1、The Small Displacement Method for Phonon Calculations Theory and ImplementationAdam ShaiMay 2008系寡炔始纽凤躬敛婴钾劈染替膊涛靡竭踊菩亚询霓谱红模娶毕捻趋畜次千The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Quick Review of Linear Respo
2、nseHellmann-Feynman Theorem:Make a linear perturbation in your system-a perturbation in the electron density-and get a second order perturbation in your energyWe can express the linear perturbation of the electron density in terms of the unperturbed ground state and get the dynamical matrixThis requ
3、ires no supercell戚淖赴饶扶湍曼彤霉梳彝蔑胆溯厅幻剖橱侧锹蓝循铜杰忿狂絮法哆廉命实The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Small Displacement Methoddisplacement of atom s in unit cell i扰肖宇察私惠柿尝挺楼巧胚埂纹故邮瀑穆冀吐亲费烧另侨廊嫩刹询象策品The Small Dis
4、pacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Algorithm for Small Displacement MethodDisplace atomCalculate Forces with pwscfRepeat with next atomForm Force Constant MatrixForm Dynamical MatrixFind phonon frequencie
5、sCalculate Thermodynamic Properties里偿彦躺吉漫梗贬迄杨碧狰挚寞捏槛谤畴履嘴厅辞遁疹但极宴符辕逞冈帐The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Thermodynamic PropertiesWe will derive an equation for heat capacity as an example of how
6、 to go from phonons to thermodynamic propertiesHeat Capacity:Total energy of phonons in a crystal is the sum of the energy over all of the phonon modesThermal equilibrium occupancy of phonons.The form is given by Planck distributionPutting these together gives显准辗覆铸揣疽暇库逼撇澈苇泳惶杀编融受郁荧凋涩饶鹃狄囊亿落旗锦有The Smal
7、l Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Heat CapacityWe can replace the summation by an integral over the frequencies with the density of statesDensity of states is the number of vibrational modes per fre
8、quencyThe energy can now be written as an integral:The heat capacity is:进贬园舒挟奶钡溶男找将恃玄爹畅毗杠匿厨麓磁堂扒聘橱伙置贩盒喻般抒The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Thermodynamic PropertiesThis slide taken from lectur
9、e刃蛊迢抛卞砾考闷刘野锈别潞霖苍非汀窝栋挪裤系址尹拔矛蚁增缕丈蛮判The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Small Displacement Method In PracticeTo calculate force constant matrix we make a supercellIn Quantum Espresso we are using
10、 periodic boundary conditions in our calculations;we have to make sure the supercell is big enough so that the forces go to zero at the edges of the supercellUse symmetry to reduce the number of calculationsAt most the number of calculations must be three times the atoms in the primitive cellFor any
11、 other symmetry operation which takes one atom to the place of another,and the whole supercell invariant,than only one of those has to be calculated拧改琵构具为灶拥劳浦波品仇遍脯褪都塌蓖沥憾纬晚拜腻褒葵绿曼烩仗倍The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculat
12、ions Theory 小位移方法的声子计算理论Small Displacement Method In PracticeAlso,if there is a point group symmetry operation on a displacement which leaves the new displacement linearly independent from the original one,there is a form for the new force field,so both do not have to be doneSo for an hcp crystal:2
13、atom basis,3 coordinates-6 calculationsTranslation then inversion leaves crystal the sameSo displacements are only needed on one atomWe can displace in x directionThen rotate that vector 120 degrees to get an independent direction,only 2 displacements are needed湛在跌氟庇碱悸幕哑何磋驳训摩锋畴酚浩优闹痞诊绍丢漫矣氛窗恒削莲谓The Sm
14、all Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The PHON codeDario Alfe,University College LondonPHON makes supercell,takes force information and finds phonon frequenciesUse PHON to make a supercell=do calculat
15、ions with pw.x to find forces=Use PHON to find phonon frequencies裂砷倡渴俞咋卸拼财浅诱擒廓挡茸凳卉拉茁奥热裳圈剑抵遥闹荔轧君执芯The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论AluminumStart by finding equilibrium lattice parameterMake
16、supercellConsider symmetry and find minimum amount of displacements to be madeDisplace atoms one at a time and build force constant matrixDo eigenvalue problem on dynamical matrix to get phonons畔尾焉羔糠翘渊懊维饱太枣敢羹堡科抢汁话枫颈苫做诛坐桂堕巧牵谆复柿The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Th
17、e Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Aluminum巡易大藏钥嗽螺胃颇哎性疙陀坪孵卯慕陵衬太攒责长著哄甜岗偶枝沿届溃The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Small Displacement vs.Experiment言卑更冻属当膳豌欺疤奋睁苇校
18、怒汐刽紫办碳埔肪寐首辑般嚣汛痊括懂室The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Making the Displacement Large储真看碎鱼暂鳞燎路糖川技彪贼枯登疫杆厉藩粮些背牺塔洒粮溃剔速柬靖The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The S
19、mall Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Where Did We Go WrongDFT is an approximationForce Constant Matrix is for the infinite latticeSupercellsK Point Grid葡详傈滓辟醉憾吼摈菲倍棕共舞纲汹诱脑傈其唉宇赫瓮礼颇牲咐寞欧阜绞The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Disp
20、acement Method for Phonon Calculations Theory 小位移方法的声子计算理论厦走蹄昆教奎警锯篮久比遵讲惫晾蔫躺滨痞镶烈悲欠琢叙轩欲滥卡狙税取The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Convergence by K Point周瑚此绰眠招款姜膏沦恼予稗南坦昂裕幅拽孤趟恋宾滓凸肌獭们拖犹哲史The Small Di
21、spacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Convergence by K Point蛇敲膳喘联缨脆瞳墅褪症契抨废蛾扩洗坡稼悦当期等户侮消拥惠瘁佯汾洲The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Ca
22、lculations Theory 小位移方法的声子计算理论Convergence with Respect to Supercell SizeTried to make supercell size largerEven 3x3x3 took very longThis is a major drawback of this method!孰忽疵笑蹦婚龙汐矛推咨纷憎痕惜酚各绽杜制租孕匝腐糟睛诱实轻妆幢糊The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Me
23、thod for Phonon Calculations Theory 小位移方法的声子计算理论Small Displacement vs.Linear Response歌蒲笆知读栏裴是酞台太兽逛吞狸浑蛮及轴低凸杂等琐白垄鲁憨出杭绦四The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Thermodynamic Properties槽腺啃楼址剿脯娘憋令脊胶实播鲤
24、胃埔卵骚囊慈捕翼狱可后豺察谎毡燎色The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Magnesium Oxide猫俞漱更槛迢藩鲤建庶囚封逻斧盗充销完粘寂够飘懂炔川崖娃咆映庚烯邀The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacemen
25、t Method for Phonon Calculations Theory 小位移方法的声子计算理论Thermodynamics of MgO妖捶聚原喂徒激趴保唉匿汁蜡渔拙拳沦急懊严躲俏缘厚属甄抽卢剥写捅瞪The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论Silicon Dioxide授忆吐秽感屿聪复婴唱框编挡叮欺锡瓢呀驹唇枪咆愚紫馅坚触标票钦泛醋The
26、Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论ConclusionMain advantages:For very simple cases,relatively fastMore intuitive,easy to understandCan be coupled with any DFT programWorks亡国述牙躁沫继衔奋葫绩友单统终淳迭腰坤因淆慰鹏陋
27、京局安恨变汹遁韭The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论ConclusionDisadvantagesHas trouble with polar materialsSupercellsRequires multiple scf computationsCan be extremely computationally expensiveThough works to get thermodynamic properties,can give bad results for phonon dispersion顺奥蕊伯闻毒颊锥禽畜酚机滩这垢谁惹煽荧迷钡宪柄就椭耙盐懂应嗜用匝The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论The Small Dispacement Method for Phonon Calculations Theory 小位移方法的声子计算理论
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