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

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：

2 22 2

22 2

0 0 0

ˆ2 2

1 1 1

4 8 8

i iR r

i ii e

i ji

ij i j i ji j i j i j

HM m

e Z Ze Z e

R r r r R R

Born-Oppenheimer Born-Oppenheimer approximationapproximation ：：

ˆ ˆ ˆextH T V V T

V

and depends on electronic configuration only is considered as external field

extV

Quantum Many-bodiesQuantum Many-bodies ？？

： Kinetic energy

： e—e interaction

： e — ion interaction

T V

extV

Hohenberg-KohnHohenberg-Kohn 原理原理：

Density functional theoryDensity functional theory

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 唯一地對應外勢 Vext ，而此系統 的任何觀察量 ，其基態的期望值僅是基態密度函數的唯一泛函：

)(r

O

H [ ] [ ]VextH E

ˆ ˆ ˆ[ ]

[ ] ( ) ( )

ext

HK

V ext

F

HK ext

E T V V

F r V r dr

ˆ [ ]O O

定理二：若 為 Hamiltonian ，則系統基態的總能泛函為 ，其形式如下：

O

VTEe

0 ( )HF H x

V

E T V V

0cV T T

0H HE T V

x HV V V

0 0

0 0

0

0

0

( )

( )

( )

c

x

xc

HK

V

c H H

H c H

V

H x c

V

F T V T T

T V T T

T V V V V

T V V V V

T V V V

推導推導 Hohenberg-Kohn Hohenberg-Kohn 能量泛函能量泛函：

系統的總能量泛函分別表示：精確的 (Exact) :

Hartree :

Hartree-Fock :

[ ]eE

[ ]HFE

[ ]HE

交換項 ：

關聯項：

Hohenberg-Kohn 能量泛函：

xcV 交換關聯能泛函 (exchange-correlation energy) ：

0[ ] [ ] [ ] [ ] [ ]extV H xc extE T V V V

0ˆ ˆ ˆ ˆ ˆKS H xc extH T V V V

2 22

0

( ) ˆ ˆ2 4i xc ext

e

e rdr V V

m r r

][ˆ xc

xc

VV

*

1

( ) ( ) ( )N

i ii

r r r

ˆKS i i iH

Kohn-Sham Kohn-Sham 方程式方程式：

系統的總能量泛函分別表示：

對應為 Kohn-Sham Hamiltonian ：

交換關聯勢：

Kohn-Sham 的理論表述：

有 N 個電子系統的精確基態密度：

( )i r 其單粒子波函數是 Kohn-Sham 方程式：

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交換關聯泛函之近似 ： ：

( ) ( )LDAxc xcE r r dr

( ) ( ), ( )GGAxc xcE r r r dr

解晶體電子本徵方程：解晶體電子本徵方程：

2 22

0

ˆ

( ) ˆ ˆ ( ) ( )2 4

sp

k ext k k ke

H

e rdr V V r E r

m r r

1( ) ( )

M kk m mmr C r

ESCHC

是均勻電子氣交換關聯能密度，是 的函數，而非泛函，是可嚴格求解的

( )xc r

H： Hamiltonian 矩陣

S：重疊矩陣

C：系數矩陣

V = 或 xV xcV

many electrons　 ＋many nuclei

non-uniform electron gas

one-electronproblem

Adiabaticapproximation

Density functionaltheory

First-principles molecular dynamics

applications

Because of much smaller electronic masscompared with nuclear mass, electrons canfollow the nuclear motion instantaneouslyto keep their electronic ground state.

Kohn-Sham equation

Nano-technology, reactions etc.

Local density appr. （ LDA ）Generalized gradient appr. （ GGA ）

Summary Dozens of methods have been developed to solve the resulting one-partic

le 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 augmente

d-plane-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 (m

any 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 experim

ents is at finite temperatures. We combine the ab initio calculations and classical Monte Ca

rlo 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.

Em

(mRy) J1

(mRy) J2

(mRy) J3

(mRy)bcc Fe 25.1 4.209 1.648 -1.057

.2

1

,)(

jijiijr

iMPMT JEEE σσ

The exchange parameters can be extractedFrom 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 MonteCarlo simulations to study the finite-temperature properties , i. e. phase transitions.

2

22

223

4

1TBk

EEC

M

M

LU

PRB 62 3354 2000

Classical Monte Carlo methodA Brief Introduction In statistical problems, all that we need is the canonical

ensemble .

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.

TBkxH

edxZ/)(

)(/)(1

)( xATBkxH

edxZT

xA

Tcexp (K) Tc

Stoner (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 (Thi

s 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 h

ave 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 sho

rt-range magnetic interactions. Most recently, Gambardella et al. succeeded in preparing a high density o

f 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 t

heCo 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 insid

eself-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

Consider a solid as a many-electron system in an external electric potential Vext(r) and an external magnetic field Bext(r). For simplicity, is assumed. The system Hamiltonian is

).()(

)()(||

ˆ

||1

21ˆ

21ˆ

)1.1(ˆˆˆˆ

1

1

1

2,

,

rr

rrRr

rr

BV

BVZ

H

H

H

HHHH

zext

zfield

N

i

M

iext

N

j jijiee

N

iiKe

exteeKe

electron kinetic energy

electron-electron Coulomb interaction

electron-nuclei Coulomb interaction + applied fields

[von Barth, Hedin, J. Phys. C 5 (1972) 1629; Rajagopal, Callaway, PRB 1 (1973) 1912]

zBextext ˆ)()( rrB

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.

.)(

])(),([,

|'|

)'(')(

,),()()()()( ,||

)2.1(or , )()()}(2

1{

)(,

,,2

)(,1

)(

,,,,2

r

rrr

rr

rrr

rrrrrrr

rrr

n

nnEV

ndV

NNNVVBVVn

V

xcxch

xchzexteffi

N

i

iiieff

(2) The GS n(r) and n(r) can be obtained by solving selfconsistently a set of spin-dependent Kohn-Sham equations

The number density is n(r) = (n(r) + n(r)) and spin density ism(r) = (n(r) n (r)).

)3.1()].()()()()(21 [

1, rrrrrrr

nVndVndE xcxch

N

ii E

The total energy is given by

Local spin-density approximationAssume

where the exchange-correlation (x-c) energy per electron xc(n(r), n(r)) is set to that of a spin-polarized homogeneous electron gas with the densities n(r) and n(r).

where the exchange energy density per electron is

ch(n(r), n(r)) could be calculated by perturbative many-body theories (e.g., RPA)

or by QMC.

)4.1())(),(()()](),([)()](),([ rrrrrrrrrr dnnndnnnnnE xcxcxc

)5.1()),(),(())(),(())(),(( rrrrrr nnnnnnhc

hx

hxc

)6.1()).()((1)4

3(3))(),((

3/43/43/1rrrr

nnn

nnhx

Spin-polarized GGA

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]

)7.1())(),(),(),(()](),([ rrrrrrr dnnnnnnE fxc

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 addi

ng 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 eV. The calculations were performed with three dimensional codes , and thus the system simulated was an infinite two-dimensional 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 axisof 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.

510

Interatomic Distance

0.22.2

This Other Exp. This Other Exp. This Other Exp.

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

Fe 2.252.281

2.252 Ru 2.182.241

2.213 Os 2.242.281

2.304

Co 2.15 2.181 Rh 2.252.271

2.253 Ir 2.402.301

2.344

Ni 2.18 2.181 Pd 2.432.441

2.373 Pt 2.382.401

2.484

Cu 2.29 2.272 Ag 2.66 2.571 Au 2.56 2.5721 PRB 69, 1934042 PRB 65, 2354053 PRB 68, 0354234 PRB 68, 144434

0.24.2

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

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).

]sin[sin 221

20 EEEE

Direction Ms ( ) Mo ( ) MAE（ meV）

V001 4.865 0.028

1.04100 4.862 0.173

Fe001 3.408 0.215

0.2100 3.406 0.127

Co001 2.209 0.234

1.0100 2.206 0.121

Ni001 1.167 0.473

9.8100 1.166 0.147

Zr001 1.011 0.064

7.7100 1.013 0.012

Ru001 0.940 0.055

10.6100 0.999 0.059

Rh001 0.327 0.423

6.3100 0.257 0.019

Hf001 0.251 0.300

1.2100 0.171 0.000

Ir001 0.421 0.373

11.2100 0.478 0.085

B B

Conclusion II My calculations are in good agreemen

t 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.