Transcript Document 7275250

Structural, Electronic and Magnetic
Properties of Transition Metal Atomic
Chains from First-principles
Calculations
Advisor: G. Y. Guo (郭光宇)
J. C. Tung
Department of Physics, National Taiwan University
Outline



Introduction to DFT theory
Summary
Application I




Introduction
Classical Monte Carlo method – – A Brief Introduction
Conclusion I
Application II




Introduction
Spin DFT theory
Structure and Computational Method
Results and Discussions




Interatomic Distance
Spin and Orbital moments
Magnetic Anisotropy Energy
Conclusion II
Quantum Many-bodies？
一塊固體假如有N個原子核，則我們必須處理N+ZN個電磁交互作用
粒子的量子多體問題，此系統的Hamiltonian可精確寫成：
Hˆ  

2
2
2Ri
M
i
1
4 0

ij
i

2
2
2ri
m
i
e
e2 Zi Z j
e2 Zi
1
e2
1




Ri  rj 8 0 i  j ri  rj 8 0 i  j Ri  R j
Born-Oppenheimer approximation：
Hˆ  Tˆ  Vˆ  Vˆext
Tˆ ：Kinetic energy
Vˆ ：e—e interaction
：e — ion interaction
V̂ext
Tˆ and Vˆ depends on electronic configuration only
V̂ext is considered as external field
Density functional theory
Hohenberg-Kohn 原理：
密度泛函理論的基本想法是原子、分子和固體的基態物理性質可以用粒子
密度函數來描述，淵源於H. Thomas和E. Fermi 1927年的工作，其理論基
礎是建立在1964年P. Hohenberg和W. Kohn及Sham的關於非均勻電子氣理
論基礎上的工作 它可歸結於兩個基本定理：
定理一：一個多電子系統的基態電子密度  (r ) 唯一地對應外勢Vext ，而此系統
的任何觀察量 Ô ，其基態的期望值僅是基態密度函數的唯一泛函：
 Oˆ   O[  ]
定理二：若 Ô 為Hamiltonian Ĥ ，則系統基態的總能泛函為 H [  ]  EVext [  ]
，其形式如下：
EVext [  ]   Tˆ  Vˆ    Vˆext 
FHK   
 FHK [  ]    ( r )Vext ( r )dr
推導Hohenberg-Kohn 能量泛函：
系統的總能量泛函分別表示：
精確的(Exact) : Ee [  ]
Hartree
: EH [  ]
Hartree-Fock :
EHF [  ]
Ee  T  V
Hohenberg-Kohn 能量泛函：
FHK  T  V  T0  T0
 T0  V  (T  T0 )
Vc
 T0  V  Vc  VH  VH
EH  T0  VH
 T0  VH  Vc  (V  VH )
EHF  T0  (VH  Vx )
V
Vx
 T0  VH  (Vx  Vc )
Vxc
關聯項： Vc  T  T0
交換項 ： Vx  V  VH
Vxc：交換關聯能泛函
(exchange-correlation energy)
Kohn-Sham 方程式：
Kohn-Sham 的理論表述：
系統的總能量泛函分別表示：
有N個電子系統的精確基態密度：
EVext [  ]  T0 [  ]  VH [  ]  Vxc [  ]  Vext [  ]
N
 ( r )   i ( r )* i ( r )
i 1
對應為 Kohn-Sham Hamiltonian：
其單粒子波函數  i ( r )
Hˆ KS  Tˆ0  VˆH  Vˆxc  Vˆext

2
2me
 
2
i
e2
4 0
是Kohn-Sham方程式：
 ( r)
 r  r dr  Vˆ
xc
V [  ]
交換關聯勢： Vˆxc  xc
 Vˆext
Hˆ KS  i   i  i
的N個最低能量態的解

準粒子所構成的密度等於真實電子的密度
交換關聯泛函之近似 ：
E xcLDA    ( r )  xc   ( r )  dr
ExcGGA    ( r )  xc   ( r ),  ( r )  dr
是均勻電子氣交換關聯
 xc   (r )  能密度，是  的函數，
而非泛函，是可嚴格求
解的
解晶體電子本徵方程：
2

e2
2
k 

4 0
 2me


ˆ
ˆ
dr  V  Vext  k ( r )  Ek  k ( r )
r  r

 ( r)
Hˆ sp
 k (r )  m1 Cmk m (r )
M
HC  ESC
Vˆ
=
Vˆx 或 Vˆxc
H：Hamiltonian矩陣
S：重疊矩陣
C：系數矩陣
many electrons
＋
many nuclei
Adiabatic
approximation
non-uniform
electron gas
Because of much smaller electronic mass
compared with nuclear mass, electrons can
follow the nuclear motion instantaneously
to keep their electronic ground state.
Density functional
Kohn-Sham equation
theory
one-electron
problem
Local density appr. （LDA）
Generalized gradient appr. （GGA）
First-principles molecular dynamics
applications
Nano-technology, reactions etc.
Summary
Dozens of methods have been developed to solve the resulting one-particle
Schrodinger equation of the local density approximation (LDA). The most
widely used electronic structure methods can be divided into many classes.
1.
The linear methods[1] developed by Andersen[2] from the augmentedplane-wave (APW) method, and the Korringa-Kohn-Rostocker method.
2.
The pseudopotential method based on norm-conserving ab initio
pseudopotentials invented by Hamann, Schluter, and Chiang.[3]
3.
The combination of above. (PAW)
4.
Atomic orbital expansion for saving computing time and for a huge (many
ions ) system. (Siesta)
5.
Using Gaussian basis sets to expand the full wave functions.
1. Phys. Rev. 140 B1133 (1965)
2. PRB 12 , 3060 (1975)
3. PRL 43, 1494 (1979)
Application I


DFT is restricted in Zero temperature, but any of our
experiments is at finite temperatures.
We combine the ab initio calculations and classical Monte Carlo
simulations to study the phase transition at finite temperatures.
Introduction



The simplest Ising model (1D) was solved by Ising himself and there is
no magnetic phase transition. Many years later , 2D Ising model and
1D Heisenberg model were solved analytically . Exact solution of 2D
Heisenberg , 3D Ising ,and 3D Heisenberg model was still a challenge
today.
The magnetic of materials will decrease when we raise the temperature.
So there is magnetic – nonmagnetic phase transition at a specify
temperature (Curie temperature).
In principle, to calculate the thermodynamical properties and phase
transition temperature, one should start with T  0 spin-density
functional theory of Mermin. However, so far such calculations have not
been possible. Nonetheless, it has been possible to accurately calculate
these properties from first-principles indirectly.
ET  E PM   E M 
i
1
 J r ( ij )σ i  σ j .
2 i, j
The exchange parameters can be extracted
From the ab initio calculations by mapping
the calculated total energies for various
magnetic configurations (structures) onto the
Heisenberg-like Hamiltonian. With that
parameters, we can perform a classic Monte
Carlo simulations to study the finitetemperature properties , i. e.
phase transitions.
Em
bcc Fe
J1
L
J3
(mRy)
(mRy)
(mRy)
(mRy)
25.1
4.209
1.648
-1.057
M4
U
J2
 1
3 M2
E2  E
2
C
2
k T2
B
PRB 62 3354 2000
Classical Monte Carlo method
A Brief Introduction

In statistical problems, all that we need is the canonical ensemble .
A( x)
T

Z   dx e


 H ( x) / k T
1
B A( x)
dx
e
Z
 H ( x) / k T
B
Specify the type and size of the lattice and the boundary conditions
which have to be used.
Calculating the total energy in “many” configurations. Then we have
the canonical ensemble Z to compute the desire averages.
Tcexp (K)
TcStoner (K)
TcRPA (K)
bcc Fe
1039
5300
1316
fcc Co
1390
4000
1558
fcc Ni
630
2900
642
Conclusion I


We can make classical Monte Carlo simulations more reliable.
We can use these technique in many other systems.
Application II



We perform a systemically study of transition metal atomic chain (This
report ) for a testing of PAW method and FLAPW method.
We also consider other structures of transition metal atomic chain or
nanowire and put them onto different subtracts (Later, not this time).
For possible future development of ultrahigh density hard disks, we have
also calculated the magnetic anisotropy energy of all considered cases.
Introduction
Magnetism at the nanoscale is an exciting emerging research field,
of both basic and applied relevance. The study of magnetism at the
nanometer scale has been an exceptionally active research area over
the past years. The modern methods to prepare nanostructured
systems made it possible to investigate the influence of dimensionality
on the magnetic properties. The fundamental idea is to exploit the
geometrical restriction imposed by an array of parallel steps on a vicinal
surface along which the deposited material can nucleate, a process
called the step decoration. The early experimental measurements
reported a bond length for the Au monatomic chain of 4 Å. However,
recent experimental results claim that the bond length should be 2.5 Å.
The later value is in much better agreement with theoretical calculations.
Theoretically, a great deal of research has been done both on finite and
infinite chains of atoms. Isotropic Heisenberg model calculations with
finite-range exchange interactions show that a one-dimensional (1D)
chain can not maintain ferromagnetism at finite temperature.
Nonetheless, the presence of a strong magnetic anisotropy should
substantially alter this conclusion. Monostrand nanowires of Pd ,i.e.,
nanowires consisting of a single straight line of atoms, have recently been
observed by Rodrigues et al. The monoatomic chain , being an ultimate 1D
structure, has been a testing ground for the theories and concepts
developed earlier for three-dimensional (3D) systems. The 1D characters
of nanowires cause several new physical phenomena to appear. It is of
fundamental importance to know the atomic structure in a truly 1D
nanowire and how the mechanical and electronic properties change in the
lower dimensionality. Calculations of finite chains have been performed for
chains of Ni, Pd, Pt, Cu, Ag, Au, and Na atoms. Early studies of infinite
chains of Au , Al, Cu, Ca and K have shown a wide variety of stable and
unstable configurations. Recently, the magnetic properties of transition
metal (TM) infinite chains of Fe, Co, Ni, Ru, Rh, Pd, Os, Ir, Pt have
calculated. These calculations show that the metallic and magnetic
nanowires may become important for electronic/optoelectronic devices,
quantum devices , magnetic storage, nanoprobes and spintronics. Infinite
linear atomic chains are the simplest 1D material.
Nature 395 , 780 (1998)
The existence of such linear chains, though mostly transient in nature,
has been demonstrated experimentally. A central question thereby is
how the qualities behavior will change when going from 3D to 1D systems
because it has been predicted that there is no long-range magnetic order at
finite temperature in infinitely extended one-dimensional systems with shortrange magnetic interactions.
Most recently, Gambardella et al. succeeded in preparing a high density of
parallel atomic chains along steps by growing Co on a high-purity Pt(997)
vicinal surface in a narrow temperature range(10~20K). The magnetism of the
Co wires was investigated by the x-ray magnetic circular dichroism (XMCD).
Structurally stable nanowires can be grown on stepped surfaces or inside
tubular structures, like the Ag nanowires of micrometer lengths grown inside
self-assembled organic calix[4]hydroquinone nanotubes. Short suspended
nanowires have been produced by driving the tip of scanning tunneling
microscope into contact with a metallic surface and subsequent retraction,
leading to the extrusion of a limited number of atoms from either tip or
substrate.
Nature 416,301 (2002)
Science 449,93 (2000)
For a future spin-based technology, an understanding of
nanomagnetic phenomena will be very important. But little is currently
understood about how magnetism arises and how it affects the properties
of metals at the nanoscale. The relativistic effect due to spin-orbit (SO)
interaction is important for 5-d TM’s , and also, to a lesser extend though,
for 4-d and 3-d TM’s. Here we also performed calculations including SO
interactions for all systems of 3-d , 4-d and 5-d elements under study.
Transition metals, because of their partly filled d orbitals, have a strong
tendency to magnetize. Bulk Fe, Co, and Ni are well known for their
ferromagnetic ordering. An experiment found that small Rh clusters may
possess a permanent magnetic moment ,though bulk Rh is nonmagnetic.
More recently, experiments have shown that magnetic nature of atomic
chains of transition metals such as Co, Pd, and Pt. Thus, the transition
metals atomic chains are an interesting subjects to study their magnetic
properties.
The magnetic anisotropy energy (MAE) of 1D transition metal
nanostructures has also been calculated in terms of tight-binding
techniques. We perform a first-principles calculation in all considered
cases.
Spin-Density Functional Theory
[von Barth, Hedin, J. Phys. C 5 (1972) 1629; Rajagopal, Callaway, PRB 1 (1973) 1912]
Consider a solid as a many-electron system in an external electric potential Vext(r) and an
external magnetic field Bext(r). For simplicity, B ext (r )  Bext (r ) zˆ is assumed. The system
Hamiltonian is
Hˆ  Hˆ e , K  Hˆ e  e  Hˆ ext
Hˆ e , K 
Hˆ e e
Hˆ ext
(1.1)
N
  1 i2
i 1
 1 
2 i j
2
electron kinetic energy
N

1
electron-electron Coulomb interaction
|
r

r
|
j 1 i
j
N M
Z
  
 V field (r )   z B (r )
electron-nuclei Coulomb
i  1 | ri  R |
interaction + applied fields

Vext (r )
  z B (r ).
Electrons are fermions with ½-spin, and thus their spins are either up ()
(along z-axis) or down () (against z-axis).
[In this way, we have ignored relativistic effects and diamagnetic effects]
Density functional (or Hohenberg-Kohn-Sham) theorems now read:
(1) The GS properties are a unique functional of both spin-up density n(r) and spin-down
density n(r) for given Vext(r) and Bext(r); the correct GS n0(r) and n0(r) minimizes the
energy functional E[n(r), n(r)] and this minimum is the GS energy E0.
(2) The GS n(r) and n(r) can be obtained by solving selfconsistently a set of spin-dependent
Kohn-Sham equations
{
1
2
2
  Veff , (r )} i , (r )   i ,  i , (r ) ,    or 
N
(1.2)
2
n (r )   
| i , (r ) | , Veff , (r )  Vext (r )   z B (r )  Vh (r )  Vxc , (r ),
i 1
Vh (r )   dr '
n (r ' )
| r  r' |
,
Vxc , (r ) 
E xc [ n (r ), n (r ) ]
n (r )
.
The number density is n(r) = (n(r) + n(r)) and spin density is
m(r) = (n(r)  n (r)).
N  N  N ,
The total energy is given by
E
N
  i ,
 i 1
 1  drn(r )Vh (r )  
2

 drn (r )Vxc (r )  Exc [n (r )].
(1.3)
Local spin-density approximation
Assume
E xc [ n (r ), n (r )]   n(r ) xc [ n (r ), n (r )]dr   n(r ) xc ( n (r ), n (r )) dr
(1.4)
where the exchange-correlation (x-c) energy per electron xc(n(r), n(r)) is set to that of a spinpolarized homogeneous electron gas with the densities n(r) and n(r).
h
h
h
 xc ( n (r ), n (r ))   x ( n (r ), n (r ))   c ( n (r ), n (r )),
(1.5)
where the exchange energy density per electron is
4/3
4/3
1
 3( )
(n
(r )  n
(r )).
(1.6)

4
n 
ch(n(r), n(r)) could be calculated by perturbative many-body theories (e.g., RPA)
or by QMC.
h
 x ( n (r ), n (r ))
3
1/ 3
Spin-polarized GGA
E xc [ n (r ), n (r )] 
 f ( n (r ), n (r ), n (r ), n (r ))dr
(1.7)
GGA functionals f(n(r), n(r), n(r), n(r)) were constructed under guidances of
wave vector analysis of x-c energy functional and were forced to have the same
physical asymptotic behaviors such as x-c sum rule.
[Perdew, et al., PRL 77 (1996) 3865]
Structure and Computational
method
We use the full-potential projector augmented-wave
(PAW) method as implemented in the Vienna ab initio
simulation package (VASP) which is as accurate as the
frozen-core all-electron methods. Exchange and
correlation effects were described by the local
functional due to Perdew and Zunger , employing the
spin-interpolation proposed by Vosko et. al. and adding
generalized gradient approximation (GGA). The cutoff
energy of the plane wave basis set varied depending
on the chemical elements ,we applied the default
values tabulated for the PAW potentials .
To calculate the band structure, the Gamma-centered and standard
Monkhorst-pack k points generation schemes are used with a grid of
1x1x100 points in the full Brillouin zone (BZ). The convergence with
respect to the energy cutoff and number of k points were tested. Ionic
potentials are represented by ultrasoft Vanderbit-type pseudopotentials.
The densities of states (DOSs) were calculated by Fermi Dirac method.
For the calculation of the total energy, a Fermi-Dirac-smearing
approach with sigma equals to 0.01 eV was used, and the convergence
criteria for energy is 10 5 eV. The calculations were performed with three
dimensional codes , and thus the system simulated was an infinite twodimensional array of infinite long, straight wires. A one-dimensional BZ
was used, i.e., the k points form a single line, stretching along the z axis
of the wire. The wire-wire vacuum distance was set to 8 Å, more than
three bond lengths in all studies cases, which should be wide enough
to decouple neighboring wires. To calculate the magnetic anisotropy
energy (MAE), we compare the energy differences between the z axis,
along the chain direction, and the x axis, perpendicular to the chain
direction.
Interatomic Distance
This
Other
Exp.
This
Other
Exp.
This
Ti
2.12
Zr
2.32
Hf
2.58
V
2.60
Nb
2.33
Ta
2.40
Cr
2.78
Mo
2.09
W
2.24
Mn
2.60
Tc
2.19
Re
2.22
2.25
2.281
2.252
Co
2.15
2.181
Ni
2.18
Cu
2.29
Fe
2.18
2.241
2.213
Rh
2.25
2.271
2.253
2.181
Pd
2.43
2.272
Ag
2.66
Ru
Other
Exp.
2.24
2.281
2.304
Ir
2.40
2.301
2.344
2.441
2.373
Pt
2.38
2.401 2.4  2.0
2.484
2.571
Au
2.56
2.572
Os
2.2  2.0
PRB 69, 193404
PRB 65, 235405
3 PRB 68, 035423
4 PRB 68, 144434
1
2
Spin and Orbital moments
Magnetic Anisotropy Energy
It is easy to show that the lowest non-vanishing terms of energy for a wire
can be expressed in the form
E  E0  sin 2  [ E1  E2 sin 2  ]
where  is the polar angle of magnetization away from the chain, while 
is the azimuthal angle in the plane perpendicular to the wire, measured
from the x axis. For free standing cases is zero. With this convention, total
energy differences between x and z axis are calculated. That is , where
and are the total energies with the magnetization in the [100] and [001]
directions, i.e., perpendicular or parallel to the z axis, respectively. The
convergence of the MAE is tested (not shown).
V
Fe
Co
Ni
Zr
Ru
Rh
Hf
Ir
Direction
Ms (  B )
Mo (  B )
001
4.865
0.028
100
4.862
0.173
001
3.408
0.215
100
3.406
0.127
001
2.209
0.234
100
2.206
0.121
001
1.167
0.473
100
1.166
0.147
001
1.011
0.064
100
1.013
0.012
001
0.940
0.055
100
0.999
0.059
001
0.327
0.423
100
0.257
0.019
001
0.251
0.300
100
0.171
0.000
001
0.421
0.373
100
0.478
0.085
MAE（meV）
1.04
0.2
1.0
9.8
7.7
10.6
6.3
1.2
11.2
Conclusion II



My calculations are in good agreement
with existing experiments and other
theories calculations.
My calculations show some transition
metals atomic chains exhibit magnetism
while they are nonmagnetic metals.
The MAE is small in most cases except
Ru, Ir.
Conclusions II


In DOSs curves of Mn, Fe, Co, and Ni only the
majority states are completely filled and it is
only the minority carriers that are available
for conduction at the Fermi level.
The free standing atomic chain is the simplest
case. Some results are not in good
agreement with experiment. I should consider
the effect of substrate.