Fermi Surface III

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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