What is solid state physics?

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Clyde Jennings
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Wht is solid stte physics? Explins the properties of solid mterils. Explins the properties of collection of tomic nuclei nd electrons intercting with electrosttic forces.

More progress hs been mde in understnding the behvior of crystlline solids thn tht of noncrystlline mterils since the clcultion re esier in crystlline mterils.

1 Wht is solid stte physics? Explins the properties of solid mterils. Explins the properties of collection of tomic nuclei nd electrons intercting with electrosttic forces. Formultes fundmentl lws tht govern the behviour of solids. Crystlline olids Crystlline mterils re solids with n tomic structure bsed on regulr repeted pttern. The mjority of ll solids re crystlline. More progress hs been mde in understnding the behvior of crystlline solids thn tht of noncrystlline mterils since the clcultion re esier in crystlline mterils. Understnding the electricl properties of solids is right t the hert of modern society nd technology. Electricl resistivity of three solid Crbon sttes How cn this be? After ll, they ech contin system of toms nd especilly electrons of similr density. And the plot thickens: grphite is metl, dimond is n insultor nd buckminster-fullerene is superconductor. They re ll just crbon! 4

CLAIFICATION OF OLID OLID MATERIAL CRYTAL TRUCTURE CRYTALLINE ingle Crystl

mterils (crystlline, polycrystlline, morphous) Crystllogrphy Crystl Lttice

Brvis Lttices nd the even Crystl ystem) Crystl tructure 5 INGLE CRYTAL ingle

At long rnge length scles, ech tom is relted to every other equivlent tom in

POLYCRYTALLINE OLID Polycrystlline mterils re mde up of n ggregte of mny

Polycrystlline mterils hve high degree of order over mny tomic or moleculr

The tomic order cn vry from one domin to the next.

2 CLAIFICATION OF OLID OLID MATERIAL CRYTAL TRUCTURE CRYTALLINE ingle Crystl POLYCRYTAL LINE AMORPHOU (Noncrystlline) Elementry Crystllogrphy olid mterils (crystlline, polycrystlline, morphous) Crystllogrphy Crystl Lttice Crystl tructure Types of Lttices Unit Cell Typicl Crystl tructures (3D 14 Brvis Lttices nd the even Crystl ystem) Crystl tructure 5 INGLE CRYTAL ingle crystls hve periodic tomic structure cross its whole volume. At long rnge length scles, ech tom is relted to every other equivlent tom in the structure by trnsltionl or rottionl symmetry ingle Pyrite Crystl POLYCRYTALLINE OLID Polycrystlline mterils re mde up of n ggregte of mny smll single crystls (lso clled crystllites or grins). Polycrystlline mterils hve high degree of order over mny tomic or moleculr dimensions. Grins (domins) re seprted by grin boundries. The tomic order cn vry from one domin to the next. The grins re usully 100 nm microns in dimeter. Polycrystls with grins less thn 10 nm in dimeter re nnocrystlline ingle Crystls Amorphous olid Polycrystlline Pyrite form (Grin) 7 8

AMORPHOU OLID Amorphous (Non-crystlline) olids re mde up of rndomly

Amorphous mterils do not hve ny long-rnge order, but they hve vrying

Exmples to morphous mterils include morphous silicon, plstics, nd glsses.

CRYTALLOGRAPHY Crystllogrphy is brnch of science tht dels with the

It s importnt the symmetry of crystl becuse it hs profound influence on

tructures should be clssified into different types ccording to the

Energy bnds cn be clculted when the structure hs been determined.

In crystllogrphy, only the geometricl properties of the crystl re of

3 AMORPHOU OLID Amorphous (Non-crystlline) olids re mde up of rndomly orientted toms, ions, or molecules tht do not form defined ptterns or lttice structures. Amorphous mterils hve order only within few tomic or moleculr dimensions. Amorphous mterils do not hve ny long-rnge order, but they hve vrying degrees of short-rnge order. Exmples to morphous mterils include morphous silicon, plstics, nd glsses. Amorphous silicon cn be used in solr cells nd thin film trnsistors. CRYTALLOGRAPHY Crystllogrphy is brnch of science tht dels with the geometric description of crystls nd their internl tomic rrngement. It s importnt the symmetry of crystl becuse it hs profound influence on its properties. tructures should be clssified into different types ccording to the symmetries they possess. Energy bnds cn be clculted when the structure hs been determined CRYTAL LATTICE Crystl Lttice Wht is crystl lttice? In crystllogrphy, only the geometricl properties of the crystl re of interest, therefore one replces ech tom by geometricl point locted t the equilibrium position of tht tom. An infinite rry of points in spce, Ech point hs identicl surroundings to ll others. y B b O C α A D x E Arrys re rrnged in periodic mnner. Pltinum Pltinum surfce Crystl lttice nd (scnning tunneling microscope) structure of Pltinum 11 1

Crystl tructure A two-dimensionl Brvis lttice with

obtined by ttching toms, groups of toms or molecules which

Lttices in D Unit Cell in D The smllest component of the

4 Crystl tructure A two-dimensionl Brvis lttice with different choices for the bsis Crystl structures cn be obtined by ttching toms, groups of toms or molecules which re clled bsis (motif) to the lttice sides of the lttice point. Crystl tructure = Crystl Lttice + Bsis 13 Five Brvis Lttices in D Unit Cell in D The smllest component of the crystl (group of toms, ions or molecules), which when stcked together with pure trnsltionl repetition reproduces the whole crystl. b 15 16

Three common Unit Cells in 3D Unit Cell The unit cell nd, consequently, the

nd γ. Only 1/8 of ech lttice point in unit cell cn ctully be ssigned to tht

17 18 TYPICAL CRYTAL TRUCTURE 3D 14 BRAVAI LATTICE AND EVEN CRYTAL TYPE Cubic

Monoclinic Crystl ystem (, Bse-C) Orthorhombic Crystl ystem (, Bse-C, BC, FC)

5 Three common Unit Cells in 3D Unit Cell The unit cell nd, consequently, the entire lttice, is uniquely determined by the six lttice constnts:, b, c, α, β nd γ. Only 1/8 of ech lttice point in unit cell cn ctully be ssigned to tht cell. Ech unit cell in the figure cn be ssocited with 8x1/8=1lttice point TYPICAL CRYTAL TRUCTURE 3D 14 BRAVAI LATTICE AND EVEN CRYTAL TYPE Cubic Crystl ystem (C, BCC,FCC) Hexgonl Crystl ystem () Triclinic Crystl ystem () Monoclinic Crystl ystem (, Bse-C) Orthorhombic Crystl ystem (, Bse-C, BC, FC) Tetrgonl Crystl ystem (, BC) Trigonl (Rhombohedrl) Crystl ystem () 19 Crystl tructure 0

odium Chloride tructure odium chloride lso

numbers of sodium nd chlorine ions plced t

Ech ion hs six of the other kind of ions s its

1 Atomic Pcking Fctor (APF) Number VOLUME,

metl = v = mss/unit cell volume/unit cell Plnr

6 odium Chloride tructure odium chloride lso crystllizes in cubic lttice, but with different unit cell. odium chloride structure consists of equl numbers of sodium nd chlorine ions plced t lternte points of simple cubic lttice. Ech ion hs six of the other kind of ions s its nerest neighbours. 1 Atomic Pcking Fctor (APF) Number VOLUME, PLANAR, AND LINEAR DENITY Volume density of metl = v = mss/unit cell volume/unit cell Plnr tomic density = p = # tom centers intersected selected re of plne Liner tomic density = = l # tom dimeters intersected selected length of line

Atomic Pcking Fctor: BCC R 3 Close-pcked directions: length = 4R = 3 Fce Centered Cubic tructure (FCC) Atoms touch ech other long fce digonls.

toms 4 volume unit cell ( 3/4 )3 3 tom APF = 3 volume unit cell APF for body-centered cubic structure = 0.68 (Courtesy P.M. Anderson) Adpted from Fig. 3.1, Cllister 7e.

74 toms unit cell APF = The mximum chievble APF!

7 Atomic Pcking Fctor: BCC R 3 Close-pcked directions: length = 4R = 3 Fce Centered Cubic tructure (FCC) Atoms touch ech other long fce digonls. --Note: All toms re identicl; the fce-centered toms re shded differently only for ese of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag Coordintion # = 1 Adpted from Fig. 3.(), Cllister 7e. toms 4 volume unit cell ( 3/4 )3 3 tom APF = 3 volume unit cell APF for body-centered cubic structure = 0.68 (Courtesy P.M. Anderson) Adpted from Fig. 3.1, Cllister 7e. 4 toms/unit cell: (6 fce x ½) + (8 corners x 1/8) Atomic Pcking Fctor: FCC Adpted from Fig. 3.1(), Cllister 7e. APF for fce-centered cubic structure = 0.74 toms unit cell APF = The mximum chievble APF! Close-pcked directions: length = 4R = Unit cell contins: 6 x 1/ + 8 x 1/8 = 4 toms/unit cell 4 4 ( /4 )3 3 3 ( = *R) volume tom volume unit cell Hexgonl Close-Pcked tructure (HCP) ex: Cd, Mg, Ti, Zn ABAB... tcking equence 3D Projection c Coordintion # = 1 A sites B sites A sites Adpted from Fig. 3.3(), Cllister 7e. c/ = (idel) D Projection 6 toms/unit cell APF = 0.74 Top lyer Middle lyer Bottom lyer

8 We find tht both FCC & HCP re highest density pcking schemes (APF =.74) this illustrtion shows their differences s the closest pcked plnes re builtup Theoreticl Density, Mss of Atoms in Unit Cell Density = = Totl Volume of Unit Cell = n A V C N A where n = number of toms/unit cell A = tomic weight V C = Volume of unit cell = 3 for cubic N A = Avogdro s number = 6.03 x 10 3 toms/mol Theoreticl Density, MILLER INDICE toms unit cell = volume unit cell R x 10 3 Ex: Cr (BCC) A = 5.00 g/mol R = 0.15 nm n = = 4R/ 3 = nm g mol theoreticl ctul = 7.18 g/cm 3 = 7.19 g/cm 3 toms mol DIRECTION PLANE

MILLER INDICE FOR DIRECTION Vector r pssing from the

5 + 3b (0,0) Miller indices [53] b (c) 003 Brooks/Cole

9 MILLER INDICE FOR DIRECTION Vector r pssing from the origin to lttice point: r = r 1 + r b + r 3 c, b, c fundmentl trnsltion vectors Miller indices of directions, b, nd c b (0,0) Miller indices [53] b (c) 003 Brooks/Cole Publishing / Thomson Lerning Lttice ites in n Orthogonl Coordinte ystem i.e. imple Cubic BCC FCC HCP

10 DIRECTION IN CUBIC LATTICE 1. Vector components of the direction re resolved long ech of the coordinte xes nd reduced to the smllest integers.. All prllel directions hve the sme direction indices. 3. Equivlent directions hve the sme tom spcing. 4. The cosine of the ngle between two directions is given by cos = ( h +k h h+k k +ll +l ) ( h +k +l ) Indices of Fmily or Form < 100 > < 111> [100],[010],[001],[010],[001],[100] [ 111],[ 11 1 ],[ 1 1 1],[ 1 11], [ ],[ 1 1 1],[ ],[ ] < 110 > [ 110 ],[ 011],[ 101],[ ],[ 01 1 ],[ 10 1 ] plus the six negtives Fmily of directions MILLER INDICE FOR PLANE (0,0,1) Index Number in the fmily for cubic lttice <100> 3 x = 6 (0,3,0) <110> 6 x = 1 <111> 4 x = 8 ymbol Alternte symbol [ ] Prticulr direction < > [[ ]] Fmily of directions (,0,0) Find intercepts long xes 3 1 Tke reciprocl 1/ 1/3 1 Convert to smllest integers in the sme rtio 3 6 Enclose in prenthesis (36)

11 b (110) plnes (130) plnes (-10) plnes The orienttion of plnes is best represented by vector norml to the plne. The direction of set of plnes is indicted by vector denoted by squre brckets contining the Miller indices of the set of plnes. Miller indices re lso used to describe crystl fces. (100) plnes [100] vector (111) Fmily of {111} plnes within the cubic unit cell d111 / 3 3/3 The (111) plne trisects the body digonl (111) Plne cutting the cube into two polyhedr with equl volumes (-100) fce (100) fce

Points bout (hkl) plnes For set of trnsltionlly equivlent lttice plnes will divide: Entity being divided (Dimension contining the entity)

All prllel plnes hve sme Miller indices.

46 Crystllogrphic Plnes exmple b c 1. Intercepts 1 1. Reciprocls 1/1 1/1 1/ 1 1 0 3. Reduction 1 1 0 4. Miller Indices (110) exmple b c 1.

12 Points bout (hkl) plnes For set of trnsltionlly equivlent lttice plnes will divide: Entity being divided (Dimension contining the entity) Direction number of prts Cell edge (1D) [100] h Crystllogrphic Plnes Crystllogrphic plnes re specified by 3 Miller Indices (h k l). All prllel plnes hve sme Miller indices. b [010] k c [001] l Digonl of cell fce (D) (100) [011] (k + l) (010) [101] (l + h) (001) [110] (h + k) Body digonl (3D) [111] (h + k + l) 46 Crystllogrphic Plnes exmple b c 1. Intercepts 1 1. Reciprocls 1/1 1/1 1/ Reduction Miller Indices (110) exmple b c 1. Intercepts 1/. Reciprocls 1/½ 1/ 1/ Reduction Miller Indices (00) x x c c z z b b y y Crystllogrphic Plnes exmple b c 1. Intercepts 1/ 1 3/4. Reciprocls 1/½ 1/1 1/¾ 1 4/3 3. Reduction Miller Indices (634) x c z b y 47 48

13 Fmily of Plnes Plnes tht re crystllogrphiclly equivlent hve the sme tomic pcking. Also, in cubic systems only, plnes hving the sme indices, regrdless of order nd sign, re equivlent. Ex: {111} = (111), (111), (111), (111), (111), (111), (111), (111) Intercepts 1 Plne (100) Fmily {100} 3 Intercepts 1 1 Plne (110) Fmily {110} 6 Ex:{100} = (100), (010), (001), (100), (010), (001) Intercepts Plne (111) Fmily {111} 8 (Octhedrl plne) ummry of nottions Tetrhedron inscribed inside cube with bounding plnes belonging to the {111} fmily 8 plnes of {111} fmily forming regulr octhedron Direction Plne Point ymbol Alternte symbols [ ] [uvw] Prticulr direction < > <uvw> [[ ]] Fmily of directions ( ) (hkl) Prticulr plne { } {hkl} (( )) Fmily of plnes...xyz. [[ ]] Prticulr point : : :xyz: Fmily of point A fmily is lso referred to s symmetricl set

Hexgonl crystls Miller-Brvis Indices Unknown direction [uvw] Unknown plne (hkl)

(hkl) 3 Intercepts 1 1 - ½ Plne (1 1 0) (h k i l) i = (h + k) d hkl h k l 1 The

crystllogrphiclly equivlent plnes nd directions Exmples to show the utility of

Intercepts 1-1 Miller (1 1 0 ) Intercepts 1-1 Miller (0 1 0) Intercepts 1 - -

14 Hexgonl crystls Miller-Brvis Indices Unknown direction [uvw] Unknown plne (hkl) Double digit indices should be seprted by comms (1,,3) In cubic crystls [hkl] (hkl) 3 Intercepts ½ Plne (1 1 0) (h k i l) i = (h + k) d hkl h k l 1 The use of the 4 index nottion is to bring out the equivlence between crystllogrphiclly equivlent plnes nd directions Exmples to show the utility of the 4 index nottion 3 Exmples to show the utility of the 4 index nottion 3 1 Intercepts 1-1 Miller (1 1 0 ) Intercepts 1-1 Miller (0 1 0) Intercepts Plne ( ) 1 Intercepts ½ Plne (1 1 0) Miller-Brvis ( ) Miller-Brvis ( )

Intercepts 1 1 - ½ 1 Plne (1 1 1) UMMARY OF

of directions Intercepts 1 1 1 Plne (1 0 1 1)

15 Intercepts ½ 1 Plne (1 1 1) UMMARY OF MEANING OF PARENTHEE (q,r,s) represents point note the exclusive use of comms [hkl] represents direction <hkl> represents fmily of directions Intercepts Plne ( ) (hkl) represents plne {hkl} represents fmily of plnes

Crystalline Structures The Basics

Crystalline Structures The Basics Crystlline Structures The sics Crystl structure of mteril is wy in which toms, ions, molecules re sptilly rrnged in 3-D spce. Crystl structure = lttice (unit cell geometry) + bsis (tom, ion, or molecule