3.1 The Coupled Harmonic Oscillator Model
3.1.3 Vibrational Modes in Higher Dimensions
Thesameapproachcanbeusedforhigherdimensions.Letusconsideratwo-dimensional latticeofatomsconnectedbysprings,asillustratedinFigure3.6.Tomakethingseasier,we consideronlyasimplebasiswithoneatomperunitcell,andassumethatallofthespring constantsarethesame.Thesystemismademorecomplexbythepossibilityofinteractions betweenatomsinseveraldifferentdirections,however.
Thedirectionoftheforceexertedbyaspringisalwaysalongthedirectionbetweenthe twoatomsitconnects.Ingeneral,ifbothatomshavemoved,thedirectionoftheforceon oneatomwillchange.Ifweassumethatthedisplacementofanatomissmallcompared tothedistance betweenatoms,though,thenwe cantreatthedirection ofthespringsas
(xn–1, n′+1, yn–1, n′+1) (xn, n′+1, yn, n′+1) (xn+1, n′+1, yn+1, n′+1)
(xn–1, n′–1, yn–1, n′–1) (xn, n′–1, yn–1, n′–1) (xn+1, n′–1, yn+1, n′–1) (xn–1, n′, yn–1, n′) (xn, n′, yn, n′)
(xn+1, n′, yn+1, n′)
±Fig.3.6 Nearest-neighborandnext-nearest-neighborinteractionsinatwo-dimensionallatticewithaone-atombasis.
unchanged by the motion ofthe atoms. For example, the force for the diagonalspring alongpositive x and y is
±F =−K±l
² 1
√2ˆx + 1
√2ˆy
³
, (3.1.16)
where l istheunstretchedlengthofthespring.Thechangeinthelengthisapproximately
±l = l − à
(l/√
2 + ±x)2+ (l/√
2 + ±y)2
³ l − l à
1 +√
2(±x + ±y)/l
³ 1
√2
±x + ±y
l . (3.1.17)
Thus,writing xn,n´forthedisplacementoftheatomatposition(n, n´)inthe ˆx-directionand yn,n´ forthedisplacementinthe ˆy-direction,Hooke’slawfortheeightclosestneighborsof givesus
M¨xn,n´= K(xn+1,n´− xn,n´) − K(xn,n´− xn−1,n´) +1
2K(xn+1,n´+1− xn,n´+ yn+1,n´+1− yn,n´)
−1
2K(xn,n´− xn−1,n´−1+ yn,n´− yn−1,n´−1) +1
2K(xn+1,n´−1− xn,n´− yn+1,n´−1+ yn,n´)
−1
2K(xn,n´− xn−1,n´+1− yn,n´− yn−1,n´+1), (3.1.18) andasimilarequationfor ¨yn,n´.
Asintheprevioussection,weguessasolutionoftheform x(t) = x0ei(±kã ±R−ωt)
y(t) = y0ei(±kã ±R−ωt), (3.1.19) wherewemustnowkeepaccountofatwo-dimensionalvector ±k = (kx, ky),andthelattice vector ±R = (an, an´).Substitutingtheseintotheequationsofmotion,weobtain
−ω2Mx0= Kx0(eikxa+ e−ikxa− 2) +K
2(x0+ y0)(ei(kxa+kya)+ e−i(kxa+kya)− 2) + K
2(x0− y0)(ei(kxa−kya)+ e−i(kxa−kya)− 2)
−ω2My0= Ky0(eikya+ e−ikya− 2) +K
2(x0+ y0)(ei(kxa+kya)+ e−i(kxa+kya)− 2)
− K
2(x0− y0)(ei(kxa−kya)+ e−i(kxa−kya)− 2).
(3.1.20)
165 3.1 TheCoupledHarmonicOscillatorModel
Let uspick ±k along the ˆx-direction. The equationsof motion then correspond to the matrixequation
K M
² 4cos ka − 4 0 0 2cos ka − 2
³² x0
y0
³
=−ω2
² x0
y0
³
. (3.1.21)
Thisisalreadydiagonal,witheigenvectors(1,0)and(0,1).When k islow,sothatwecan approximatecos ka ³ 1 −12(ka)2,theeigenvaluesare
ωL=√ 2ω0ka
ωT = ω0ka, (3.1.22)
where ω0 =√
K/m.Herewehavelabeledthetwomodes L and T for longitudinal and transverse.Inthe one-dimensionallinear-chainmodelwithtwo atomsperunitcell, we hadtwoeigenmodes,whichcorrespondedtoanacousticmodeandanopticalmode,and theoptical modecorrespondedto motionofthetwo atomsrelativeto eachother, inthe low-frequency limit.In the present case,the two eigenvectorshave different meanings.
Bothareacousticmodes,withdispersioninthelow-frequencylimitgivenby ω = vk,but thetwomodeshavetwodifferent polarizations.Thelower-frequencymodehas x0 = 0 and y0 ²= 0,whichmeansthatthemotionoftheatomsisperpendicularto ±k,ortransverse, asillustratedinFigure3.7(a).Thehigher-frequencymodehas x0²= 0and y0= 0;inother words,themotionoftheatomsisinthesamedirectionas ±k,orlongitudinal,asillustrated inFigure3.7(b).
k
(a)
(b)
(c)
k
k
±Fig.3.7 (a)Atomicmotionforapurelytransverseacousticwave.(b)Atomicmotionforapurelylongitudinalwave.(c)Atomic motionforamixed-polarizationwave.
In three dimensions,there are threepossible polarizations for acoustic waves,unlike electromagneticwaveswhichcanonlyhavetransversemodes.Ingeneral,thethreepolar- izationmodesdonothavetobepurelylongitudinalortransverse;thepolarizationvector canbeinsomeotherdirection,asillustratedinFigure3.7(c).Alongasymmetrydirection ofthe Brillouin zone,however, itis common tohave twodegenerate, purelytransverse modes,andonelongitudinalmodewithfastervelocity.Awayfromsuchasymmetrydirec- tion, thetwo lower modescontinueto bemostly transverse,and the highmodemostly longitudinal,andsowestillcallthem L and T modes.
Noticethatthetransversemodeinourmodelhaslowerspeedeventhoughallthespring constantswerethesame.Thisisbecauseinthelongitudinalmode,anatompushesdirectly againstthespringconnectingittoitsnearestneighbor,whilethetransversemoderelieson restoringforcefromatoms furtheraway.It isgenerallythecasethatlongitudinalmodes havehigherfrequencyinsolids,forthisreason.
Exercise3.1.4 UsethematrixsolverinaprogramlikeMathematicatosolve(3.1.20)for a general ±k, and plot ω versus k, for ±k in two different directions in the plane.
Determinethepolarizationvectorofthetwomodesineachcase.
The vibrational spectra of real, three-dimensional solidscan beaccurately calculated usingathree-dimensionalmodelthatisessentiallythesameasthoseabove.Ifthecrystal structureisknown,thenthepositionsofalltheatomsintheunitcellatzerodisplacement canbewrittendown,andtheinteractionsbetweentheatomscanbemodeledassprings, justasinthelinear-chainmodel.Thematrixequationinthiscasewillhaveadimension equaltothenumberofatomsintheunitcelltimesthree,sinceeachatomhasthreespatial degreesof freedom.Sinceamoleculein asolidisnot normallyfreetospin,1rotational degreesoffreedomareusuallynottakenintoaccount.
Inthelinear-chainmodelofSection3.1.2,wehadaunitcellwithtwodegreesoffree- dom,and wefound twoeigenmodes for each k.Inthetwo-dimensional model,thetwo degreesoffreedomcorrespondedtodifferentdirectionsofmotionofthesingleatominthe unitcell.Ingeneral,sincethedimensionofthematrixequationforthevibrationisequalto thenumberofdegreesoffreedom,thenumberofeigenmodesatagiven ±k equalsthenum- berofdegreesoffreedomintheunitcell.Furthermore,becauseazero-frequencyacoustic modecorrespondstoasimpletranslation,acrystalwillalwayshaveanumberofacoustic modesequalto thenumber of simple translationsitcanundergo. In particular,in three dimensions,therewillalwaysbeexactlythreeacousticmodes, whichcanhavedifferent polarizations,thatis,directionsoftheoscillationoftheatomsrelativetothe k-vector,and differentspeeds.
Forexample,aunitcelloftwoatomsinthreedimensionswillhavesixdegreesoffree- dom,andsixvibrationalmodes.Threeofthesewillbeacousticmodes,andthreewillbe optical.Acrystalwithalargerunitcellcanhavemanymoremodes.Forexample,thecrys- talSrF2,whichhasafluoritelatticewithaunitcellofthreeatoms(seeTable1.1),which givesninedegreesoffreedom.Threeoftheseareacousticmodes,andthesixremaining
1 “Buckyball” solids, that is, solids composed of 60 atom carbon spheres, are one notable exception – at high temperatures buckyballs can spin freely while keeping their position in a lattice.
167 3.1 TheCoupledHarmonicOscillatorModel
[THz]
k
] 1 1 1 [ ]
0 1 1 [ ]
1 0 0 [ 12
8
4
0 0.4 0.8 0.8 0.4 0 0.2 0.4
1 1
5 2
4 1
5 2
1 1
5 2
±Fig.3.8 VibrationalmodesofSrF2,whichhasafluoritelattice.Thedatapointsaretheexperimentalresultsofneutron scatteringexperiments,whilethesolidlinesaretheresultofamulti-parameterspring-constantmodel.Thecurves alongthe[001]directionarelabeledusingthegrouptheorynotationdiscussedinChapter6;thesuperscriptsinfront ofthesymbolsgivethedegeneraciesofthemodes.FromElcombe(1972).
modesareoptical modes.Figure3.8 showsthevibrational modesofSrF2,measuredby neutronscattering(fordetailsonthismethod,seeSection3.2).
Thelinear-chainmodelofSection3.1.2includedonlynearest-neighbor interactions– each atom was connected by springs only to its nearest neighbors, while in the two- dimensional model we also included next-nearest-neighbor interactions. Models of the vibrationalmodescanbegivengreatersophisticationbyincludinghigh-orderinteractions toevenfurtheratoms.Somemodelsalsosplitanatomintothenucleusandashellofelec- trons aroundit, whichcan move separately.This allows polarization ofthe atom asan additionaldegreeoffreedom.Theelectronshellcanthenbecoupledtoneighboringshells byadditionalsprings.
Eachadditionalspringconstantgivesanotherparameterbywhichthetheoreticalmodel canbefit tothe data.Figure 3.8 shows afit ofa springmodel to thevibrational mode dataofSrF2.Thesemodelsareusefulinpredictingthefrequenciesofvibrationalmodes thatcannotbedirectlymeasured.Byfittingtothefrequencieswhichareknown,themodel givespredictionsoftheentirevibrationalspectrum.
An ab initio modelof the ground state ofa crystal (see Section1.9.5) canalso give apredictionofthevibrational spectrumbycalculatingthesecond derivativeof thetotal energywithrespecttoperturbationsoftheatomic positions.Thespringconstantsofthe
atomicinteractionsarenot,afterall,freeparameters,butcomefromthesameHamiltonian asusedfortheelectronicbandstructure.