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Elementary Semiconductor Elementary Semiconductor Physics for Transition Metal Physics for Transition Metal
Oxide Oxide HeterostructureHeterostructure
Seiji Yunoki (UT & ORNL)[email protected]
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Content of lecturesHomogenous semiconductors
Intrinsic (pure) semiconductorsExtrinsic (impurity doped) semiconductors
Inhomogenous semiconductorsHomopolar junction (p-n junction, rectification, …)
Hetero junction (inversion layer, …)
See, for example, “Solid State Physics” by Ashchroft & Mermin
Heterojunction made of correlated electronicsystems
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Metal, Insulator, and Semiconductor
gap
unoccupied
occupied
InsulatorSemiconductorE
unoccupied
occupied
MetalE
gE
FE
g(E) g(E)Density of States Density of States
gInsulator: relatively large (~several eV)
gEESemiconductor: relatively small (~1eV or less)
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Semiconductor: small energy gap
Density of States g(E)
E
gEsmall
unoccupied (conduction band)
occupied (valence band)
relatively easy to excite electrons form valence to conduction bands
at finite temperature (T)
finite amount of electrons in C.B.
( )
( )( )
-
35
210 0
exp 2
10 =4.
.25 e
0 eV
V
gc
B
g
g
En
E
T k T
E−
−
=
∼
∼
∼
( )at room 0.025 eVBT k T ≈
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Pure (Intrinsic) Semiconductor
Semiconducting elements: IV
Si, Ge, …
Semiconducting compounds: III-VGaAs, GaP, InSb, …
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Extrinsic Semiconductor
( ) ( )2 2Ge: 4 4s p( ) ( )2 3As: 4 4s p
extra electron
( ) ( )2 1Ga: 4 4s p
extra negative charge
missing electron(extra hole)
((donor impurity)donor impurity)
Covalent bonding
extra positive charge
((acceptor impurity)acceptor impurity)
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Very small binding energy4
2 13.6 eV2HmeE = − −∼
Hydrogen atom:2
82 0.5 10 cmBa
me−= ×∼
binding energy
Bohr radius
Inside solid:2 2e er rε→ screening 10 20ε ≈ −
*0.1 1.0m
m ≈ −*m m→ band
*
2
1 10 100 meVHEmm E
ε= ⋅ ⋅ −∼ very small
binding energy
* 100B Bmr a am
ε= ⋅ ⋅∼ very large radius
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Energy band for extrinsic semiconductorsAs-doped Ge (n-type semiconductor)
Ga-doped Ge (p-type semiconductor)
density of states g(E)
E
gE
conduction band
valence band
cE
vE
dε
density of states g(E)
E
gE
conduction band
valence band
cE
vEaε
( ) ( )exp 2 1 c dc
B
En T k Tε− −
∼ ∼ at room T
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# of conduction electrons vs TAs-doped Ge (n-type semiconductor)
E
gE
conduction band
valence band
cE
vE
dε
( )Tµ
0
cEdε
vE
T
( )cn T
0
extrinsic regimeintrinsic regime
dNdensity of As impurities
2gE
g(E)density of statesT
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Conductivity (metal vs semiconductor)
2nemτσ =
carrier density
relaxation time
Drude formula:
(effective) mass
( )Tτ dominatesMetal:
( )n T dominatesSemiconductor:
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Antimony (Sb) doped Germanium (Ge)
T
( )cn T
0
extrinsic regimeintrinsic regime
dN
T-dependence of resistivity with varying impurity concentrations
H. J. Fritsche (1958)
n-type semiconductor
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Inhomogeneous Semiconductorspnpn junctionjunction (pn diode): homopolar junction( heterojunction)
pp--typetypesemiconductorsemiconductor
nn--typetypesemiconductorsemiconductor
Ga-doped Ge As-doped Ge at equilibrium (no current flow)
Schematic energy band (not yet in contact): saturation regime
pp--typetype nn--typetype
impurity ioncharge: +e
impurity ioncharge: -e free conduction
electrons embedded in the back ground of immobile positive ionized donors
free valence holesembedded in the back ground of immobile negative ionized acceptors
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pn junction in contact ( in equilibrium)p nµ µ µ= =
impurity ioncharge: +e
impurity ioncharge: -e
pp--typetype nn--typetype
- - + + non neutralized charge
bφ
electron redistributionnon-neutralized charge regionsfinite electronic field E(x) only at the interface: ( ) ( )E x xφ= −∇
x
( )xφelectronic potential b n pφ µ µ= −
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Schematic energy band of p-n junctionpp--typetype nn--typetype
impurity ioncharge: +eimpurity ion
charge: -e
( ) ( )c cE x E e xφ= −
( ) ( )v vE x E e xφ= −
deN
aeN−0
ndpd
d n a pN d N d⋅ = ⋅space charge distribution
total charge neutral:
no free carriers in the depletion region
+−
space charge space charge (depletion) region(depletion) region
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More quantitative treatment
pp--typetype nn--typetype
+e-e
depletion regiondepletion region
dNaN
ndpd
( ) ( ) ( ) ( ) ( )a dx en x ep x eN x eN xρ = − + − +
density of electronsin conduction band
density of acceptorimpurities Charge density:
density of holesin valence band
density of donorimpurities
non zero local charge distribution
finite electronic field
Poisson eq.:
( ) ( )2 ex xφ ρε
∇ = −
static dielectronic const. (~10-20)
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Numerical model calculations 1d tight binding model
(instructive for model calculations of heterojunctions made of correlated materials)
2 band = 2 orbitals per site(unit cell)
saturation region (all impuritiesionized)
valence band
conduction band
pp--typetype nn--typetype
- charged acceptors + charged donors
gap
negatively charged acceptors
positively charged donors
gap
( )in xρ
( ) ( )2in
ex xφ ρε
∇ = −
Schrodinger eq.
( )out xρ
( ) ( )out inif x xρ ρ≠1D 2 orbital tight binding model with Coulomb interactions
self-consistent calculationpp--typetype nn--typetype
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Numerical calculations: results 1d tight binding model
pp--typetype nn--typetype
- charged acceptors
+ charged donors
pp--typetype nn--typetype
depletion regiondepletion region
net chargenet charge
to compensateto compensatedifferencedifferencen pµ µ−
finite electronic fieldfinite electronic field( ) ( )E x xφ= −∇
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Schematic picture pp--typetype nn--typetype
- charged acceptors
+ charged donors
pp--typetype nn--typetype
depletion regiondepletion region
beφpp--typetype nn--typetype
depletion regiondepletion region
ndpd
beφ
beφ
at equilibrium, no current flow
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No current flow ?? at equilibrium, no current flow
From Boltzmann eq.
( ) ( )n n nen x eD n xµ= + ∇J E
( ) ( )h p pep x eD p xµ= − ∇J E
diffusion coefficientmobility
for conduction electrons
for valence holes
(see Ashcroft & Mermin, p601)diffusion currentdrift current
Exact cancellation
( ) ( ) ( )n nen x x eD n xµ = − ∇E
( ) ( )x xφ= −∇EB
n nk TDe
µ= ⋅(Einstein relation)
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Conduction electrons
( ) ( )( )Const. expB
e xn x k T
φ −= × −
pp--typetype nn--typetypedepletion regiondepletion region
ndpd
beφ
( ) ( ) [ ]expp n b Bn d n d e k Tφ= ⋅ −
( ) ( ) [ ]expn p b Bp d p d e k Tφ= ⋅ −
beφ
beφ
free electrons in conduction band in p-type semiconductor (minority carriers):
similarly, free holes in valence band in n-type semiconductor (minority carriers):
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I-V characteristic of p-n junction ((rectifying rectifying response)response)
( )0eV > ( )0eV <
0eV >
b eVφ −
eV
non zero bias voltagenon zero bias voltageb n pφ µ µ= −
( )0eV =zero bias voltagezero bias voltage
reversereverse forwardforward
pp--typetype nn--typetype
pp--typetypenn--typetype
nonnon--linear Ilinear I--V characteristicV characteristic
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I-V characteristic of p-n junction
( )0eV > ( )0eV <
bφ
bφ
b eVφ −
b eVφ −
b eVφ −
b eVφ −
ener
gy b
and
V.B.C.B.
( )xφ ( )xφ ( )xφ
depletion regiondepletion region depletion regiondepletion region depletion regiondepletion region
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Numerical model calculations 1D 2 orbital tight binding
depletion regiondepletion region
forwardforward
reversereverse
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Rectifying feature in I-V curve
reversereverse forwardforward“Oversimplified” understanding
0eV >
depletion region decreases
electronic potential decreases
increases
0eV < decreases
depletion region increases
electronic potential increases
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reversereverse forwardforward
Rectifying feature in I-V curve
( )0eV >( )xφ
b eVφ −
b eVφ −
ener
gy b
and
pp--typetype nn--typetype
ndpd
( ) ( ) ( )expp n b e Bn d n d e V k Tφ= ⋅ − −
free electrons in conduction band in p-type semiconductor (minority carriers):
# of minority carriersdepends on Vediffusion current
# of electrons injected into C.B. in p-type semiconductor
( ) ( ) ( )( )
0
e 1e B
p p p
eV k Tn
n d n d n d
n d
∆ = −
= −
( ) e 1e BeV k TpI n d∆ ∝ −∼
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Semiconductor Heterojunctions homopolar junction
1GaAs Al Ga Asx x−gap: 1.42 eV 2.17eV (x=1.0)
GaAs 1Al Ga Asx x−
1p-GaAs n-Al Ga Asx x−p-n junction made of
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Schematic energy band
1p-GaAs n-Al Ga Asx x−
donor ioncharge: +e
acceptor ioncharge: -e
finite electric fieldin contact non-neutralized charge appears
x
( )xφelectronic potential b n pφ µ µ= −
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Schematic energy band (in contact)
p-GaAs 1n-Al Ga Asx x−
depletion regiondepletion region
Band discontinuity preserved& c vE E∆ ∆
( ) ( )c cE x E e xφ= −
( ) ( )v vE x E e xφ= −
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Inversion Layer By tailoring the band off set
Thin two dimensional layer of electrons
Inversion layer of electronsInversion layer of electrons
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Heterojunction made of transition metal oxides
Rapidly growing, very promising new research field
Many experimental groups (S. Pennycook, H. Christen, J.Shen, …) and a theory group (Dagotto) at UT and ORNL
Good for your career
Transition metal oxides (cuprates, manganites, …)Strongly correlated systems (Coulomb and electron-lattice interactions)
Many degrees of freedom (charge, spin, orbital, phonon)
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Heterojunction made of transition metal oxides
Transition metal oxides (cuprates, manganites, …)Complex phase diagramComplex phase diagram and large response
Next generation electronic device with rich functionality
large response
T
Oxide Electronics (Strongly Correlated Electronics)Oxide Electronics (Strongly Correlated Electronics)
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A Simplest Example: Titanate Superlattices(Ohtomo et al, Nature 419, 378 (‘02))
superlattice film4 33 3SrTi O LaTi O+ + Sr, La
O
TiPerovskite structure
2+ 4 03Sr Ti O : d+ band insulator
Mott insulator3+ 3 13La Ti O : d+
ge
2gt
[ ]( ) ( )2 2Ti: Ar 3 4d s
atomic d level
d
spherical symmetrycubic symmetry (octahedral)
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Titanate Superlattices 4 33 3SrTi O LaTi O+ +
(Ohtomo et al, Nature 419, 378 (‘02))
Sr, LaO
Ti
LaSrdark filed image (scanning transmission electron microscopy)
atomicatomic--scale fabricationscale fabrication
Insulator + insulator = metalmetale.g.,
2T dependence of ( )Tρ
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Titanate Superlattices 4 33 3SrTi O LaTi O+ +
(Ohtomo et al, Nature 419, 378 (‘02))electron energy loss spectra (EELS) - atomic column by atomic column -
2,3Ti L edge spectra:
Ti: 2 3p d→
How to analize data:
( ) ( ) ( ) ( )3+ 4+33LaTi O SrTi O
1I II ω α ωω α= + −
measured spectra depending on position
reference spectra
# of d electronsα
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Titanate Superlattices 4 33 3SrTi O LaTi O+ +
(Ohtomo et al, Nature 419, 378 (‘02))3+Ti fraction = # of d electrons
# # of d electrons < 1of d electrons < 1
charge leakagecharge leakage
d electron density
2~3 unit cells
To recover the bulk valcue, at least 5 layers of La-ions are needed.
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Model Calculations of Titanate SuperlatticesPerovskite structuresuperlattice4 3
3 3SrTi O LaTi O+ +
Top viewSr, La
O
Ti
2+ 4 03Sr Ti O : d+ = reference, 3+ 3 1
3La Ti O : d+
Extra +1 charge on La site (positively charged back ground) very similar to
semiconductors# of total d electrons = # of total La ions (donor impurities)
Ti d orbitals: electrically active orbital
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Model Calculations of Titanate Superlatticessuperlattice4 3
3 3SrTi O LaTi O+ +
positively charged back ground
d band Hubbard long rangeH H H H− −= + +
d bandH − : Ti d-electron band (cubic lattice)
HubbardH : on-site short range electron-electron interaction (on-site U)
long rangeH − : long-range Coulomb Interactionbetween electrons and electron-ion (positive back ground charge)( ) 1.0n r+ =( ) 0.0n r+ = ( ) 0.0n r+ =
where
total # of d electrons = total # of La ions = total # of positively charged back ground charge
self-consistent mean-field calculations
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Oxide heterostructure interface: mean-field results
LaTiO3/SrTiO3
z
LaTiO3 SrTiO3
charge leakagecharge leakage55--7 7 layers needed to reach n=1layers needed to reach n=1
15ε =dielectronic constant:
single band approximation
electron-positively charged ion interaction crucial to produce potential wells
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Oxide heterostructure interface
LaMnO3/SrMnO3
2gt
ge
3d 4d
4+3SrMn O 3+
3LaMn O
z
SrMnO3 LaMnO3
Ferromagnetic halfFerromagnetic half--metallic interfacemetallic interface
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Oxide heterojunction: magnetic tunnel junctionmagnetic tunnel junction
La2/3Sr1/3MnO3/SrTiO3/La2/3Sr1/3MnO3
A. Fert’s group (2003)Tunneling magnetoTunneling magneto--resistance (TMR)resistance (TMR)
TMR ratio ~1800%((La,La,SrSr)MnO)MnO33 ((La,La,SrSr)MnO)MnO33
SrTiOSrTiO33
current
hor
half-metal
LaMnOLaMnO33
( ) ( ) ( )0 1 TMR ratiohρ ρ = +
calculations for
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Summary:
Transition Metal Oxide Transition Metal Oxide HeterostructureHeterostructure::
Very promising new research field
Next generation electronic devices
Oxide ElectronicsOxide Electronics
Emergence of new exotic quantum states in heterostructure interface
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