Temperature dependent free energy barrier for magnetization

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					  Temperature dependent free
energy barrier for magnetization
    reversal in nanomagnets
           Thomas C. Schulthess
           schulthesstc@ornl.gov
Computer Science and Mathematics Division &
  Center for Nanophase Materials Sciences
Collaborators

 •   Cheggang Zhou, Don Nicholson, Markus Eisenbach, and
     Paul Kent, Oak Ridge National Laboratory
 •   David Landau, University of Georgia, Athens GA
 •   Oleg Mryasov, Seagate Research, Pittsburgh PA
 •   Greg Brown, Tallahassee FL
         Free Energy in biophysics and nanomagnetism
                                                                                                     5

  CBHI from Trichoderma reesei
          CBH I
                                                                                                         H
us on the linker domain
          Linker                                                                                             Θ
27 residue
   CBM
 olypeptide
9 glycosylation                                                                                                  M
 ites
 - 21 mannose
                                    Catalytic
   moleculesCellulose
                                    domain

                                                    ξ                                                                         particle volume
                                                                                                         H   Θ
                                   M M M
                               M M M M
                               M M M M
                                                                    M M
                                                                M M M
                                                                M M M           M M
                                                                                                                           KV > kB T
                               | | | |                          |   |   |       | |
o-Pro-Gly-Gly-Asn-Pro-Pro-Gly-Thr-Thr-Thr-Thr- Arg-Arg-Pro-Ala-Thr-Thr-Thr-Gly-Ser-Ser-Pro-Gly-Pro

                                                                                                                                    Θ
                                                                                                         M

                                                                                                         material dependent parameter
  Studying free energy in transition metal magnets
 H                            H
     Θ                            Θ
                                           FePt
         M
                         Θ
                              M

 Consider atomic degrees of
 freedom {m1 , m2 , ..., mN }
                   N
         M = 1/N         mi
                   i=1

F (T, M ) = E(T, M ) − kB T ln W (E, M )

Realistic model or LDA/GGA         With the density of states we would
based DFT calculation. In          know the free energy at all
most cases E(T, M ) ≈ E(T = 0, M ) temperature

             Can we compute the density of states?
  Metropolis Method                            Wand-Landau Method
  Metropolis et al, JCP 21, 1087 (1953)        Wang and Landau, PRL 86, 2050 (2001)


  Z=          e−E[x]/kB T dx                     Z=        W (E)e     −E/kB T
                                                                                dE
 Compute partition function and    If configurations are accepted with
 other averages with               probability 1/W all energies are visited
 configurations that are weighted  equally (flat histogram)
 with a Boltzmann factor
Sample configuration where Boltz- 1. Begin with prior estimate, eg W (E) = 1
mann factor is large.
                                  2. Propose move, accepted with probability
1. Select configuration
                                    Ai→f = min{1, W (Ei )/W (Ef )}
      Ei = E[xi ]
2. Modify configuration (move)    3. If move accepted increase DOS
      Ef = E[xf ]                       W (Ef ) → W (Ef ) × f f > 1
3. Accept move with probability   4. Iterate 2 & 3 until histogram is flat
                            β(Ei −Ef )
  Ai→f = min{1, e                         }   5. Reduce f → f =            f and go back to 1
  Generalization to continues models with
  applications to magnetism, proteins, ...
   Zhou, Schulthess, Torbrügge, and Landau, Phys. Rev. Lett. 96 120201 (2006)


                                                 Heisenberg model:
                                                     W (E, M )

                       PRL 96, 120201 (2006)        magnetizationC A L
                                                         PHYSI                  REVIE

                                                                                   ke
 Chain with                                                                        mo
                                                                                   wa
 torsional forces                                                                  ma
                                                                                   ori
                                                                                   CP
     W (E, ξ)                                                                      stu
                                                                                   wi
end-to-end distance                                                                res
                       FIG. 5 (color online).   (a) JDOS g…; E† of a chain with
Metropolis Method              Wand-Landau Method

         −E[x]/kB T                             −E/kB T
Z=       e            dx         Z=      W (E)e           dE

         Sample configuration space with probability
     −E[x]/kB T
     e                                  1/W (E[x])


Samples                                         Samples all
mainly regions                                  energies
around energy                                   equally
minima
The main features of our model for FePt
            Mryasov, et. al. Europhys. Lett. 86, 805 (2005)

    Standard Heisenberg form with uniaxial anisotropy
          E({mi }) = −           Jij,α mi,α mj,α −       Ki m2
                                                             i,z
                         i<j,α                       i

 Only Fe spins are free variables (induced moments on Pt)
       Truncate exchange interaction at the surface


                                                 FePt


     When all spins are aligned, energy has the standard
     form             E = KV sin Θ2
Test the model for bulk FePt (L=30) with standard
Metropolis Monte Carlo
                                                       Bulk FePt
                                     0.8

                                     0.7

            Magnetization (<M>/Ms)
                                     0.6

                                     0.5

                                     0.4

                                     0.3

                                     0.2

                                     0.1

                                      0
                                       200   300   400 500 600 700     800   900
                                                     Temperature (K)



    Curie temperature within about 10% of experiment
    Ising type phase transition
Temperature dependent magnetization for
nanoparticles
                                    R = 2.2 nm
                       2.2 nm
                       3.0 nm
                       30, bulk




  • Small effect on Curie temperature
  • Magnetization suppressed in surface region
    - Expected due to truncation of exchange interactions at surface
  • Surface region does have structure
    -   Due to shape of nanoparticle (truncated octahedron)
Joint density of states and free energy

                        Numerically integrate density of states


                 F (Mz , T ) = −kB T ln     W (Mz , E) exp(−E/kB T )dE
W




            E
Metropolis Method                Wand-Landau Method

         −E[x]/kB T                                  −E/kB T
Z=      e             dx           Z=       W (E)e             dE

        Sample configuration space with probability
     e−E[x]/kB T                            1/W (E[x])

Samples                                             Samples all
mainly regions                                      energies
around energy                                       equally -
minima

 Check validity of Wang-Landau method by estimating barrier
 hight from Metropolis MC and fitting to KV sin2 Θ
    Quantitative test for bulk FePt

                                         Exact value at T=0 (determined analytically) 3

                              15
                                                                       Extended WL algorithm
                                                                       Metropolis Monte Carlo
                                                                       Monte Carlo histogram
               " f(T)/N (K)




                              10




                               5




                               0
                                   0   100   200    300          400        500      600        700
                                                          T(K)


e with    FIG. 3: Free energy barrier of a bulk system with peri-
rier is   odic boundary conditions calculated by two different meth-
t have    ods. The data point at T = 0 is the energy barrier. Both
arrier.   data sets approach the T = 0 energy barrier at low temper-
Temperature and particle size dependent
magnetic free energy barrier
                                                                Free energy barrier
                                                 14
                                                                        R = 1.9 nm
                                                                        R = 1.5 nm
                                                 12                     R = 1.1 nm

              Free energy barrier per spin (K)
                                                                        R = 0.7 nm
                                                                        Bulk (L=30)
                                                 10

                                                  8

                                                  6

                                                  4

                                                  2

                                                  0
                                                      0   100   200 300 400 500       600   700
                                                                  Temperature (K)


  •   Quantitative reduction of barrier hight in nanoparticles.
  •   Linear temperature dependence of barrier in
      nanoparticles.
Deviation from idealized Stoner-Wolfarth
behavior.
        Plot the deviation of the barrier hight from the
                       Deviation from behavior KV sin
        idealized Stoner-Wolfarth Stoner-Wolfarth behavior Θ
                                                         2

                                                                           R = 1.9 nm
                                                       2                   R = 1.5 nm
             Deviation from ideal barrier hight (K)                        R = 1.1 nm
                                                                           R = 0.7 nm

                                                      1.5



                                                       1



                                                      0.5



                                                       0
                                                        100   200     300    400        500   600
                                                                    Temperature (K)

         Incoherent modes are important in smallest
         particles - large surface to volume ratio.
Summary and outlook

 •   Temperature dependent free energy barrier in nanomagnets:
     immediate relevance to data storage applications
 •   Direct computation of the joint density of states is possible
     for systems with ~103 microscopic degrees of freedom
 •   Reliable computation of temperature free energy barrier
 •   Preliminary model calculations show interesting nano-scale
     effect that need to be investigated with first principles
     methods (LDA/GGA)
 •   Outlook: combine Wang-Landau with ab initio DFT (LSMS)
     -   will scale perfectly to 106 cores!
     -   WL/LSMS code will run at >80% of peak floating point performance
   Wang-Landau method is NOT embarrassingly
   parallel

   Metropolis MC acceptance:                                  β(Ei −Ef )
                                      Ai→f = min{1, e                      }

   Generalized Wang-Landau acceptance:

                  Ai→f = min{1, W (Ei , Xi )/W (Ef , Xf )}

random walker 1

                     ... global update of joint DOS at every WL step
random walker 2

                                     limited by latency ~ microseconds
local calculation of energy and observable ~ millisecond to minutes
  Hybrid communication model

   Not embarrassingly parallel!
                                              Master node controlling gWL
                                              acceptance, DOS, and histograms.
Communication at every
Monte Carlo step: low
bandwidth required.                                               Application at
                                                                  the core of
 Potentially                                                      simulation:
 higher band-                                                     domain de-
 width                                                            composed;
 requirements                                                     turn into
 in core of                                                       function call;
 simulation.                                                      run in local
                                                                  communicator
     Very general as Wang-Landau method can be used in conjunction with
     any simulation method (DFT, MD/MM, spin models ...)
Strong scaling results for hybrid WL-LSMS code

                           128 Fe atoms and 800
                           Monte Carlo samples
                           running on Cray XT4
                           (Jaguar)
Efficiency of gWL/LSMS prototype code


                                        Efficiency (LIZ radius=12.5, lmax=3)
                               90
                                                             Short simulation: includes startup
                                                             Long simulation: subtract startup
                                                                                     estimated
                                                                                     estimated
     FLOPS (percent of peak)




                               85




                               80




                               75




                               70
                                    0   5000        10000          15000                  20000
                                                 Number of cores
 Extensions to the communication model


1. Take
advantage of                             2. Consider
multi-core                               and prepare
nodes                                    for fault
                                         tolerance.

                                         3. Parallelize
                                         master -
                                         asynchronous
                                         WL algorithm?