ener
gy ¡
n(k) [
eV]
Ferromagnetic bcc−Fespin−upspin−up
-9 -6 -3 0 3 6 9Energy (eV)
-4
-2
0
2
4
Desn
ity o
f st
ate
s
total
Ferromagnetic bcc-Fe
Mn: 25 electrons = 1s2 2s2 2p6 3s2 3p6 4s2 3d5
l = -2 -1 0 +1 +2
-8 -6 -4 -2 0 2 4 6 8Energy (eV)
-4
-2
0
2
4
De
nsi
ty o
f st
ate
s
total
Antiferromagnetic bcc-Mn
Net moment = spin up – spin down = 0 !
Difference 1: Net moment
Mn: 25 electrons = 1s2 2s2 2p6 3s2 3p6 4s2 3d5
l = -2 -1 0 +1 +2
-8 -6 -4 -2 0 2 4 6 8Energy (eV)
-4
-2
0
2
4
De
nsi
ty o
f st
ate
s
totalMn1 3d
Antiferromagnetic bcc-Mn
Net moment = spin up – spin down = 0 !
Moment at Mn1 site = 2.8μB
Difference 1: Net moment
Mn: 25 electrons = 1s2 2s2 2p6 3s2 3p6 4s2 3d5
l = -2 -1 0 +1 +2
-8 -6 -4 -2 0 2 4 6 8Energy (eV)
-4
-2
0
2
4
De
nsi
ty o
f st
ate
s
totalMn1 3dMn2 3d
Antiferromagnetic bcc-Mn
Net moment = spin up – spin down = 0 !
Moment at Mn1 site = 2.8μB Moment at Mn2 site = -2.8μB
Difference 1: Net moment
Difference 2: No. of bands
−5.0
0.0
5.0
¡F
K K U Y Z T C P
Ener
gy ¡
n(k) [
eV]
Antiferromagnetic bcc−Mn
spin−upspin−up
−5.0
0.0
5.0
¡F
K K U Y Z T C P
Ener
gy ¡
n(k) [
eV]
Antiferromagnetic bcc−Mn
spin−downspin−down
Unit cell of AFM = double of FM
à double the number of bands
Similarity Mn: 25 electrons = 1s2 2s2 2p6 3s2 3p6 4s2 3d5
l = -2 -1 0 +1 +2
-8 -6 -4 -2 0 2 4 6 8Energy (eV)
-4
-2
0
2
4
De
nsi
ty o
f st
ate
s
totalMn1 3dMn2 3d
Antiferromagnetic bcc-Mn
Net moment = spin up – spin down = 0 !
Moment at Mn1 site = 2.8μB Moment at Mn2 site = -2.8μB The value of the moment is reduced compared to the atomic value à due to bond formation
MnO
Invoking magnetic ordering opens a gap
Non magnetic fcc MnO
-8 -6 -4 -2 0 2 4 6 8Energy (eV)
0
3
6
9
12
De
nsi
ty o
f st
ate
s
total
fcc - MnO
−5.0
0.0
5.0
¡F
K X W K K L W U X
Ener
gy ¡ n(k
) [eV
]fcc MnO
-8 -6 -4 -2 0 2 4 6 8Energy (eV)
0
3
6
9
12D
ensi
ty o
f st
ate
stotalMn 3d
fcc - MnO
-8 -6 -4 -2 0 2 4 6 8Energy (eV)
0
3
6
9
12D
ensi
ty o
f st
ate
stotalMn 3dO 2p
fcc - MnO
AFM-MnO
AFM-MnO
AFM-MnO
−8.0
−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
¡F
K L U T W Ux X K
Ener
gy ¡ n(k
) [eV
]
AFM MnO
spin−upspin−up
−8.0
−6.0
−4.0
−2.0
0.0
2.0
4.0
6.0
¡F
K L U T W Ux X K
Ener
gy ¡ n(k
) [eV
]
AFM MnO
spin−downspin−down
AFM-MnO
-8 -6 -4 -2 0 2 4 6Energy (eV)
-9
-6
-3
0
3
6
9D
ensi
ty o
f st
ate
stotal
AFM MnO
AFM-MnO
-8 -6 -4 -2 0 2 4 6Energy (eV)
-9
-6
-3
0
3
6
9D
ensi
ty o
f st
ate
stotalMn1 3d
AFM MnO
AFM-MnO
-8 -6 -4 -2 0 2 4 6Energy (eV)
-9
-6
-3
0
3
6
9D
ensi
ty o
f st
ate
stotalMn1 3dMn2 3d
AFM MnO
AFM-MnO
-8 -6 -4 -2 0 2 4 6Energy (eV)
-9
-6
-3
0
3
6
9D
ensi
ty o
f st
ate
sO 2ptotalMn1 3dMn2 3d
AFM MnO
AFM-MnO
Did you notice an interesting aspect to the individual manganese moments ? How is it different from AFM bcc-Mn ?
AFM-MnO
Did you notice an interesting aspect to the individual manganese moments ? How is it different from AFM bcc-Mn ? à the manganese sites each have a net moment of 5 μB !
Cs2AgF4
Cs2AgF4
Non magnetic
Ferro magnetic
Cs2AgF4
Non magnetic
Ferro magnetic
Cs2AgF4
Orthorhombic distortion
Cs2AgF4
Ferro magnetic
Non magnetic
Cs2AgF4
Ferro magnetic
Non magnetic
Cs2AgF4
Other types of magnetic order