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Chapter Nine Fermi surfaces and Metals
To get band structure of real crystals, turns on weak periodic potential.Band gaps open up at BZ edges.
To calculate electronic properties, put in electrons (Fermions).
fill them up to Fermi energy F.At T=0, the Fermi surface separates the unfilled orbits from the filled orbits.The electrical properties of the metal are determined by the shape of theFermi surface, because the current is due to change in the occupancy of
states near the Fermi surface.Aluminum (v=3)Copper (v=1)
Zone 2 Zone 3Zone 1Zone 1
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-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0k(2/a)
(22/2ma2)
1
9
25
1D chain
4
16
Extended-zone scheme
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1 1 2 3 4 525 4 3
k
/a-/a
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
k(/a)
(22/2ma2)
1
49
16
25
Reduced-zone scheme
Free electrons (k) :modulated by latticeperiodicity
All in the first BZ.
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-1.0-0.5
0.0
0.51.0
0.0
0.5
1.0
1.5
2.0
-1.0-0.5
0.00.5
1.0
E(2
h2/2
ma
2)
ky(/a
)kx(/a)
Two dimensional square lattice
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Construction of Brillouin zones : bisect all G
1 1 2 3 4 525 4 3
/a-/a1Dk
2D
1
2a
2d2b
2c
3
3a
3 3
3d
3
3 3
kx
ky translate region into 1st zone by Gto form reduced zones
1st zone
3a
3d
3rd zone
2a
2c
2d 2b
2nd zone
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Construction of free electron Fermi surface
1
2
22
2
3
3
3 3
3
3
3 3
kx
ky
kF
1st zone
Fully occupied
2nd zone
3a
3d
The shaded regions are filled withelectrons and are lower in energythan the unshaded regions.
3rd zone
electron-likehole-like
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[010]
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Fermi surface
[010]
[100]
Fermi surface is distortedfrom a sphere
near the zone boundary.
BCC Li
A cusp is caused by interaction
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Two electrons per primitive cell v=21st band Fermi surface
Free electron
[110] 2nd band Fermi surface
[100]
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(1) kF
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(3) k = kBZ Bragg scatterings open energy gap
0(k)1
v(k)k
==h at zone boundary. Standing waves
(4) k > kBZ electron states in second or higher bandscorresponding to higher order Brillouin zones of k space.
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Nearly free electrons
The interaction of the electron with periodic potential of the crystal
causes energy gaps at the zone boundary.Fermi surfaces will intersect zone boundaries perpendicularly.
The crystal potential will round out sharp corners in the Fermi surface.
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Fermi surface: surface in k-space separates filled and unfilled states
Only metals have Fermi surfaces.Important because electronic properties depend on electron states near F
within kBT
T)kD(3C 2BF
2
e=
)D(ve3
1F
2
F
2=
TL=
Heat capacity
Electric conductivity
Thermal conductivity
Volume of Fermi surface only depends on conduction electron density.Shape of Fermi surface depends on strength of periodic potential andsize of kF relative to kBZ
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h
rr v kg
=Density of states depends on (k) , actually
State number between and +d in bandIn two dimensions, uniform in k-spaceL)k(D
=
2
2
kdk2)kD(dk)dkkD()dD( yxrr
== +d
=
=
=
gvd
dk
dd
dkdk
dkk
h
Area of k-space
=
1dk
2L2)D(
k
2
Therefore, path integral along constant contour
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In three dimensions, uniform in k-spaceL
)k(D
=
3
2
=
1dS
2
L2)D(
k
3
area integral over constant surface
P
Q
R
D()
PQ
R
D()multiple peaks
Crystal is not cubic,
[100]
[010]Crystal is cubic,
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How to determine the Fermi surface?
Magnetic field response direct probe of the Fermi surface
dt
kdBvqF
rh
rrr== 0vF =
rr
Magnetic field drives electrons in k-space along constant contours.and
Nearly free electron
Br
dtkdF
rr
v kr
electron orbit
Br
dtkdF
rr
electron orbit
v kr
Free electron
hole orbit
dtkdFrr
Br v k
r
Free hole
hole orbit
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Period of orbit
Lorentz force
Period ==
===
1dk
qBdtT
dkBvq
dtBvqdt
kdF
k
2h
rrhrr
r
hr
constant
k
k
k
k
k
k
1
=
=
=
)k,S(qB
kdk
qBk
dk
qBz
2212
==
hhhTPeriod
and
where S is the k-space area enclosed by the orbit in its plane
Free electron
FF v
m
kv ==
h ( )
m
eB
T
2
eB
m2k2
eBvT
c
F
F
==
== h
cyclotronfrequency
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De Haas-van Effect : oscillation of the magnetic moment of a metalas a function of magnetic field (1930)
-M/H (106)Bi (1930)&(1932)
H is along [111] direction noble metal
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H is along [111] direction noble metal
Bi
R()
Magnetoresistance of Ga
T=1.3K
A111(belly)/A111(neck)=51 Ag
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Calculation of energy bands
The tight-binding method The Wigner-Seitz method
The pseudopotential method extension of the OPW method
Orthogonalized plane-wave
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Push isolated atoms together to form crystal
An isolated atom
very far away
two isolated atoms
Two atoms move closer to each other.
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Two atoms move closer to each other.
Two energy levels
A-B
A+ B
-6 -4 -2 0 2 4 6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4-0.2
0.0
0.2
0.4
0.6
0.81.0
1.2
A+ B
A-B
r
S lid ith N t h N ll d t t
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Solid with N atoms has N allowed energy states.
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When more atoms are brought together,the degeneracies are further split to form
bands ranging from fully bonding to fullyantibonding.
Different orbitals can lead to band overlap.
There are two idealized situations for which wave functions can be
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expressed in a simple manner andan energy band calculation can be carried out with relative case.
Energies are far above the maxima of potential energy.Nearly Free Electrons
Energies are deep within the potential wells at nuclei.
Tightly Bound Electrons
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Influence of the periodic potential depends both onthe magnitude of this potential and on the opportunities for atoms
to interact which varies with the interatomic spacing.
Tight binding method (Linear Combination of Atomic Orbitals)For an interatomic spacing which permits some overlaps between atoms(but not very much), the bands can be stimulated.
It is quite good for the inner electrons of atoms, but not for theconduction electrons.
the d bands of the transition metalsthe valence band of diamond-like materialsinert gas crystals
Ti ht b i d i method
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Free atoms s Overlapping s
)r(E)r()rU(2m
)r(Hkkk
22
katom
rrrhrrrr =
+=
Let is the ground state of an electron moving in the potential U(r)of an isolated atom.
)r(k
rr
19521905~198
Tight binding method introduced by Bloch in 1928
( )[ ]
=
=
jjkj
jjkkk
)rr(rkexpN
1
)rr(C)r( j
rrrr
rrrrrr
iN linear combinations
Let is for the electron moving in the whole crystal that containsN isolated atoms. Atoms are at lattice sites (j=1,.,N)jr
r)r(
k
rr
A trial wavefunction
satisfying Bloch condition
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( )[ ]
( ) ( )( )[ ]
( ) )r(Tkexp
)rTr(TrkexpTkexpN
1
)rTr(rkexp
N
1)Tr(
k
jjkj
j
jkjk
rrr
rrrrrrrr
rrrrrrr
r
r
rr
=
+=
+=+
i
ii
i
satisfying Bloch condition
r-(rj-T)
[ ] )r()r()rU(H)r(H kkkatomkcrystalrrrr
rrr
=+=
Schrdinger equation
Where contains all corrections to the atomic potential required toproduce the full periodic potential of the crystal.
)rU(r
First order energy
( ) ( ) =j m
jcrystalmmjk
crystalk
Hrkexprkexp
N
1H
rrrrrr ii
where and jj rr rr=( )mm rr
rr=
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)rrU( nrr
)r(U latticer
= nm mlatticen )rr(U)rrU( rrrr
[ ]
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[ ] )rr()rU(H)rr(dVH jatommjcrystalmrrrrr
+=
( )( ) ( )( )[ ]
( ) ( )[ ] )r()rU(HrdVkexp
)r()rrU(HrrrdVrrkexp
N
1H
atommm
m
jatomj m
jmmjkcrystalk
rrrrrr
rrrrrrrrrrr
+ =
++ =
i
i
Rewrite the first order energy
)r()Hr(dV)r()Hr(dV atomatommrrrrr = on the same atom
)r()rU()r(dV)r()rU()r(dV)r()rU()r(dV mrrrrrrrrrrr
+=
-Overlap, up to nearest neighbors -m=0 m0, n.n.
)r()Hr(dV crystalrr
==n.n.
kcrystalkk kiexpH rr
Therefore,
For a simple cubic
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For a simple cubic,the nearest neighbor atoms = (a, 0,0), (0, a, 0), (0, 0, a)
( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )[ ]ak2cosak2cosak2cos
aik-expaikexpaik-expaikexpaik-expaikexp
kiexp
zyx
zzyyxx
n.n.k
++=
+++++= =
rr
66 k +Along L, [111]
-12
An energy band width is 12
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An energy band width is 12 .The weaker the overlap is, the narrower the energy band is.
Constant energy surfaces in the BZ of a SC lattice
( ) ( )akcosakcosakcos2 zyxk ++= Reduced zone scheme Periodic zone scheme
142
+ xxBy series expansion
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For ka
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the nearest neighbor atoms = .5a(1, 1,0), .5a(0, 1, 1), .5a(1, 0, 1)
+
+
=
+
+
+
+
+
=
+
+
++
+
+
+
+
++
+
+
+
+
++
+
=
=
2
akcos
2
akcos
2
akcos
2
akcos
2
akcos
2
akcos4
2
aik
exp2
aik
exp2
ak
cos2
2
aikexp
2
aikexp
2
akcos2
2
aikexp
2
aikexp
2
ak2cos
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
2
aikaikexp
kiexp
xzzyyx
xxz
zzyyyx
xzxzxzxz
zyzyzyzy
yxyxyxyx
n.n.k
rr
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d states
Energy gap of the "metallic" single-walled carbon nanotubesM d Ph L tt B18 769 (2004)
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Based upon the Slater-Kostertight-binding calculations, we investigated
electronic properties of the "metallic" single-walled carbon nanotubes(SWNTs) in detail. Our results show that tube curvature may produce anenergy gap at the Fermi level for zigzag and chiral "metallic" SWNTs, andthis effect decreases with the increasing of either the radius or the chiral
angle. Our calculated results are in good agreement with experiments
Mod. Phys. Lett. B18, 769 (2004)
Calculations and applications of the complex band structure forcarbon nanotube field-effect transistors PRB70, 045322 (2004)
Using a tight binding transfer matrix method, we calculated the complexband structure for armchair and zigzag carbon nanotubes (CNTs). The
imaginary part of the complex band structure connecting the conduction andvalence band forms a loop, which can profoundly affect the characteristics ofnanoscale electronic devices made with CNTs. We then study the quantumtransport in carbon nanotube field-effect transistors (CNTFETs) with the
complex band structure effects. A complete picture of the complex bandstructure effect on the performance of semiconductor zigzag CNTFETs isdrawn.
Wigner-Seitz method
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The cellular method by Wigner and Seitz in 1933polyhedron structure in real space
quite successful for the simple alkali metals
1935 Kimball extended to nonmetallic materials such as diamond, Si, Ge,
1963
1902~199
(r)(r)U(r)2m kkk2
2
=
+ h
(r)(r) krk
k ue
i rr
=and Bloch function
( ) (r)(r)U(r)k2m
1kkk
2uui =
++ hh
start with the easiest-found solution at k=0, uo(r) within a single primitive cell
then, construct the approximation solution Wigner-Seitz
(r)(r) ork
k uei rr
= boundary condition: , are continuous
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The first approximation of the cellular method is the replacement of
periodic potential U(r) within the WS primitive cell by a potential V(r)with spherical symmetry about the origin.
Radial functions for
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Radial functions for
r (Bohr units)
3S orbital of free Na atom
3S conduction band in metal Na
k=0, metal Na
as r eigenenergy -5.15eV (atom)
-8.20eV (k=0)In real metal Na,
2m
kand)r(u
22
oko
rk
k
hrrr
+== ie
average energy-6.3eV
0eVFermi level
metal
1.15eV less
Metal is morestable than
free atom.
As we know, F=3.1eV for Na.The average KE per e
-
is0.6 F=1.86eV -8.2eVk=0 state
Two major difficulties with the cellular method :
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The computational difficulties involved in numerically satisfying
the boundary condition over the surface of the WS primitive cell.
The cellular method potential has a discontinuous derivative midway
between lattice points but the actual potential is quite flat there.
Later, a modification
muffin-tin potential
Pseudopotent ia lmethods
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The orthogonalized plane-wave (OPW) kvalence electrons + core electrons
The theory of pseudopotential began as an extension of OPW method.
+= c
c
kc
rk
k (r)b(r) i
e
satisfying Bloch condition w/. wavevector k
Outside the core, the potential energy that acts on conduction electron is
relatively weak. Potential due to the singly charged positive ion cores isreduced markedly by the electrostatic screening of other conductionelectrons.
by C. Herring (1940)
(r))r()r((r)(r) cc
vk
cvvkkkk = rd exact valence wave function
and vk
v
k
v
kH =
( ) ( )
= c
c
v
k
cv
k
c
c
v
k
cv
kkkkkk)r()r(H)r()r( rdrdH
and ck
c
k
c
kH =
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( ) vpseudo22
vvv
R kkkkV
2
VH
+==+
m
h
adding VR to U : partial cancellationcancellation theorem( ) c
c
k
ccvv
kkkkk)r()r( = rdVR
effective Schrdinger eq.
The pseudopotential for a problemis neither unique nor exact.
On Empty Core Model (ECM)Unscreened potential :
V(r)={ 0, for rReNa
Later, screening effect :Thomas-Fermi dielectric function
Empirical Pseudopotential Method (EPM) band structure
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Coefficients V(G) are deduced from theoretical fits to
measurements of the optical reflectance and absorption of crystal.Chapter 15.
Charge density map can be plotted from the wavefunctionsgenerated by the EPM, in excellent agreement with X-raydiffraction determination,
giving an understanding of the bonding and have greatpredictive value for proposed new structures and compounds.
Numerical calculation of band structures using the first-principalnon local pseudo potential
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non-local pseudo-potential.A. Zunger and M.L. Cohen, Phys. Rev. B20, 4082 (1979).
Si W
Numerical calculation of Density of states
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Some physical quantities obtained fromnumerical calculation and experimentalmeasurement , respectively.
4.334.43
8.107.35
3.603.57
Diamond
0.73
0.77
4.26
3.85
5.66
5.65Ge
0.98
0.99
4.84
4.63
5.45
5.43Si
Bulkmodulus(Mbar)
Cohesiveenergy
(eV)
Latticeconstant
()
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APPROACHESF t M t lli t l
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Free atoms
Atomic s
Metallic crystals
Free electrons
Tight-Binding model
Overlapping s
Nearly Free electron model
Free electrons + periodic potential
Essence ofenergy gaps /transport
Essence ofcrystal binding
Many body Treatments
PseudopotentialsBand structures
Full treatment
Experimental methods in Fermi Surface studies
H is along [111] direction noble metal
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H is along [111] direction noble metal
de Haas-van effect :
Oscillation of the magnetic momentof a metal as a function ofmagnetic field (H). (1930)
A111(belly)/A111(neck)=51 Ag
Quantization of orbits in a magnetic fieldReview:
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a charge q of mass m in a magnetic field B
AB where,Ac
qp
2m
1H
2 rrrr=
=Hamiltonian
Ac
q
kppp fieldkinetictotal
rrh
rrv
+=+=Total momentum
Bohr-Sommerfeld relation orbits are quantized in a magnetic field
+=
+=
rdAc
qrdk
2)2
1n(rdp total
rrrrh
h
rr
Brc
qk
Bdtrd
cqBv
cq
dtkd
rrrh
rr
rrr
h
=
==
( )
B==
==
c
q2n2(area)Bc
q
rdrBc
qrdBr
c
qrdk
r
rrrrrrrrh
BadB
adArdA
==
=
r
r
rrvr
the first termthe second term
hrr
22
1n
c
qrdp Btotal
+==
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2c
27
B mTesla1014.42
1n
q
c2
2
1n
+=
+=
h
magnetic flux through the orbit in real space
How about in k- space ?
n
2
n S
qB
cA
=
hk
qB
cr = h
q
c2
2
1nS
qB
cBBA n
2
nB
hh
+=
==
Therefore,the area of an orbit in k space Sn is quantized in magnetic field B
Bcq2
21n
B1
cqB
qc2
21nS
2
nhh
h
+=
+=
Different orbits can have the same area by changing magnetic field B.For instance
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For instance,
The nth orbit in magnetic field BnThe (n+1)th orbit in magnetic field Bn+1 cq2
2
1nB
S
n h
+=
c
q2
B
1
B
1S
n1n h
=
+
Equal increments of 1/B reproduce orbits w/ the same area.
Bi, T=1.6K
Steele andBabiskin,
PR98, (1955)
-M/H (106) Bi (1930)&(1932)
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oscillation of the magnetic moment of a metal as a function of magnetic fieldLow temperature and high magnetic field
De Haas-van Effect
Onsager: the change in 1/B through a single period (1/B)was determined by
where S is any extremal cross-sectional area of the Fermisurface in a plane normal to magnetic field.
(1952)S
1
c
q2
B
1
h
=
Bc
q2
2
1nSn
h
+= The normal line of the orbital area is along thedirection of magnetic field B
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c2 h direction of magnetic field B.
Quantized of the closed orbits in a magnetic field B. (L.D. Landau)
Free electrons modelan electron in a cubical box of side L in magnetic field B z
cyclotron frequencyn=1,2,positive integermc
eBwhere
2
1nk
2m)k( cc
2
z
2
zn =
+= hh
only need to consider kx and kyThe number of levels with energy for a given n and kz
( )c
eB2m2m2kk2k c
2
2
hhh
====
2
2
LD(k)
=
Bc2c
eB2
2
L 22
hh
eL=
depending on B
the area between
successive orbitsthe number of level per unit areain k-space
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in the absence of B in the presence of B
The number of orbital levels on a circle is a constant,
independent of n.
Does Fermi level change with magnetic field B?States w/. k kF are occupied at T=0, N is conserved.
F i h l i k i d
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Fermi sphere volume in k-space occupied
by electrons in the ground states
F
B=0 B1
h
c
s+1Fpartlyfilled
s-1
sempty
B2
splitting into many Landau levels
g()
s
0.5hc
increasingslightly
F At the critical fields Bs,no partly filled level
andc
2
Bc2
sNh
eL=
How to put electrons in the energy levels?
BeB2L 2
2 eL
=
The number of levels with energy for a given n
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c2c2 hh
The number of levels with energy for a given n= 0.5B (assumption)
w/o. consideration of spinIn a 2D system with N=50
When all levels are fully occupied from n=1 to s, total energies of e-
eLs1eL 22s 2
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cc
h
h
h
h
Bc2
eL
2
s)
2
1B(n
c2
eL
1n
=
=When s+1 level is partly occupied by decreasing B slightly,
Energy for e- in s+1 level
Energy for e- in the lower levels
)2
1
s(Bc2
eL
sN
2
+
c hhU(B)
c
h
h
B
c2
eL
2
s 22
Oscillation of total
electronic energy
Magnetic moment at T=0K
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B
U
=
cS
e2
B
1
h
=
where S is the extremalarea of the Fermi surfacenormal to the direction of B
Information of the Fermi surface
shape and size
oscillates.
w/. period
Oscillation of the magnetic moment
De Haas van Alphen effect
What are the extremal areas ?
When magnetic field is along z-axis
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When magnetic field is along z axis,
the area of a Fermi surface cross section at height kz is S(kz), andthe extremal area Se are the values of S(kz) at the kz wheredS/dkz=0, stationary wrt. small changes in kz.
Along k1-axis,
three extremal orbits :(1),(2) area peaks and (3) area dip
Along k2-axis,
only one extremal orbit : (4) area peak
Fermi surface
FCC lattice BCC reciprocal lattice
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Cu, Ag, Au, monovalent metal w/. FCC structure
( )a
4.90
a
43n3k
1/3
3
23/12
F =
==
FCC lattice BCC reciprocal lattice
The distance between hexagonal faces isa
10.883
a
2=
The distance between square faces is a
12.572
a
2=
The Fermi surface does not neck out to meet these faces.
The Fermi surface neck out to meet these faces.
Experimental data on Au by Shoenberg
Period of 1/B for the magnetic moment
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e od o / o t e ag et c o e t
1/B111=2.0510-9 gauss-1 S=4.66 1016 cm-2 (belly)
1/B111=6.10-8 gauss-1 S=1.6 1015 cm-2 (neck)
1/B100=1.9510-9
gauss-1
S=4.90 1016
cm-2
a: a closed particle orbit
b: a closed hole orbitc: an open orbit
Sbelly/Sneck=29
10 nm, GaAs Cap
15 nm, - doping layer, Si
8 nm, spacer AlGaAs
2DEG
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-6 -4 -2 0 2 4 6
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Rxx
(k)
T=0.4K
Rxx
=45/(LT), 5k/(RT)
n=1.4x1011
(cm-2)
=0.99x106
( cm
2
/Vsec)
Rxy
(h/e)
H (Tesla)
0.0 0.2 0.4 0.6 0.80.00
0.02
0.04
0.06
0.08
0.10
0.120.14
2DEGE1
energy
Ec
EF
0.2eV
15 nm, doping layer, Si
60 nm, spacer AlGaAs
1500 nm, GaAs
buffer layer
1 2 3 4 50.0
0.1
0.2
0.3
T=0.4K
Rxx
()
H-1 (Tesla-1)
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0.3m
0.45m
0.6m
Source Drain
Quantized Conductance Transport
GaAs/AlGaAs heterostructures (Dr Umansky provided)
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GaAs/AlGaAs heterostructures
n=1.41011
/cm2
=2.2106cm2/Vs (0.3K)
(Dr.Umansky provided)
Mean free path l=13.6 m
-1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4
0
1
2
3
45
6
7
8
9
10
11
12
G
2e
2/h
VSG (V)
Split gates confined QPC
dgap=0.3m and lchannel=0.5 m Rh=
=29001
N;2e
NG2
-1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0
0.0
2.0k
4.0k
6.0k
8.0k
10.0k
12.0k
14.0k
16.0k
18.0k
20.0k
R
VSG (V)
1D2D
T=0.3K