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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 εF.
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 ofstates near the Fermi surface.
Aluminum (v=3)Copper (v=1)
Zone 2 Zone 3Zone 1Zone 1
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 k(2π/a)
ε (π2h2/2ma2)
1
9
25
1D chain
4
16
Extended-zone scheme
1 1 2 3 4 525 4 3k
π/a-π/a
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 k(π/a)
ε (π2h2/2ma2)
1
4
9
16
25
Reduced-zone scheme
Free electron’s ε(k) :
modulated by lattice periodicity
All in the first BZ.
-1.0-0.5
0.00.5
1.0
0.0
0.5
1.0
1.5
2.0
-1.0-0.5
0.00.5
1.0
E(π2 h
2 /2m
a2 )
k y(π/a)
kx(π/a)
Two dimensional square lattice
Construction of Brillouin zones : bisect all G
1 1 2 3 4 525 4 3
π/a-π/a1D
k
2D
1
2a
2d2b
2c
3
3a
3 3
3d
3
3 3
kx
ky translate region into 1st zone by G to form reduced zones
1st zone
3a
3d
3rd zone2a
2c
2d 2b
2nd zone
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 with electrons and are lower in energy than the unshaded regions.
3rd zone
electron-likehole-like
One electron per primitive cell v=1
Free electron
+ weak periodic Potential
So the Fermi surface is extendedtoward the zone boundary as it get closer.
constant ε
With the crystal potential, the energy inside the first Brillouin zoneis lower close to the zone boundary.
Fermi surface
[010]
[100]
Fermi surface is distorted from a sphere
near the zone boundary.
BCC Li
A cusp is caused by interaction
Two electrons per primitive cell v=21st band Fermi surface
Free electron
[110] 2nd band Fermi surface
[100]
(1) kF << kBZ No dispersion Constant ε contours are spheres.
kF
aπ
−aπ
aπ
−aπ
BZ boundary
kF
(2) kF <~ kBZ Fermi surface bulges toward BZ boundary
free e
lectro
n
ε
k0 π/akF
k > kF
K bulges toward BZ boundary due to dispersion.a
π− a
π
(3) k = kBZ Bragg scatterings open energy gap
0ε(k)1v(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.
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.
Fermi surface: surface in k-space separates filled and unfilled statesOnly metals have Fermi surfaces.Important because electronic properties depend on electron states near εF
within kBT
T)k ε D(3πC 2
BF
2
e =
) ε D( τve31
F2F
2=σ
T L σκ =
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 and size of kF relative to kBZ
h
rr εv k
g∇
=Density of states – depends on ε(k) , actually
State number between ε and ε+dε in band
In two dimensions, uniform in k-spaceL)k(D
=
2
2π
kdk 2π )kD(dk)dkkD()dε ε D( yx
rr== ε+dεε
∫=
∫=
∫=
gvdεdk
dεdεdkdk
dk∆k
h
Area of k-space
∫∇
=
ε1dk
2πL2) ε D(
k
2
Therefore, path integral along constant ε contour
In three dimensions, uniform in k-spaceL)k(D
=
3
2π
∫∇
=
ε1dS
2πL2) ε D(
k
3
area integral over constant ε surface
P
Q
R ε
D(ε)
PQ
R
ε
D(ε)multiple peaks
Crystal is not cubic,
[100]
[010]Crystal is cubic,
How to determine the Fermi surface?
Magnetic field response – direct probe of the Fermi surface
dtkdBvqFr
hrrr
=×= 0ε vF =∆⇒⊥rr
Magnetic field drives electrons in k-space along constant ε contours.
and
Nearly free electron
Br
dtkdFrr
∝
εv k∇∝r
electron orbit
Br
dtkdFrr
∝
electron orbit
εv k∇∝r
Free electron
hole orbit
dtkdFrr
∝
Br εv k∇∝
r
Free hole
hole orbit
Period of orbit
Lorentz force
Period ∫∫ ∇==
×=⇒×==
ε1dk
qBdtT
dkBvq
dt BvqdtkdF
k
2h
rrhrr
r
hr
constant ε
∆ε∆k
kε ∆k
kε∆k
kεε
1
=
∂∂
→
∂∂
=•∂∂
=∆−
⊥⊥
)k , S(εεqB∆ε
∆kdkqBk
εdkqB z
2212
∂∂
→=
∂∂
= ∫∫−
⊥
hhhTPeriod
and
where S is the k-space area enclosed by the orbit in its plane
Free electron
FF v
mkv ==h ( )
meB
T2πω
eBm 2πk 2π
eBvT
c
FF
==
==h
cyclotronfrequency
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
Bi
R(Ω)
Magnetoresistance of Ga
T=1.3K
A111(belly)/A111(neck)=51 Ag
Calculation of energy bands
The tight-binding method
The Wigner-Seitz method
The pseudopotential method – extension of the OPW method
Orthogonalized plane-wave
Push isolated atoms together to form crystal
An isolated atom
very far away
two isolated atoms
Two atoms move closer to each other.
Two energy levels
ϕA-ϕB
ϕA+ ϕB
-6 -4 -2 0 2 4 6
-1.4-1.2-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.2
ϕA+ ϕB
ϕA-ϕB
r
Solid with N atoms has N allowed energy states.
When more atoms are brought together, the degeneracies are further split to form bands ranging from fully bonding to fully antibonding.
Different orbitals can lead to band overlap.
There are two idealized situations for which wave functions can be expressed in a simple manner and
an energy band calculation can be carried out with relative case.Energies are far above the maxima of potential energy.
Nearly Free ElectronsEnergies are deep within the potential wells at nuclei.
Tightly Bound Electrons
Influence of the periodic potential depends both on the 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 the conduction electrons.
the d bands of the transition metalsthe valence band of diamond-like materialsinert gas crystals
Free atoms ϕ’s Overlapping ϕ’s
)r(E)r()rU(2m
)r(H kkk2
2
katomrrrhr
rrr ϕϕϕ =
+∇−=
Let is the ground state of an electron moving in the potential U(r) of an isolated atom.
)r(k
rrϕ
19521905~1983
Tight binding method introduced by Bloch in 1928
( )[ ]∑ −•=
∑ −=
jjkj
jjkkk
)rr(rkexpN1
)rr(C)r(j
rrrr
rrrrrr
ϕ
ϕ
iN linear combinations
Let is for the electron moving in the whole crystal that contains N isolated atoms. Atoms are at lattice sites (j=1,….,N)jr
r)r(k
rrψ
ψA trial wavefunction
( )[ ]
( ) ( )( )[ ]( ) )r( Tkexp
)rTr(Trkexp TkexpN1
)rTr(rkexpN1)Tr(
k
jjkj
jjkjk
rrr
rrrrrrrr
rrrrrrr
r
r
rr
ψ
ϕ
ϕψ
•=
∑ −+−••=
∑ −+•=+
i
ii
i
satisfying Bloch condition
r-(rj-T)
[ ] )r()r()rU(H)r(H kkkatomkcrystalrrrr
rrr ψεψψ =∆+=
Schrödinger equation
Where contains all corrections to the atomic potential required to produce the full periodic potential of the crystal.
)rU(r∆
First order energy
( ) ( )∑ ∑ •−•=j m
jcrystalmmjkcrystalk HrkexprkexpN1H ϕϕψψ
rrrrrr ii
where and ( )jj rrrr
−= ϕϕ( )mm rrrr
−= ϕϕ
)rrU( nrr
−
)r(Ulatticer
∑ −=−∆≠nm
mlatticen )rr(U)rrU( rrrr
[ ] )rr()r∆U(H)rr(dVH jatommjcrystalmrrrrr
−+−∫= ∗ ϕϕϕϕ
( )( ) ( )( )[ ]( ) ( )[ ] )r()r∆U(HrdVkexp
)r()rr∆U(HrrrdVrrkexpN1H
atommm
m
jatomj m
jmmjkcrystalk
rrrrrr
rrrrrrrrrrr
ϕρϕρ
ϕϕψψ
+−∑ ∫•−=
++∑ ∑ ∫ −−−•=
∗
∗
i
i
Rewrite the first order energy
)r()Hr(dV)r()Hr(dV atomatommrrrrr
ϕϕϕρϕ ∗∗ ∫=−∫ on the same atom
)r()rU()r(dV)r()rU()r(dV)r()rU()r(dV mrrrrrrrrrrr
ϕρϕϕϕϕρϕ ∆−∫+∆∫=∆−∫ ∗∗∗
≡ -γOverlap, up to nearest neighbors
≡ -αρm=0 ρm≠0, n.n.
)r()Hr(dV crystalrr
ϕϕ ∗∫
( )∑ •−−−==n.n.
kcrystalkk kiexpH ργαψψεrr
Therefore,
For a simple cubic, the nearest neighbor atoms ρ = (±a, 0,0), (0, ±a, 0), (0, 0, ±a)
( )( ) ( ) ( ) ( ) ( ) ( )[ ]( ) ( ) ( )[ ]ak2cosak2cosak2cos
aik-expaikexpaik-expaikexpaik-expaikexp
kiexp
zyx
zzyyxx
n.n.k
++−−=
+++++−−=
∑ •−−−=
γα
γα
ργαεrr
γαεγα 66 k +−≤≤−−Along ΓL, [111]
-α12γ
An energy band width is 12 γ. The weaker the overlap is, the narrower the energy band is.
Constant energy surfaces in the BZ of a SC lattice
( ) ( ) ( )[ ]akcosakcosakcos 2 zyxk ++−−= γαεReduced zone scheme Periodic zone scheme
For ka<<1,
...!4!2
1cos42
−+−=xxx
22
22
22z
22y
22x
k
ak 6
2ak32
2ak1
2ak
12ak12
γγα
γα
γαε
+−−=
−−−=
−+
−+
−−−=
By series expansion
The constant energy surfaces in the neighborhood of k=0 are spherical.
[ ]
22
2
2
2222
2k
22
a2
dkak 6d
dkd
γ
γγαε
h
hh
=
+−−==∗mEffective mass
The weaker the overlap is, the narrower the energy band is and the higher the effective mass is.
For a face-centered cubic, the nearest neighbor atoms ρ = .5a(±1, ±1,0), .5a(0, ±1, ±1), .5a(±1, 0, ±1)
( )
+
+
−−=
−
+
+
−
+
+
−+
−−=
−−
+
−
+
+−
+
+
+
−−+
−+
+−+
+
+
−−+
−+
+−+
+
−−=
∑ •−−−=
2akcos
2akcos
2akcos
2ak
cos2ak
cos2akcos4
2aikexp
2aikexp
2akcos2
2aikexp
2aikexp
2ak
cos22
aikexp
2aik
exp2ak2cos
2aikaikexp
2aikaikexp
2aikaikexp
2aikaikexp
2aikaik
exp2
aikaikexp
2aikaik
exp2
aikaikexp
2aikaik
exp2
aikaikexp
2aikaik
exp2
aikaikexp
kiexp
xzzyyx
xxz
zzyyyx
xzxzxzxz
zyzyzyzy
yxyxyxyx
n.n.k
γα
γα
γα
ργαεrr
r- spaceThe energies of an s-band in a FCC crystals
k- space
Along ΓΧ [ kx=µ2π/a, (0≤µ≤1), ky=kz= 0 ]
( )[ ]µπγαε cos214 +−−=
Along ΓL [ kx= ky=kz= µ2π/a, (0≤µ≤0.5) ]
( )µπγαε 2cos12−−=
Along ΓK [ kx= ky= µ2π/a, (0≤µ≤3/4), kz= 0 ]
( ) ( )[ ]µπµπγαε cos2cos4 2 +−−=
quite successful for the weakest type of interaction bet. neighboring atoms
d states
Based upon the Slater-Koster tight-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, and this effect decreases with the increasing of either the radius or the chiralangle. Our calculated results are in good agreement with experiments
“Energy gap of the "metallic" single-walled carbon nanotubes”Mod. Phys. Lett. B18, 769 (2004)
“Calculations and applications of the complex band structure for carbon nanotube field-effect transistors” PRB70, 045322 (2004)Using a tight binding transfer matrix method, we calculated the complex band structure for armchair and zigzag carbon nanotubes (CNTs). The imaginary part of the complex band structure connecting the conduction and valence band forms a loop, which can profoundly affect the characteristics of nanoscale electronic devices made with CNTs. We then study the quantum transport in carbon nanotube field-effect transistors (CNTFETs) with the complex band structure effects. A complete picture of the complex band structure effect on the performance of semiconductor zigzag CNTFETs is drawn.
The cellular method by Wigner and Seitz in 1933
polyhedron structure in real space
quite successful for the simple alkali metals
1935 Kimball – extended to nonmetallic materials such as diamond, Si, Ge, …
1963
1902~1995
Wigner-Seitz method
(r)(r)U(r)2m kkk
22
Ψ=Ψ
+∇− εh
(r)(r) krk
k ueirr
•=Ψand Bloch function
( ) (r)(r)U(r)k2m1
kkk2 uui ε=
++∇− 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•=Ψ boundary condition: ϕ, ∇ϕ are continuous
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
r (Bohr units)
3S orbital of free Na atom3S conduction band in metal Na
k=0, metal Na
as r ∞eigenenergy -5.15eV (atom)
-8.20eV (k=0)In real metal Na,
2mk and )r(u
22
okork
khrrr
+== • εεψ ie
average energy-6.3eV
0eV Fermi levelmetal
1.15eV less
Metal is more stable than free atom.
As we know, εF=3.1eV for Na.
The average KE per e- is 0.6 εF=1.86eV -8.2eV
k=0 state
Two major difficulties with the cellular method :
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
Pseudopotential methods
The orthogonalized plane-wave (OPW) φkvalence electrons + core electrons
The theory of pseudopotential began as an extension of OPW method.
∑+= •
c
ckc
rkk (r)b(r) ψφ ie
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 is reduced markedly by the electrostatic screening of other conduction electrons.
by C. Herring (1940)
( ) (r) )r()r((r)(r) c
c
vk
cvvkkkk
ψφψφψ ∑ ′′∫ ′−= ∗rd exact valence wave function
and vk
vk
vkH ψεψ =
( ) ( )
′′′−=′′′− ∑ ∫∑ ∫ ∗∗ c
c
vk
cvk
c
c
vk
cvkkkkkk
)r()r(H )r()r( ψφψφεψφψφ rdrdH
and ck
ck
ckH ψεψ =
( ) vpseudo22
vvvR kkkk
V2
VH φφεφ
+∇−==+
mh
adding VR to U : partial cancellationcancellation theorem( )( ) c
ck
ccvvkkkkk
)r()r( ψψψεεφ ∑ ∫ ′′′−= ∗rdVR
effective Schrödinger eq.
The pseudopotential for a problem is neither unique nor exact.
On Empty Core Model (ECM)Unscreened potential :
V(r)=0, for r<Re
-e2/r, for r>ReNaLater, screening effect :Thomas-Fermi dielectric function
Empirical Pseudopotential Method (EPM) band structure
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-ray diffraction determination,
giving an understanding of the bonding and have great predictive value for proposed new structures and compounds.
Numerical calculation of band structures using the first-principal non-local pseudo-potential.
A. Zunger and M.L. Cohen, Phys. Rev. B20, 4082 (1979).
Si W
Numerical calculation of Density of states
Some physical quantities obtained from numerical calculation and experimental measurement , respectively.
4.334.43
8.107.35
3.603.57
Diamond
0.730.77
4.263.85
5.665.65
Ge
0.980.99
4.844.63
5.455.43Si
Bulk modulus (Mbar)
Cohesive energy
(eV)
Lattice constant
(Å)
APPROACHESFree atoms
Atomic ϕ’s
Metallic crystals
Free electrons
Tight-Binding model
Overlapping ϕ’s
Nearly Free electron model
Free electrons + periodic potential
Essence of energy gaps / transport
Essence of crystal binding Many body Treatments
PseudopotentialsBand structures
Full treatment
Experimental methods in Fermi Surface studiesH is along [111] direction – noble metal
de Haas-van effect :
Oscillation of the magnetic moment of a metal as a function of magnetic field (H). (1930)
A111(belly)/A111(neck)=51 Ag
Quantization of orbits in a magnetic fieldReview: a charge q of mass m in a magnetic field B
AB where,Acqp
2m1H
2 rrrr×∇=
−=Hamiltonian
Acqkppp fieldkinetictotal
rrh
rrv +=+=Total momentum
Bohr-Sommerfeld relation – orbits are quantized in a magnetic field
∫∫
∫
•+•=
+=•
rdAcqrdk
2)21n(rdptotal
rrrrh
hrr
π
Brcqk
Bdtrd
cqBv
cq
dtkd
rrrh
rrrrr
h
×=⇒
×=×=
( )
BΦ−=•−=
ו−=•×=• ∫∫∫
cq2n2(area)B
cq
rdrBcqrdBr
cqrdk
r
rrrrrrrrh ( )
BadB
adArdA
Φ=•=
•×∇=•
∫∫∫
rr
rrvr
the first termthe second term
hrr
π221n
cqrdp Btotal
+=Φ−=•∫
27B m Tesla1014.4
21n
qc2
21n −×
+=
+=Φ
hπ
magnetic flux through the orbit in real space
How about in k- space ?
n
2
n SqB
cA
=
hkqB
cr ∆=∆h
qc2
21nS
qBcBBA n
2
nBhh π
+=
==Φ
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 ππ
+=
+=
Different orbits can have the same area by changing magnetic field B.For instance,The nth orbit in magnetic field BnThe (n+1)th orbit in magnetic field Bn+1
cq2
21n
BS
n h
π
+=
cq2
B1
B1S
n1n h
π=
−
+
Equal increments of 1/B reproduce orbits w/ the same area.
Bi, T=1.6K
Steele and Babiskin,
PR98, (1955)
-M/H (106) Bi (1930)&(1932)
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 Fermi surface in a plane normal to magnetic field.
(1952)S1
cq2
B1
h
π=
∆
Bcq2
21nSn
h
π
+= The normal line of the orbital area is along the
direction of magnetic field B.
Quantized of the closed orbits in a magnetic field B. (L.D. Landau)
Free electrons model an electron in a cubical box of side L in magnetic field B z
cyclotron frequencyn=1,2,…positive integer
mceB where
21nk
2m)k( cc
2z
2
zn =
−+= ωωε h
h
only need to consider kx and ky
The number of levels with energy ε for a given n and kz
( )ceB 2m 2m 2kk2k c2
2
hhh
πωπεπππ ==∆=∆=∆
2
2LD(k)
=
π
Bc 2c
eB 22L 22
hh ππ
πeL
=
depending on B
the area between successive orbits
the number of level per unit area in k-space
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.
Fermi sphere – volume in k-space occupiedby electrons in the ground states
ε
εF
B=0 B1
hωc
s+1εFpartly filled
s-1
sempty
B2
splitting into many Landau levels
g(ε)
s
0.5hωc
increasing slightly
εF At the critical fields Bs,
no partly filled level
and c
2
Bc 2
sNhπ
eL=
How to put electrons in the energy levels?
Bc 2c
eB 22L 22
hh ππ
πeL
=
The number of levels with energy ε 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-
cc ωπ
ωπ
hh
hh
Bc2
eL 2s)
21B(n
c2eL 22s
1n
2
=−∑=
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
)21s(B
c2eLsN
2
+
− cω
πh
hU(B)
cωπ
hh
Bc2
eL 2s 22
Oscillation of totalelectronic energy
Magnetic moment at T=0K
BU
∂∂
−=µ
cSe2
B1
h
π=
∆
where S is the extremalarea of the Fermi surface normal to the direction of B
Information of the Fermi surface
shape and size
oscillates.
w/. period
Oscillation of the magnetic momentDe Haas – van Alphen effect
What are the extremal areas ?When magnetic field is along z-axis,
the area of a Fermi surface cross section at height kz is S(kz), and the extremal area Se are the values of S(kz) at the kz where dS/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
Cu, Ag, Au, monovalent metal w/. FCC structure
( )a
4.90a43n3k
1/3
323/12
F =
== ππ
Fermi surface
FCC lattice – BCC reciprocal lattice
The distance between hexagonal faces is a10.883
a2
=π
The distance between square faces is a12.572
a2
=π
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
1/B111=2.05×10-9 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=1.95×10-9 gauss-1 S=4.90 ×1016 cm-2
a: a closed particle orbitb: a closed hole orbitc: an open orbit
Sbelly/Sneck=29
-6 -4 -2 0 2 4 6-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
R xx (k
Ω)
T=0.4K
Rxx=45Ω/ (LT), 5kΩ/ (RT)
n=1.4x1011 (cm-2)
µ=0.99x106( cm2/Vsec)
Rxy
(h/e
2 )
H (Tesla)
0.0 0.2 0.4 0.6 0.80.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
2DEGE1
energy
Ec
EF
0.2eV
10 nm, GaAs Cap
15 nm, δ- doping layer, Si
60 nm, spacer AlGaAs
1500 nm, GaAs
buffer layer
8 nm, spacer AlGaAs
1 2 3 4 50.0
0.1
0.2
0.3
T=0.4K
Rxx
(Ω)
H-1 (Tesla-1)
2DEG
0.3µm
0.45µm
0.6µmSource Drain
Quantized Conductance TransportGaAs/AlGaAs heterostructures
n=1.4×1011/cm2
µ=2.2×106cm2/Vs (0.3K)
(Dr.Umansky provided)
Mean free path l=13.6 µm
-1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.40123456789
101112
G (2
e2 /h)
VSG (V)
Split gates confined QPC
dgap=0.3µm and lchannel=0.5 µm RhΩ
=
=
29001N ;2eNG2
-1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.00.0
2.0k
4.0k
6.0k
8.0k
10.0k
12.0k
14.0k
16.0k
18.0k
20.0k
R (Ω)
VSG (V)
1D2D
T=0.3K