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