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H. Buhmann H. Buhmann Hartmut Buhmann Physikalisches Institut EP3 Physikalisches Institut, EP3 Universität Würzburg Germany
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Transcript of Physikalisches Institut EP3Physikalisches Institut, EP3 ... · PDF filePhysikalisches Institut...

  • H. BuhmannH. Buhmann

    Hartmut Buhmann

    Physikalisches Institut EP3Physikalisches Institut, EP3Universitt Wrzburg

    Germany

  • H. BuhmannH. Buhmann

    Outline

    Hall effects and topological insulatorsp g

    HgTe quantum well structures

    experimental observations quantized conductivity non-locality spin polarization

  • H. BuhmannHall Effects

    1879M.I. Dyakonov and

    V.I. Perel (1971)

    V = 0VH = 0

    ( )BvEeF rrrr +=

    E j syjxy y xE j =x

    sysH E

    j

    (dissipationless spin current)

  • H. BuhmannQuantum Hall Effect

    1 h

    Nobel Prize K. von Klitzing 1985

    Hall resistance 21

    eh

    ienB

    sxy ==

    xy

    semiconductor 2DEG

    xx

    zero longitudinal resistance

  • H. Buhmann2DEG in a Magnetic Field

    magnetic field

    -

    g

    -

    -

    - -

    -

    hi l d t tchiral edge states

  • H. BuhmannEdge Channels

    magnetic field

    -

    -

    -

    -

    -

    -

    1d edge channelsD.B. Chklovskii, B.I. Shklovskii, L.I. Glazman, Phys. Rev. B 46, 4026 (1992)

  • H. BuhmannQuantum Hall Effect

    0=3=Edge Channel Picture

    0

    0

    0=I +-

    0=

  • H. BuhmannQuantum Spin Hall Effect

    two copies of QH statestwo copies of QH states, one for each spin component, each seeing the opposite magnetic field,

    are united in one sample and form

    helical edge states. This new state does not break the time reversal symmetry, and can exist without any

    g

    and can exist without any external magnetic field.

    Quantum Spin Hall StateQ p

    consisting of two counter propagating spin polarized edge channels.protected by time reversal symmetry (Kramers pair),p y y y ( p ),and an insulating bulk

  • H. BuhmannQuantum Spin Hall Effect

    Stability of Helical Edge States: 4 = 2 + 2

    + sksk ,, ,backscattering between Kramers doublets is forbidden

    x

    heGQSHE

    22=

    x

    backscattering is only possible by time reversal symmetry breaking processesbackscattering is only possible by time reversal symmetry breaking processes (for example external magnetic fields)

    or if more edge states exist

    + sksk ,, 21, x

    x

    xx

  • H. BuhmannQuantum Spin Hall Effect

    no magnetic fieldmagnetic field

    - -- spin up

    g

    -

    -

    -

    --

    -

    - - --spin down

    chiral edge states helical edge states

  • H. BuhmannQSHE in Graphene

    Graphene edge states

    gap

    C.L.Kane and E.J.Mele, PRL 95, 226801 (2005)

    Graphene spin-orbit coupling strength is too weak gap only about 10-3 meV. p p p g g g p y

    not accessible in experiments

  • H. BuhmannQSHE in HgTe

    Helical edge statesfor inverted HgTe QW

    E

    H1

    E1

    k 0

    B.A Bernevig, T.L. Hughes, S.C. Zhang, Science 314, 1757 (2006)

  • H. BuhmannH. Buhmann

  • H. BuhmannQuantum Well Structures

    E

    quantizedenergylevels in

    di iEg2 E z-directionEg1

    x quasi freeelectron gas

    z (growth direction) y

    gin the xy-plane

  • H. BuhmannMBE

    Quantum Well Growthbyby

    Molecular Beam Epitaxy

    QW

  • H. BuhmannMBE

    Integrated in UHV transport system with 6 chambers Riber 3200 MBE Horizontal system 8 cell ports Maximum substrate dimension 2 wavers Pumped by a kryo pump, three nitrogen coldshields and a Ti-Pumped by a kryo pump, three nitrogen coldshields and a Ti

    sublimation pump

  • H. BuhmannHgTe Quantum Wells

    MBE-Growth

    6.0

    5 5

    Bandgap vs. lattice constant(at room temperature in zinc blende structure)

    4 0

    4.5

    5.0

    5.5

    )

    3.0

    3.5

    4.0

    ap e

    nerg

    y (e

    V)

    1.5

    2.0

    2.5

    Ban

    dga

    1.0

    0.5

    0.0

    5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.76.6-0.5

    lattice constant a []0 CT-CREW 1999

  • H. BuhmannHgTe Quantum Wells

    free electron gas in the QW

    by donor doping of the barriers

  • H. BuhmannH. Buhmann

  • H. BuhmannHgTe

    band structuresemi-metal or semiconductorsemi metal or semiconductor

    1000

    0

    500

    eV) 8

    E

    -500E(m

    e

    6

    Eg

    -1 0 -0 5 0 0 0 5 1 0-1500

    -1000

    7

    fundamental energy gap

    1.0 0.5 0.0 0.5 1.0k (0.01 )

    D.J. Chadi et al. PRB, 3058 (1972)

    fundamental energy gap

    meV 30086 EE

  • H. BuhmannHgTe-Quantum Wells

    BarrierQW

    1000HgTe Hg0.32Cd0.68Te

    1000

    500

    8

    5006

    -500

    0

    E(m

    eV) 8

    6 -500

    0

    E(m

    eV)

    8VBO

    -1000

    7

    -1000

    7

    -1.0 -0.5 0.0 0.5 1.0k (0.01 )

    -1500

    7

    -1.0 -0.5 0.0 0.5 1.0k (0.01 )

    -1500

    7

    VBO = 570 meVVBO = 570 meV

  • H. BuhmannHgTe-Quantum Wells

    Typ-III QW

    d

    HgTe

    6

    HgCdTeHgCdTe

    g

    HH1E1

    8

    band structure

    invertednormal

  • H. BuhmannBand Gap Engineering

    H CdTHgTe

    6Zero Gap for 6.3 nm QW structures

    HgCdTeHgTe

    6

    H1E1

    HgCdTe

    8

    E1H1

    8

    8normal

    inverted

    direct band gaps between80 ... 0 and 0 ... - 30 meV

  • H. Buhmann8 band k.p calculation

    finite gap

    E (m

    eV)

  • H. BuhmannSimplified Picture

    normal

    m > 0 m < 0

    insulator QSHE

    bulkbulk

    bulkinsulating

    entire sampleinsulating

  • H. BuhmannH. Buhmann

  • H. Buhmann

    Experiment

    k k

  • H. BuhmannSmaller Samples

    LL

    W

    (L x W) m

    2.0 x 1.0 m1.0 x 1.0 m1.0 x 0.5 m

  • H. BuhmannQSHE Size Dependence

    106(1 x 1) m2

    non-inverted10

    non inverted

    105

    2

    Rxx

    /

    104G = 2 e2/h

    (1 x 1) m2(2 x 1) m2

    -1.0 -0.5 0.0 0.5 1.0 1.5 2.0103

    (1 x 0.5) m2

    (VGate- Vthr) / VKnig et al., Science 318, 766 (2007)

  • H. BuhmannQSHE Temperature Dependence

    Q2367 Microhallbar 1 x 1 m2

    I(2 8) U(9 11) B 0 T T i b l

    16.0

    18.0 T = 1.8 K T = 2.5 K

    I(2-8), U(9-11); B=0 T; T= variabel;

    12.0

    14.0T = 4.2 K T = 6.4 K T = 9.8 K T = 19 K

    6 0

    8.0

    10.0 T = 28 K T = 48 K

    R2-

    8, 9

    -11(

    k)

    2.0

    4.0

    6.0R

    -2.0 -1.6 -1.2 -0.8 -0.4 0.0-2.0

    0.0

    UG(V)

  • H. BuhmannH. Buhmann

  • H. BuhmannConductance Quantization

    20

    in 4-terminal geometry!?

    V

    16

    18

    0 V

    12

    14 G = 2 e2/h

    k

    I

    8

    10

    Rxx

    / k

    2

    4

    6

    -1.0 -0.5 0.0 0.5 1.0 1.5 2.00

    2

    (V V ) / V(VGate- Vthr) / V

  • H. BuhmannMulti-Terminal Probe

    Landauer-Bttiker Formalism normal conducting contacts

    no QSHEno QSHE

    ( ) e2

    100012

    ( )

    =

    ijjijiiiii TRMh

    eI 2

    =001210000121100012

    T

    214

    43 2

    2t

    I eGh

    = =

    heG t

    2

    exp,4 2

    210001121000012100

    T 3 22

    142

    4 1

    23t

    I eGh

    = =

    3exp4

    2 t

    t

    RR

    210001

    generally22 2)1(

    ehnR t

    +=

  • H. BuhmannQSHE in inverted HgTe-QWs

    40

    30

    35G = 2/3 e2/h

    I1

    2 3

    4

    V

    25

    30

    (k

    ) 2 3V 56

    15

    20

    G = 2 e2/h

    Rxx

    ( I1 4

    56

    5

    10(2 x 1) m2

    32 tRR-1.0 -0.5 0.0 0.5 1.0 1.5 2.00

    (V V ) (V)

    (1 x 0.5) m2

    4tR(VGate- Vthr) (V)

  • H. BuhmannNon-locality

    Q2308: 250 | 90: 0.1 | 400 | 90 | 400 | 90: 0.1 | 1000 | ns= 3.1x1011, =143 000

    6 8 9

    1 m2 m

    3 2 11

  • H. BuhmannNon-locality

    1 2 3

    1 m2 m

    6 5 4

  • H. BuhmannNon-locality

    I: 1-41 2 3

    V2 3V

    Vnl:2-3

    6 5 4

    I1 4

    56

    20

    2214

    @6 mK Q2308-02

    12

    14

    16

    18

    8

    10

    12

    Rnl nA

    )

    6

    8

    10

    4

    6

    8 (k

    )I (n

    -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

    2

    4

    0

    2

    UGate [ V ]

    A. Roth, HB et al.,Science 325,295 (2009)

  • H. BuhmannNon-locality

    I: 1-31 2 3

    I: 1 3Vnl:5-4

    6 5 4

    LB 8 6 k

    10

    12VLB: 8.6 k

    6

    8

    G=3 e2/h

    4-terminal

    4

    6

    -1 0 1 2 3 40

    2

    -1 0 1 2 3 4

    V* (V)A. Roth, HB et al.,Science 325,295 (2009)

  • H. BuhmannNon-locality

    1 2

    I: 1-4Vnl:2-3

    34

    V

    34

    G=4 e2/h5

    6

    7

    LB: 6.4 k

    3

    4

    R14

    ,23

    / k

    1

    2

    A. Roth, HB et al.,Science 325,295 (2009)

    -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.00

    Vgate / V

  • H. BuhmannNon-locality

    LB 4 3 k1V

    LB: 4.3 k1 2 3

    6 5 4

    5

    6

    3

    4

    nl (k

    W)

    1

    2

    Rn

    -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.00

    A. Roth, HB et al.,Science 325,295 (2009)

  • H. BuhmannBack Scattering

    potential fluctuations introduce areas of normal metallic (n- or p-) conductancepotential fluctuations introduce areas of normal metallic (n or p ) conductancein which back scattering becomes possible

    QSHE

    The potential landscape is modified by gate (density) sweeps!