Modern Research and Computational Science Modern Research and Computational Science 科學研究和計算科學科學研究和計算科學

Guang-Yu Guo (Guang-Yu Guo ( 郭光宇郭光宇 ))Graduate Institute of Applied PhysicsGraduate Institute of Applied Physics

National Chengchi University, Taipei, TaiwanNational Chengchi University, Taipei, Taiwan

(Science Talk <(Science Talk < 理院論壇理院論壇 >, College of Science, >, College of Science,

National Chengchi University, Nov. 2, 2009)National Chengchi University, Nov. 2, 2009)

((國立政治大學應用物理研究所國立政治大學應用物理研究所 ))

I. Roles of computational science in modern research

Plan of this Talk

II. Discovery of mechanism of Verwey transition in magnetite

1. Computer program discovers laws of physics2. Roles of computation in scientific research3. Ab initio calculations in materials research

1. The classic charge ordering problem2. Discovery of a t2g-orbital order3. Entropy change at Verwey transition4. Confirmation by resonant x-ray scattering experiments

I. Roles of computational science in modern research

1. Computer program discovers laws of physics

[Science 324 (April 3, 2003) 81]

科學研究的重要目的是尋找錯綜複雜的自然和社會現象背後的運動規律 .

2. Roles of computation in scientific research

計算科學在當代科學發展中的重要性計算科學在當代科學發展中的重要性

前高速電腦時代科研的兩大支柱前高速電腦時代科研的兩大支柱

問題 :(1)理論模型往往無解析解；(2)模型要大大簡化才得以解答；(3)簡化模型嚴重失真無法和實驗或事實作定量比較；

后高速電腦時代科研有三大支柱后高速電腦時代科研有三大支柱

優點 :(1)數值解答复雜真實理論模型；(2)理論得以和實驗做定量詳盡比較；(3)探討無法做到環境或無法到達地方發生的現象 , 如：高温高壓的地球內部或遙遠天體之表面 ; 921大地震或總統大選或 2008金融海嘯等。(4)預言新現象、開發新材料等

計算科學與尖端材料研發及設計計算科學與尖端材料研發及設計

由於高速電腦，先進數值算法和量子理論，材料科學己邁入一個全新的時代。第一原理量子力學計算可以補充，有時甚至取代，傳統的嘗試錯誤實驗法。利用第一原理計算，科學家可以引導尖端材料的研發，可以理解材料是怎樣形成的，外界條件會產生怎樣的影響以及可以如何改造材料使其具有更佳的性能。目前，量子理論計算已成為世界上科學家探討材料新穎性質及研發和設計新功能材料不可缺少的工具。

從遠古的銅器時代到今日的半導體資訊時代，人類文明一向是以新材料來命名並透過掌握和操控新材料而進步。材料的發展推動了經濟、社會和科學的進步，進而改變了人們日常生活的方式。

3. Ab initio calculations in materials research

第一原理理論計算發明密度泛函理論 , W. Kohn 獲 1998 年諾貝爾獎

[first principles (ab initio calculations]

[PRL 68 (1992) 1355]

Si (111) surface reconstruction

第一原理理論計算 确定矽 (111) 表面重構后原子結構

For example, the structure of AuFor example, the structure of Au2020-- determined determined

jointlyjointly by photoelectron spectroscopy and by photoelectron spectroscopy and ab ab initio calculationsinitio calculations

[Science 299, 864 (2003) 1355]

量子理論計算對研究奈米材料新奇特性尤其重要。例如，量子理論計算在確定奈米材料的原子結構上扮演不可或缺的角色。

[predicted and subsequently [predicted and subsequently synthesized by Berkeley groups synthesized by Berkeley groups in the mid-90’s]in the mid-90’s]

zigzag (10,0) zigzag (10,0) nanotubenanotube

chiral (7,3) nanotubechiral (7,3) nanotube

armchair (5,5) armchair (5,5) nanotubenanotube

(Iijima, 1991)(Iijima, 1991)

碳奈米管碳奈米管

Here is an example how ab initio calculations lead to a discovery that solves a long standing (60 yrs old) problem in solid state physics.

[Rubio et al., PRB 1994;[Rubio et al., PRB 1994; Chopra, et al., Science, 1994]Chopra, et al., Science, 1994]

硼化氮硼化氮 (BN)(BN) 奈米管奈米管

磁鐵的用途由其矯頑力决定 , 而矯頑力來自由相對論造成的磁晶異向能

Hard disk (~100GB)Flopy disk (~1.4MB)

Storage capacity vs time相對論量子力學計算能恊助尋找具高磁晶異向能材料 .

Large SOC effect magnetism in 4d and 5d TM atomic chains.

3d TM linear atomic chains

[Tung, Guo, PRB76(2007)094413][Tung, Guo, PRB (submitted 2009)]

過度金属奈米線之磁晶異向能

Magneto-crystalline anisotropy energy in bulk metals

[Guo, et al., Physica B 172 (1991) 61]

[Kittel]

L10 ordered Fe(Co)Pt alloys

001 111fcc N 2.7μeV/atomiE

001 111bcc Fe 1.4μeV/atomE

100 001hcpCo 65μeV/atomE

100 001FePt 1.2meV/f.u.E 100 001CoPt 0.8meV/f.u.E

FePt has the largest MAE among TM alloys and is the hottest alloy.

The limit based on FePt is ~1 Tbit/in2

with perpendicular recording .

5SmCo 2.0meV/CoE

4[1eV 1.16 10 K, 300K 25.9meV]

1. The classic charge ordering problemis that of magnetite (1941).

(1) What is magnetite?

Inverted spinel structure above TV

1/3: tetrahedral (A-site) ) Fe3+

2/3: octahedral (B-site) Fe3+, Fe2+

Ferrimagnetic A-site , B-site , TC ~ 860K

M-I (Verwey) transition, TV = 122 K

Fe(A)3+

Fe(B)3+ Fe(B)2+

O2-The oldest magnet known to mankind.

(1913)

Fe3O4

magic stones (2500 yrs)

II. Discovery of mechanism of Verwey transition in magneite

(2). Verwey charge ordering model for Verwey transition (1941):

A mixed-valence compound: FeA3+[Fe2+Fe3+]BO4

2-; The M-I transition is due to a charge ordering of the Fe2+ and Fe3+ states on the B sublattice, resulting in an orthorhombic structure.Therefore, this is the oldest and classic charge ordering problem.

[PRB 47 (1993) 5564]

However, simple Verwey model was later disproved [Walz, JPCM 2002] and nature and mechanism of Verwey transition remains unresolved despite intensive investigations over the past 60 years.

Each corner-sharing B Fe tetrahedron should contain two Fe2+ and two Fe3+ ions in order to minimize the electrostatic energy.

(3). Anderson criterion [PR102 (1956) 1008]

R.G. Melko et al. PRL 87, 067304

S. T. Bramwell et al. Science, 294, 1495

Ho2Ti2O7

Yet, another long review two years later in the same JPCM 16 (2004) R145 .

“In this review, we will refute all the arguments advanced by that author. “

Issues: the existence and origin of the charge ordering, [Siratori et al. 1998, Novak et al. 2000, Garcia et al 2000]

the low temperature structure [Wright et al. 2001], the nature of the insulating gap, etc.have remained unresolved over the past 60 years.

Many theoretical models have been proposed, including(1) purely electronic [Cullen-Callen], (2) Wigner glass-Wigner crystal

transition [Mott], (3) electron-phonon [Mott, Yamada]

and (4) bond dimerization [Seo].

In 1979, Sir N.V. Mott called for an international workshop completely dedicated to the Verwey transition [Philos. Mag. B 42 (1980)].

Recently, a long review paper appears in JPCM 14 (2002) R285 (Walz) which states that many experimental aspects are clear now but the fundamental issues remain.

(4). Other interesting properties

A half-metal for spintronics?

[Jeng, Guo, PRB 65 (2002) 94429]

2. Discovery of a t2g-orbital order

Nature of the electronic structurein low temperature phase from LDA+U calculations)

Low T monoclinic structure [Wright et al. 2001, 2002]

Space group P2/c, 56 atom/cell, a=5.9444 Å, b=5.925 Å, c =16.775Å, =90.2368º. fcc lattice constant a = 8.394 Å.

Fe(B1a,b)

Fe(B2a,b)

Fe(B3)

Fe(B4)

LDA+U band structure of (a) cubic and (b) monoclinic phases

[Jeng, Guo, Huang, PRL 93, 156403 (2004)]

Charge ordering

Expt.[1]charge

LDA+U

charge spin

3d 3d

Fe (B1)Fe (B2)Fe (B3)Fe (B4)

5.6 5.4 5.4 5.6

5.57 3.45 5.41 3.90 5.44 3.81 5.58 3.39

4.41 0.97 4.54 0.65 4.51 0.71 4.39 1.01

Table I: Valence charge (e), spin moment (B), and spin-decomposed 3d charge (e) of the B Fe in monoclinic Fe3O4.

Furthermore, the charge (spin) ordering can be considered as the superposition of [001]c and [00 ]c charge (spin) density wave modulations.

2

1

[1] Wright et al., 2002.

(b) (c) Calculated charge distribution indicating t2g-orbital ordering on B-site sublattice.

(a) Low T monoclinic structure [Wright et al. 2001, 2002]

(a)

(b)

(c)

Central finding (result)

Orbital ordering

(a) Density of states projected onto the B Fe d-orbitals. The Fermi level Ef is at the zero energy.

(b) Schematic energy level diagram for the spin-down B Fe d-orbitals.

3. Entropy change at the Verwey transitionis another long standing enigma.

[Shepherd et al. PRB 31 (85) 1107]

SV = Rln2 = 5.76 (J/mol-K) [R = 8.314 (J/mol-K)]

(Rln2 = 0.693R)

+ +

+

+

+

+

－

－－ －

－

－

(1) Completely disorder to fully order transition:

a) high-T disorder phase

N=2nNA B-sites (FeB3+ : ＋ , FeB

2+ : － ): W = 2N states;Thus, ShT = klnW = kln2N = R2ln2 per mol.

b) low-T long-range order phase

SlT = klnW = kln1= 0.Thus, SV = 2Rln2 = 11.52 (J/mol-K).

(2) Short-range order:[similar to the residual entropy water ice and spin ice]

Short-range 2(Fe3+)-2(Fe2+) order:N-sites, N/4 disconnected tetrahedra (2 sets)

W = (3/8)N/4 *6N/4 = (3/2)N/2

S = kln(3/2)N/2 = Rln(3/2) = 0.405R per mol.

(C42)N/4 = 6N/4 states

Possibility of correct states = (6*34)/(64) = 3/8

Anderson’s 2-2 charge ordering model

－

－

－

－+

+ +

+

++

(3) Short-range order model of new charge-orbital state :Short-range 3(Fe3+)-1(Fe2+) and 1(Fe3+)-3(Fe2+) order:N-sites, N/4 disconnected tetrahedra (2 sets)

W = (1/2)N/4 *8N/4 = (2)N/2

S = kln(2)N/2 = 0.693R per mol.

(2C41)N/4 = 8N/4 states

Possibility of correct states = (2*4*(3+1)4)/(24*44) = 1/2

(4) Summary :

a) Disorder-order transition SV = 1.386R per mol.b) SRO-order (Anderson) SV = 0.405R per mol.c) SRO-order (new) SV = 0.693R per mol.d) Experimental result SV = ~0.693R per mol.

Present 3-1:1-3 charge-orbital ordering model[Jeng, Guo, Huang, PRB 74, 195115 (2006)]

Our opinion: If you don’t see it, doesn’t mean it is not there!

Using Fe K-edge absorption

4. Confirmation by resonant x-ray scattering experiments

[D.J. Huang, et al., Phys. Rev. Lett. 96, 096401 (2006)]

Unoccupied O 2p orbital ordering pattern

O K-edge X-ray absorption and resonant scattering spectra from Fe3O4.

[D.J. Huang, et al., Phys. Rev. Lett. 96, 096401 (2006)]

[Cond-mat/0605096 (2006)]

Fe K-edge RXRD at 50 K

Fe K-edge RXRD at 90 K.

Thank you for your attention