Highly Effective Polarized Electron Sources Based on Strained by sanmelody

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									Optimization of Semiconductor Superlattice
    for Spin-Polarized Electron Source

                        Workshop on Sources of
                        Polarized Electrons and High
                        Brightness Electron Beams


                                      Leonid G. Gerchikov
       Laboratory of Spin-Polarized Electron Spectroscopy
                      Department of Experimental Physics
                              State Polytechnic University
                                    St. Petersburg, Russia
                     Collaborators
Department of Experimental Physics, St. Petersburg State
Polytechnic University, Russia, Yuri A. Mamaev, Yuri P.Yashin, Vitaly
V. Kuz’michev, Dmitry A. Vasiliev, Leonid G. Gerchikov

A.F. Ioffe Physicotechnical Institute RAS, Russia, Viktor M. Ustinov,
Aleksey E. Zhukov, Vladimir S. Mikhrin, Alexey P. Vasiliev

Stanford Linear Accelerator Center, Stanford, CA, USA, James E.
Clendenin , Takashi Maruyama

Institute of Nuclear Physics, Mainz University, Mainz, Germany, Kurt
Aulenbacher, Valeri Yu. Tioukin
                       OUTLINE
•    Introduction
    –   Goals of optimization
    –   Problems of optimization
    –   Best photocathodes
•    Calculations of SL parameters
    –   Energy spectrum
    –   Photoabsorption
    –   Transport
•    AlInGaAs/AlGaAs SL with strained QW
    –   Optimized design
    –   Results
•    Summary&Outlook
   Goals of optimization



High maximal P at large QE
   High polarization of electron emission from
strained semiconductor SL at the expense of QE
                                                                             Spectra of electron emission: Polarization P and Quantum Efficiency QE
                        InGaAs AlGaAs                                                                                  QE                    Polarization
                                                           electron
                                                           emission
                                                                                                                                                              100
 Ec                                                                                                 1
                                                                                               10

                                                                                                                                                              80
                                        electron                                               10
                                                                                                    0
               heavy hole              generation
                miniband




                                                                                                                                                                    Polarization, %
                                                                                                -1
                                                                                               10                                                             60
 Ev




                                                                                       QE, %
                                                                                                -2
                                                                                               10
                                                                                                                                                              40
                light hole                    valence band                                      -3
                miniband                                                                       10

                                                                                                -4                                                            20
                                                                                               10
                                                                  band
                                                                  bending                       -5
                                                                  region                       10                                                             0

                                                                                                        550    600    650    700      750   800      850    900
                1.6                                                                                                           , nm
  Energy, eV




                              e1                                                                        SL Al0.2 In0.155 Ga0.65As(5.1nm)/Al0.36Ga0.64As(2.3nm)
                1.5

                1.4
                                                                      •Polarization is maximal at photoabsorption
           -0.02             hh1                                      threshold where QE is small.
           -0.04             lh1
                                                                      •Strain relaxation does not allow to produce thick
                                0.00                0.04              photocathode with high QE.
                              -1
                      k001, A                  k100, A-1              •Rise of the vacuum level increases P and
                                                                      decreases QE
         To get the best
          P                    QE
•Large valence band       •Thick working layer
splitting > 60 meV        • High NEA value
• High strain splitting   • Heavy doped BBR
and offsets in valence    layer
band
• Effective electronic
transport along SL axes
• High quality SL,
uniform layer
composition and
thickensses
• Low doping in SL
             Best photocathodes
Sample          Composition          Pmax   QE(max)      Team

SLSP16 GaAs(3.2nm)/                  92%     0.5%        Nagoya
       GaAs0.68P0.34 (3.2nm)                            University,
                                                          2005
SL5-777 GaAs(1.5nm)/                 91%     0.14%     SPbSPU, 2005
        In0.2Al0.23Ga0.57As(3.6nm)


SL7-307 Al0.4Ga0.6As(2.1nm)/         92%     0.85%     SPbSPU, 2007
        In0.19Al0.2Ga0.57As(5.4nm)
              Calculations of SL’s energy spectrum and
             photoabsorption within 8-band Kane model
Miniband spectrum: ˆ
                   Hk ,q  E(k, q)k ,q
             1 ik iqz              2 
   k ,q 
              d
                e       
                         ,n
                             u exp  i
                                     d
                                        nz  An , (k , q )
                                           
Photoabsorption coefficient:




 Polarization:
               
        P0 
               
              P&QE of electron emission
  P( )  P0 ( ) K t K e               P0 - initial polarization,
                                          Kt , Ke – depolarization factors
  P0        /              on stages of transport in SL
  K t   s /  s   t , K e  0.95    and emission through BBR,
                                          s,t – transport and spin
                                          relaxation times in SL.




QE ( )  (1  R)   ( )  d   B       R – reflection from GaAs ,
                                              B – probability of electron
R  0.3, B  0.1                              emission through BBR.
                                   Initial electron polarization P0
                            Polarization spectrum                                                                    Photoabsorption spectrum
                                                                                                                    Photoluminescence spectrum
                                                                                                                        Energy band spectrum
                  100
                                                                                                                           GaAs(9nm)/Ga Al As(9nm)
                                                                                                                                              GaAs(9nm)/Ga0.6Al0.4As(9nm)
                                                                                                                                                                0.6 0.4
                                                                                                                               1,7
                                                    = 5meV




                                                                            -1
                                                                                                                 4,0                  Total




                                                                            Photoabsorption Coefficient, 10 cm
                                                                                                                               1,6    Up          e2
                                                    =10meV




                                                                            4
                   80                                                                                            3,5                  Down
                                                    =20meV                                                                           Partial     e1
Polarization, %




                                                                                                                               1,5
                                                                                                                                      contributions
                                                                                                                 3,0
                   60                                                                                                                                 hh1
                                                                                                                 2,5




                                                                                                                 Energy, eV
                        hh1-e1                                                                                   2,0                                        lh1
                   40                                                                                                         -0,05              hh2
                                                                                                                 1,5
                                                                                                                                                e1-hh1
                                 lh1-e1    hh2-e2          lh2-e2                                                                                        hh3
                   20                                                                                            1,0
                                                                                                                              -0,10                                           e2-hh2
                                                                                                                                                                                            e2-lh2
                                                                                                                                                                         e2-lh1
                                                                                                                 0,5                                        lh2
                                                                                                                                                         e1-hh1
                                                                                                                                                      e1-lh1
                    0                                                                                            0,0                                  hh4
                    1.45    1.50    1.55    1.60    1.65      1.70   1.75                                          1,40
                                                                                                                     -0,15 1,45                1,50     1,55      1,60   1,65    1,70   1,75
                                                                                                                                                  Photon Energy, eV
                                 Photon Energy, eV                                                                               Kz, A
                                                                                                                                         -1   0,00              0,05                        -1       0,10
                                                                                                                                                                                   K||, A
                                                                                                                  Maximal P0 is limited by:
Maximal P0 is determined by:                                                                                      mixture optical transitions losses
                                                                                                                    Main polarization
                                                                                                                 Initial of hh and lh states due to
• Valence band splitting Ehh-                                                                                    smearing up
                                                                                                                 amount e1of toband e1edge and
                                                                                                                    hh1 –       lh115% depending
                                                                                                                  broadening of -
lh= Ehh1 - Elh1
                                                                                                                                     hole spectrum
                                                                                                                  caused e2 quality and design-
                                                                                                                      structure lh2 - e2
                                                                                                                 onhh2 -by doping and fluctuations
• Broadening of hole spectrum                                                                                   ofand Ehh-lh
                                                                                                                     layer composition
          Initial electron polarization P0

Enlarge Ehh-lh to increase maximal P0
                                                                                        AlInGaAs
                                                                         Evh
                                                                          vh
                                                                                                       AlGaAs              hh1
                                                                         Evl
                                                                                                                                Ehh-lh
by increase of QW deformation                                                                                             lh1
                                                                                                                                Ehh-lh
                                                                            Evl
                                                                                                                          lh1

 Large strain deformation leads to
structural defects and strain relaxation
                                                                                    100
                                                                                    100

                                                                           80
Optimal      combination   of     strain                                   70
                                                                                        80         Total




                                           Valence Band Splitting, meV
                                                                                        80
                                                                                                   Quantum effect
                                                                                                                                                  60
deformation and quantum confinement




                                                                  Polarization, %
                                                                           60                      QW width




                                                                 Polarization, %
                                                                                        60
                                                                                        60
                                                                           50
effect to provide maximal valence band                                     40
                                                                                        40
splitting with minimal risk of strain                                      30
                                                                                        40
                                                                                                                                                  40


relaxation     and   good     transport                                    20           20
                                                                                        20


properties                                                                 10
                                                                                         0
                                                                                    0    01.45                1.50                 1.55
                                                                                                                                                  20
                                                                                        0 1.45    10       20 1.50   30         40 1.55 50   60
                                                                                                 Photon Energy, eV
                                                                                                 Photon Energy, eV meV
                                                                                                  Deformation splitting,
             Electronic transport in SL

            Ballistic electron tunneling though SL


       I0                                                   I=I0T

 RI0                                                         BBR

                    Tres  exp(-b)                Tf  exp(-2b)

                                      Tf << Tres


Tunneling probability        Tunneling time
T = I/I0                      = ∫ |Ψ(x)|² dx/I
                                         Ballistic transport
    1.0                                                         100          1.0                                                       100

                                                                             Tunneling resonances
    0.8                                                                      0.8

                                                                10
                                                                             En = E0 − ∆E/2Cos(qnd)                                    10
                                                                             qn = πn/d(N+1)




                                                                                                                                            ps
                                                                 ps
    0.6                                                                      0.6




                                                                         T
T




    0.4                                                                      ∆E – width of e1 miniband
                                                                             0.4
                                                                1                                                                      1

    0.2
                                                                             N – number of QW in SL
                                                                             0.2


    0.0                                                       0.1            0.0                                                       0.1
          60   62   64   66   68   70   72   74   76   78   80               Time of resonant tunneling
                                                                                   60   62   64   66   68   70   72   74   76   78   80
                              E, meV                                                      E, meV
                                                                             SL = ħ/∆E  exp(b)
      b                            b                        b                Transport time
                                                                              = ħ/Γ  exp(2b)
                                                                             Γ << ∆E ,  >>  SL

                                                                             Optimal choice: bf = b/2
             Electronic diffusion in SL

     Kinetic equation       - electronic density matrix
                            H – effective Hamiltonian of SL
                            in tight binding approximation
    
        H  St 
     ˆ                      St{} – collision term including:
  i       ˆˆ       ˆ        • collisions within each QW in
    t                      constant relaxation time , p,
                            approximation
                            • tunneling through last barrier
 Stationary pumping         to BBR
 Approximate solution       • optical pumping

    ( N  1 / 2)( N  1)          N – number of QW in SL
                        N f   V = E/4 – matrix element of
          6V  p
               2
                                  interwell electron transition
                Electronic diffusion in
                SL          bulk GaAs
    ( N  1 / 2)( N  1)                L2 L
                        N f         
          6V  p
               2
                                        3D S
                                    D = 40 cm2/s – diffusion
N  1, d  SL period               coefficient
                                    S = 107 cm/s – surface
D  2V 2 d 2 p , S  d /  f
                                    recombination velocity

For SL Al0.2In0.2Ga0.6As(5.4nm)/ Al0.4Ga0.6As(2.1nm)
D = 12 cm2/s , S = 3*106 cm/s
Pulse response of SL Al0.2In0.16Ga0.64As(3.5nm)/
        Al0.28Ga0.72As(4.0nm) 15 periods

                          Time dependence of electron emission

                          1.0                                  *
                                                     Experiment
                                                     Calculations
        Intensity, a.u.




                          0.8
                                                                         D = 16 cm2/s ,
                          0.6                                            S = 3.4*106 cm/s
                          0.4


                          0.2


                          0.0
                                   0       5        10        15    20
                                               Time, ps
                      * K.   Aulenbacher et al, Mainz, 2006
                        Strained-well SL

        GaAs BBR
     Unstrained barrier
          ab = a0
       Strained QW
          aw > a0
SL
       Strained QW
          aw > a0
     Unstrained barrier
          a b = a0

     Buffer Layer
                            Large valence band splitting
     a0 - latt. const
                            due    to  combination    of
                            deformation and quantum
       GaAs Substrate
                            confinement effects in QW
MBE grown AlInGaAs/AlGaAs strained-well superlattice


          Composition                Thickness        Doping
                                   As cap
             GaAs QW                        60 A   71018 cm-3 Be
       Al0.4Ga0.6As                         21 A
                              SL                   31017 cm-3 Be
   In 0.19Al 0.2Ga 0.65As                   54 A
      Al0.35Ga0.65As        Buffer      0.3 m     61018 cm-3 Be
                            p-GaAs substrate


   Eg = 1.536 eV, valence band splitting Ehh1 - Elh1 = 87 meV,
   Maximal polarization Pmax= 92% at QE = 0.85%

                       SPTU & FTI, St.Petersburg
                Choice of SL parameters




                                                                                                   Valence Band Splitting, meV
                                          Evh 0.00                                            90

• y - In concentration in QW




                                     Miniband Edges, eV
                                                                                              80
                                                                                    Ehh-lh
                                                          -0.05
• x - Al concentration in QW                               Evl
                                                                                   hh1
                                                                                   lh1
                                                                                              70


• z - Al concentration in barrier                         -0.10
                                                                                   hh2        60


• a – QW width                                                                                50


• b – barrier width                                       -0.15                               40

                                                                                              30
                                                          -0.20
                                                                     2        4         6

                                                                  QW (AlInGaAs) width, nm


  AlxInyGa1-x-yAs - QW                                    AlxGa1-xAs - Barrier

y = 0.2, Ev = 76 meV               z = 0.4, Uhh = 332 meV,
x = 0.19, Eg = 1.536 eV             Ulh = 258 meV, Ue = 234 meV,
a = 5.4 nm, Ehh-lh = 87 meV        b = 2.1 nm, Ee = 31 meV
SL Al0.19 In0.2 Ga0.61As(5.4nm)/Al0.4Ga0.6As(2.1nm)
                          QE, Experiment             P, Experiment
                          QE, Theory                 P, Theory
                                                                     100
                  1
             10

                  0
             10                                                      80




                                                                           Polarization, %
              -1
             10
                                                                     60
     QE, %




              -2
             10

              -3
             10                                                      40

              -4
             10
                                                                     20
              -5
             10

              -6
             10                                                   0
                  650      700      750        800    850       900

                                       , nm

                        Pmax= 92%, QE = 0.85%
                                   SL Al0.19 In0.2 Ga0.61As(5.4nm)/Al0.4Ga0.6As(2.1nm)
                                          Dn, Theory                   Up, Theory
                                          Dn, Experiment               Up, Experiment                       QE, Experiment             P, Experiment
                                                                                                              QE, Theory                 P, Theory
                                                                                                                                                         100
                                                                                                        1
Photoabsorbtion coefficient,m-1




                                                                                                   10
                                   5
                                                                                                        0
                                                                                                   10                                                    80

                                   4




                                                                                                                                                               Polarization, %
                                                                                                    -1
                                                                                                   10
                                                                                                                                                         60




                                                                                           QE, %
                                                                                                    -2
                                   3                                                               10

                                                                                                    -3
                                                                                                   10                                                    40
                                   2
                                                                                                    -4
                                                                                                   10
                                   1                                                                                                                     20
                                                                                                    -5
                                                                                                   10

                                                                                                    -6
                                   0                                                               10                                                 0
                                   650      700         750       800        850          900           650    700      750        800    850       900

                                                          , nm                                                            , nm

                                        = 25 meV, P0max = 97%
                                       Polarization losses at
                                       • photoabsorption – 3%
                                       • transport and emission – 5%
                         Photocathode with DBR
 Goal: considerable increase of QE at the main polarization maximum
 and decrease of cathode heating
 Method: Resonance enhancement of photoabsorption in SL integrated
 into optical resonance cavity

                                                                                  h
Photoabsorption in
the working layer:
L << 1,                      GaAs                                        e
 - photoabsorbtion          Substrate      Buffer      SL        BBR
coefficient,
L - thickness of SL
                                         RDBR= 1                    RGaAs= 0.3
Resonant                                                                                h
enhancement by
factor
2/(1-(RDBRRGaAs) 1/2)2                                                        e
                           GaAs
Heating is reduced        Substrate   DBR      Buffer        SL     BBR
by factor L
                   Resonant enhancement of QE
                             QE enchancement                        P-4, SL QT 1890 non DBR
                                                                    P-2, SL QT 1890 DBR
                  10
                                                                                              80


                   8                 SPTU data
                                                                                              60
QE Enchancement




                                                                                                   Polarization, %
                   6

                                                                                              40

                   4

                                                                                              20
                   2


                                                                                              0
                   0
                       550     600      650      700   750   800    850     900     950
                                                   Wavelength, nm

                             Accepted for publication at Semiconductors, 2008
         Summary & Outlook
Photocathode       based        on    optimized
 AlInGaAs/AlGaAs        strained-well         SL
 demonstrates Pmax= 92% at QE = 0.85%.
Maximal initial photoelectron polarization P0 =
 97%. To increase P0 the higher fabrication
 quality SL is needed.
Optimization of polarization losses and QE on
 the stage of electron transport and emission
 needs an additional investigations.
DBR can considerably increase QE and reduce
 cathode heating.
This work was supported by
• Russian Ministry of Education and Science under grant N.P. 2.1.1.2215 in the frames
   of a program “Development of the High School scientific potential”
• Swiss National Science Foundation under grant SNSF IB7420-11111




         Thanks for your attention!
              SL In0.155Al0.2Ga0.645As(5.1nm)/Al0.36Ga0.64As(2.3nm)


                    o QE                            P
                                                                            100



                                                                            90




                                                                                  Polarization, %
         1
                                                                            80
QE, %




                                                                            70



                                                                            60
        0,1


                                                                            50
              2    4      6     8       10     12       14   16   18   20
                                    thickness, pairs
                         Reproducibility
        QE-1, SL 7-389 T=300K Tht=540C 21,04,2007                 P-1, SL 7-389 T=300K Tht=540C 21,04,2007
        Y7-307 Heating 540K                                       P 7-307 Heating 540K 30.11.2006




                                                                                                    100
             1
        10


             0                                                                                      80
        10




                                                                                                          Polarization, %
         -1
        10                                                                                          60
QE, %




         -2
        10                                                                                          40


         -3
        10                                                                                          20

         -4
        10
                                                                                                    0

                 550    600        650        700           750         800        850        900
                                                    , nm
                                  Polarization losses
                                                              100
                                                              100                              two interval calc.*0.98
                                                                                               two interval calc.*0.98
                                                                                               basis calc.*0.945 delta=10 meV
                                                                                               basis calc.*0.945 delta=10 meV
                                                                                               basis calc.*0.945 delta=20 meV
                                                                                               basis calc.*0.945 delta=20 meV
     Initial electron polarization




                                            Polarization, %
                                                                                               tail 30 meV, delta=25 meV




                                            Polarization, %
         1                                                    80
                                                              80                               tail 30 meV, delta=25 meV
                                                                                               tail 30 meV, delta=45 meV
                                                                                               tail 30 meV, delta=45 meV
     P0 for different values of                                                                tail 11 meV, delta=25 meV
                                                                                               tail 11 meV, delta=25 meV
       0.1                                                                                     tail 11 meV, delta=7 meV
                                                                                               tail 11 meV, delta=7 meV
     hole spectrum broadeningmeV
                        =30 meV, =25
                                                              60                               tail 11 meV, delta=7 meV,
                                                                                               tail 11 meV, delta=7 meV,
                        =30=30 meV, =45 meV                 60                               gauss=40meV
     and
      0.01      smearing=30=11 meV, of meV
                                     =25
                                                                                               gauss=40meV
                                                                                               SPEC T=540K
                                                                                               SPEC T=540K
                        =30=11 meV, =7 meV
     absorption edge SPEC T=540K
QY




                                                                                               SPEC T=570K
                                                                                               SPEC T=570K
      1E-3                 SPEC T=570K                        40
                                                              40

      1E-4
                                                              20
                                                              20
      1E-5
             1.4     1.5    1.6     1.7    1.8
                                                                       1.4     1.5       1.6        1.7        1.8         1.9
                Photon energy, eV                                      1.4     1.5       1.6        1.7        1.8         1.9
     Polarization losses caused by                                                   Photon energy, eV
                                                                                     Photon energy, eV
     mixture of hh and lh states due to
     smearing of band edge and                                      GaAs0.83P0.17/Al0.1In0.18Ga0.72As (5x4nm)x20
                     for hole spectrum
     QE spectrum of different values
     broadening

                                                                QY ( 0((R) s s/   ext 
     valence band tails  = 10-30meV
     amount 5-10% depending on
     structural quality,  = 10-30meV.
                                                                P( ) ) P 1  ) B B  (s  )  d
   Потери поляризации при транспорте

             P( )  P0 ( ) Bs s /  s   ext 
s – время спиновой релаксации , при T=300K, Na=4*1017cm-3
s = 7*10-11s
ext – время выхода электронов из СР в область BBR,
ext = d/S + d2/12D,
d – ширина СР, S – скорость поверхностной рекомбинации на
границе BBR, D – коэффициент диффузии.
При T = 300K S = 1/4<v> = 107cm/s.
Для тонкого рабочего слоя d = 100nm ext = d/S,
ext = 7*10-13s
Поляризационные потери ext /s за время транспорта не более 1%
     High-Energy physics requirements
• High electron polarization, P > 80% is better
                                  90%

• High QE for large beam currents

•Large electronic current requirement Beam
  Accelerator     P, %
•Light energy limitations:
•Surface charge      85%
                  saturation
    MAMI
•Heating
                                High QE
                                     QE > 1%

eRHIC at BNL        70%       50-250 mA, QE > 0.5%

     ILC            80%            QE > 0.5%
                                 Reflectivity

                       Experiment, SL 7-396 DBR
                       Theory, SL 7-396 DBR
               1.0



               0.8
Reflectivity




               0.6



               0.4



               0.2



               0.0
                 750          800           850      900   950

                                    Wavelength, nm
 Spectra of electron emission, P(), QE()

                   QE, SL 7-396 DBR                         P, SL 7-396 DBR
                   QE, SL 7-395 no DBR                      P, SL 7-395 no DBR

             1
        10                                                                       80

             0
        10




                                                                                      Polarization, %
                                                                                 60
         -1
        10
QE, %




                                                                                 40
         -2
        10

         -3
        10                                                                       20

         -4
        10
                                                                                 0

             550    600      650         700   750    800      850       900
                                          Wavelength, nm
 Spectra of electron emission, P(), QE()

                   QE, Experiment SL 7-396 DBR         P, Experiment SL 7-396 DBR
                   QE, Theory SL 7-396 DBR             P, Theory SL 7-396 DBR

             1
        10                                                                          80

             0
        10




                                                                                         Polarization, %
                                                                                    60
         -1
        10
QE, %




                                                                                    40
         -2
        10

         -3
        10                                                                          20

         -4
        10
                                                                                    0

             550     600      650       700      750   800       850      900
                                         Wavelength, nm
                   Resonant enhancement of QE

                        Experiment SL 7-396 DBR
                        Theory SL 7-396 DBR
                  30
QE Enchancement




                  20




                  10




                  0
                  750             800              850   900
                                        Wavelength, nm
  Optimization of Photocathode structure

• SL structure: layers composition and                                                    BBR

  thickness are chosen to assure                                                           SL
    Eg=  for P()=Pmax
    Ehh-lh > 60meV for high polarization
                                                                                          Buffer
    Ee1 > 40meV for effective electron transport
• DBR structure: 20x(AlAs(/4)/                                                           DBR
  (AlxGa1-xAs(/4))
    Layer thickness l = /4n() for Bragg reflection                                  GaAs
    x  0.8 for large reflection band width  = 2n/n                              Substrate

• Fabry-Perot resonance cavity: BBR                                  1.0

  + SL + buffer layer                                                0.8

    Effective thickness = k /2 for QE() = QEmax
                                                      Reflectivity
    Effective thickness of BBR+SL  /4
                                                                     0.6
                                                                                                   
                                                                     0.4


                                                                     0.2
                                                                                                   
                                                                     0.0
                                                                        760   780   800    820   840    860   880   900
                                                                                     Wavelength, nm
Simulation of resonant photoabsorption

•   SL’s energy band structure, photoabsorption coefficient,
    polarization of photoelectrons.
    Method: kp – method within 8-band Kane model.
    A.V. Subashiev, L.G. Gerchikov, and A.I. Ipatov. J. Appl. Phys., 96, 1511
    (2004).


•   Distribution of electromagnetic field in resonance cavity,
    reflectivity, QE.
    Method: transfer matrixes.
    M.Born and E.Wolf. Princeples of Optics, Pergamon Press, New York,
    1991
                                             Ballistic transport
    1.0                                                       100          1.0                                                       100


    0.8                                                                    0.8

                                                              10                                                                     10




                                                                   ps
    0.6                                                                    0.6




                                                                                                                                          ps
T




                                                                           T
    0.4                                                                    0.4
                                                              1                                                                      1

    0.2                                                                    0.2


    0.0                                                       0.1          0.0                                                       0.1
          60   62   64   66   68   70   72   74   76   78   80                   60   62   64   66   68   70   72   74   76   78   80

                              E, meV                                                                 E, meV

								
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