Electronic Materials Research Lab in Physics, Ch3. Crystal Bonding Prof. J. Joo Department of Physics, Korea University Solid State Physics Electronic Materials Research Lab in Physics,

3.1 Introduction  결정 격자를 유지시키는 힘 : 핵의 positive charge 와 전자의 negative charge 와 의 “ 잡아당기는 Coulomb force” Magnetic force → (very) weak Gravitational force → negligible  새로운 용어 : “Van der Waals force” “exchange energy” “covalent bonds”  “Cohesive energy” : that must be added to the crystal to separate its components into neutral free atoms at rest, at infinite separation  “Lattice energy” (for ionic crystals) : that must be added to the crystal to separate its component ions into free ions at rest, at infinite separation  Tables provide an important information e.g.) Table 1 : cohesive energy Table 2 : melting points Table 3 : bulk modulus ( 부피 탄성률 ) : [N/m 2 ] & compressibilities ( 압축률 ) : [m 2 /N] closely related to cohesive energy

Electronic Materials Research Lab in Physics, Reference.. (Fig.1) The Principle Types of Crystalline Binding

Electronic Materials Research Lab in Physics, Reference.. Holliday 일반물리학 (p762)

Electronic Materials Research Lab in Physics, Crystals of Inert Gases  He, Ne, Ar, Kr, Xe (Properties : Table 4)  Important properties 1.Weakly bound → low melting temp. (< 170K) 2.Very high ionization energy (≈10~25eV) ( ∵ outmost el. Shells are filled) 3. Fcc structure except He 3 and He 4  He, Ne 등 결정 고체를 이루기 위해서 주변 atom 의 el. 의 분포에 의한 미시적 영향력 Van der Waals-London Interaction or “ Fluctuating dipole forces”  Consider two identical inert gas atoms “Van der Waals force” e- r 1 p r 2 p R The atoms induce dipole moments each other → 결과적으로 두 개의 atom 사이에 attractive interaction 유발 → 뒤의 유도과정 … ( R > r )

Electronic Materials Research Lab in Physics, Van der Waals – London Interaction (1) atom 1 atom 2 R X1X1 (K 1 ) X2X2 (K 2 ) → 엄밀하게 양자역학적으로 에너지를 계 산하여야 하지만, 현상론적으로 (phenomenologically), 고려되는 계는 “coupled oscillation” → Due to Coulomb interaction

Electronic Materials Research Lab in Physics, Van der Waals – London Interaction (2)  “Coupled Oscillation” → due to Coulomb interaction 즉, 원래 진동수의 변화를 유발시킴 ( 가정 ) 두 개의 동일 atom → k 1 = k 2 = k, x 1 ≈ x 2 ≡ x ∴ The energy of the coupled system is ≡k' 새로운 진동수와 관련 : Taylor 전 개

Electronic Materials Research Lab in Physics, Van der Waals – London Interaction (3) 상수 : positive 중요한 의미 (due to induced dipole) 1.Quantum effect 2.Dipole coupling 에 의해서 계 (system) 의 energy 가 낮아짐 3. 두 개의 원자의 charge density 의 “overlap” 에 의해 이루어진 것이 아님

Electronic Materials Research Lab in Physics, Repulsive Interaction (1)  Consider 2 H-atoms and push together until the protons are almost in contact → similar to atomic He el. configuration a) Total spin 0 Total el. Energy eV → the electrons are ↑↓ (antiparallel) the Pauli principle has no effect b) Total spin 1 Total el. Energy eV Pauli exclusion principle Increases the repulsion : eV repulsive potential = : 실험적 사실 H H He 1S ↑ 1S ↑ 1S ↓ + → H H He 1S ↑ 1S ↑ 2S ↑ + → 1S 2S “Lennard–Jones Potential” for the crystal of inert gas

Electronic Materials Research Lab in Physics, Repulsive Interaction (2) repulsive potential van der Waals potential “ 일반적 표현 ” : “ 두 개의 atom 이 가까울 수록 repulsive potential 이 exponential 하게 증가 ”

Electronic Materials Research Lab in Physics, Equivalent Lattice Constants  If neglecting the kinetic energy, the cohesive energy of an inert gas is given by the sum of all pairs of atoms # of atomsdistance to nearest neighbor where : distance between reference atom i and any other atom j : depending on the structure ( 예 ) fcc structure  안정된 위치, 즉 atomic vibration 에 대해서 평행 상태의 위치 “R 0 /σ=1.09” → 고유한 값 (fcc inert gas) (ref.)  Cohesive energy :

Electronic Materials Research Lab in Physics, Ionic Crystals  Positive ion 과 negative ion 사이의 정전기적 상호작용 → “ ionic bonding”  ( 예 ) NaCl, Li F 1S 2 2S 2 2P 5 → F - : 1S 2 2S 2 2P 6 1S 2 2S 1 → Li + : 1S 2  Some distortion near the region of contact with neighboring atoms – Fig.7  Ionic bonding 의 세기 should know ① ionization energy ( 이온화 에너지, Table 5) ② el. affinity ( 전자친화력, Table 6)  ( 예 ) NaCl (quick estimation) the energy per molecule unit of crystal sodium chloride is = 6.4eV Lattice energy More exact calculation : next slide

Electronic Materials Research Lab in Physics, Electronic or Madelung Energy (1)  Van der Waals 상호작용 → small contribution (1-2% in cohesive energy)  Main contribution to the binding energy : electrostatic and called by Madelung energy  만약 U ij : 이온 i 와 이온 j 사이의 상호작용이라면 “central field repulsive interaction” “Coulomb potential” ① λ,ρ : 실험적 상수 (lattice constant, 압축률에 의해 결정 ) ② R -12 (inert gas) 대신 exponential 감소가 더욱 general representation

Electronic Materials Research Lab in Physics, Electronic or Madelung Energy (2)  Suppose N-molecules or 2N-ions U tot = N·U i : each pair of interactions  Introduce P ij such that r ij ≡ P ij R n – n separation 구조와 관련  가정 : repulsive 상호작용은 단지 n - n 만 있다면, # of n.n.

Electronic Materials Research Lab in Physics, Electronic or Madelung Energy (3)  At the equilibrium separation ; i.e., ∴ Total energy at R=R 0 where : Madelung energy (attractive energy & repulsive Coulomb inter. → ionic bonding 을 표현 ) from repulsive term

Electronic Materials Research Lab in Physics, Evaluation of the Madelung Constant The first calculation of the Coulomb energy constant ‘α’ was made by Madelung → Madelung constant (α) n.n. distance the distance of jth ion from the ref. i ion 3 Dimensional calculation : very difficult (hardly converge) (E.g.) 3D Ionic crystal (based upon until charge and referred to the n.n. distance) NaCl : CsCl : ZnS : !! Table.7 : properties of Alkali Halide crystal with the NaCl structure For KCl

Electronic Materials Research Lab in Physics, Covalent Crystals (1)  Covalent bond : one or more pairs of electrons are shared by the two atoms → strong bond such as in the ionic crystals → strong directional properties : 1.C, Si, Ge : directional properties such as tetrahedral angles 2.CH 4 (tetrahedral methane) → the electrons forming the bond tend to be partly localized in the region between two atoms joined by the bond → the spins of the two electrons are antiparallel H H H H C

Electronic Materials Research Lab in Physics, Covalent Crystals (2)  One of the simple e.g. for covalent bond : “H 2 molecule” → the strongest binding → two electrons are “antiparallel” → spin-dependent Coulomb energy “exchange interaction”  Pauli pr. ① strong repulsive interaction between atoms with filled shells ② if the shells are not filled, el. overlap can be accommodated and the bond will be shorter (e.g.) Cl 2 → ~2Å solid Ar → ~3.76Å  C, Si, Ge (hybridization) → for C : 1s 2 2s 2 2p 2 → to form “tetrahedral system” of covalent bonds → promote to the electronic configuration 1s 2 2s2p 3 (sp 3 hybridization) 전자친화도 : 진공 중에서 중성원자가 전자와 결합할 때 방출하는 에너지 = 음이온에 서 전자하나를 떼어내는데 필요한 에너지 “Negative energy corresponds to binding”

Electronic Materials Research Lab in Physics, Metals  Metals 1.high electrical conductivity (≥10 4 Ω -1 cm -1 ) 2.large number of el. (high electron density) → free to move 3.el. available to move → “conduction electron” “valence electrons” of the atom became the “conduction el.” of the metal  The attractive interaction between the electron gas and the positive ions → “metallic bonding” : ~1eV “not stronger than covalent bond, ionic bond”

Electronic Materials Research Lab in Physics, Hydrogen Bonds  Under certain condition, a hydrogen atom is attracted by strong force to two atoms → forming hydrogen bond : ~0.1eV → for electronegative atoms : F. O. N Note> 극단적인 case : bare proton ( H + ) forms the hydrogen bond → hydrogen connects two atoms ( HF 2 - )

Electronic Materials Research Lab in Physics, Reference..