采用VASP计算晶体的弹性常数

æ^VASPXÛOŽ¬N��5~êCij �nµûι (zfhou@fudan.edu.cn) E�ŒÆÔnXƬ� 2006 8 � 21 F Á ‡ �©ò±8Ƭ‚Ǒ~0�ÏLVASPOŽ¬N��5~êCij" ™²NŒ§žÕò�©SN€�–?Û/ª�чԥœœœ 8 ¹ §1 �5~ê�Vg [1, 2] 1 §2 8ƬN��5ACU†AC'X 2 §3 äNOŽÚ½ 3 §1 �5~ê�Vg [1, 2] �5~ê£ã ¬Né \ACǫ�ǑA�fÝ"3ACé��œ¹e§NX�SU†A C�Œ�3�g‚5'X£�Ž½Æ¤§�5~êCijÒ´£ãù«�g‚5'X§=�g‚ 5‘�Xê"æ^VoigtIPµxx→ 1, yy → 2, zz → 3, yz → 4, xz → 5Úxy → 6"ACÜþǫ½ ÂǑµ ǫ =   e1 1 2e6 1 2e5 1 2e6 e2 1 2e4 1 2e5 1 2e4 e3   . (1) AåÜþσ½ÂǑµ σi = 1 V [ ∂E(V, ǫj) ∂ǫi ] ǫ=0 , (2) ��ý9�5~êǑµ Cij = 1 V [ ∂2E(V, ǫk) ∂ǫi∂ǫj ] ǫ=0 , (3) 1 2 www.quantumchemistry.net 3AC���œ¹e§AC�NX�oUE(V, ǫ)UACÜþǫ?ŒU�V?ê�mǑµ E(V, {ǫi}) = E(V0, 0) + V0 6∑ i=1 σiei + V0 2 6∑ i,j=1 Cijeiej + . . . (4) Ù¥E(V0, 0)´AC NX�oU§V0´AC �œ�NÈ", §AC�Ä¥ −→ R ′ †AC � Ä¥ −→ Rƒm�'XǑµ −→ R ′ = −→ R • (I + ǫ) (5) Ù¥IǑü Ý " ÏdÀ�A½�ACǫ = e = (e1, e2, e3, e4, e5, e6)§OŽÑ3˜|ØÓÌݞAC �NX oU�Cz(△E = E(V, ǫ)−E(V, 0))§2ŠâoU�Cz¨ACÌÝéA�˜|êâ:§?1 �g¼ê[Ü���g‘Xê"=Œ��¬N�,‡�5~ê½�5~ê�|Ü"éØÓ� ¬X�¬N§ÏǑé¡5�'X§§Õá��5~ê´(½�"'Xé8ƬX�¬N§§Õ á��5~êǑµC11, C12, C13, C33ÚC44" §2 8ƬN��5ACU†AC'X é8ƬX�¬N§Ù�œÄ¥Œ±�Ǒµ −→ R =   a1 a2 a3   =   √ 3 2 a 1 2a 0 − √ 3 2 a 1 2a 0 0 0 c   . (6) Ù¥aÚc´¬N�¬‚~ê" Œ±–\ù��ACe = (δ, δ, 0, 0, 0, 0)5OŽC11 + C12 [3]µ ∆E V0 = (C11 + C12)δ 2 (7) –\ACe = (0, 0, 0, 0, 0, δ)5��C11 − C12µ ∆E V0 = 1 4 (C11 − C12)δ 2 (8) éuC33§–\ACe = (0, 0, δ, 0, 0, 0)5��µ ∆E V0 = 1 2 C33δ 2 (9) éuC44§–\ACe = (0, 0, 0, δ, δ, 0)5��µ ∆E V0 = C44δ 2 (10) �§–\ACe = (δ, δ, δ, 0, 0, 0)��C11!C12!C13ÚC33�|ܵ ∆E V0 = (C11 + C12 + 2C13 + C33/2)δ 2 (11) ddŒ„§ÏL–\ʇA½�AC§À�˜X�ÌÝδ�AC§��∆E ∼ δêâ:§2©O Uþ¡Ê‡'XªéƒA�∆E ∼ δ?1[Ü���g‘X꧁�éᐧ���Ñ8Ƭ X¬N�Õá�5~ê" 2 3 www.quantumchemistry.net §3 äNOŽÚ½ 3OŽž§kA:‡AO5¿�µa)§�œS�f3AC´Ä¶þ¶b)§k:�‚Œ�´Ä v §ÏǑ3AC�§�œ�é¡5¬u)Cz§=�Ó��k:�‚§3{�Ùp�«�) �k:ê8´ØÓ�"Ïd§k:�‚Œ�‡��v §±�y�5~êOŽ�°(5¶c)§A CÌÝδ‡��·¥§XJ���{§���ACU£AC �NX�Cz¤é�§3OŽ�5 ~ꞧ¬ÚåOŽØ�" e¡±OŽ8ÆAlN£n ¶(�¤Ǒ~§3OŽL§Ä AC��f� ˜I‡¶þ§ äNÚ½Xeµ • ké8ÆAlNNáÆ�¬‚ë꣬‚~êÚ�f ˜¤` z�§ù���™ACž�POSCAR§ ¿r§€�¤e¡defvector.fI‡^��˜‡�Ñ\©‡OLDPOS§¿éOLDPOS‰˜‡ ?n"ÏǑOLDPOS3‚ªþkAχ�µ a !3OLDPOS�1˜1§3title�iÎGƒ�§–�˜˜‚2\þOLDPOS¥�f�« aê8§'XAlN¥küa�f§title�ǑAlN§�oÒOLDPOS�1˜1Ò´”AlN 2”" b !OLDPOS�‚ª†POSCAR�aq§�´§�´©ê‹I5�Ñ�f� ˜" ±8ÆAlNǑ~§ù‡OLDPOS�SNXeµ AlN 2 3.11553 1.000000 0.000000 0.000000 -0.500000 0.866025 0.000000 0.000000 0.000000 1.605000 2 2 Direct 0.00000000 0.00000000 0.00000000 0.333333333 0.666666667 0.50000000 0.00000000 0.00000000 0.381483673 0.333333333 0.666666667 0.881483673 • éA½�AC§3e¡�defvector.f¥”Define the strain”Ü©§rA½�ACÏL‰strain(i)Ý DŠ"'Xéþ¡J��e = (δ, δ, 0, 0, 0, 0)^5OŽC11 + C12§�oÒédefvect.f¥ �”Define the strain”U�¤Xe�/ªµ C%%%%%%%%% Define the strain %%%%%%%%%%%%%% strain(1)=delta strain(2)=delta strain(3)=0.0 strain(4)=0.0 strain(5)=0.0 strain(6)=0.0 Ù¥�‡defvector.f´^5��,‡AC�§#�POSCAR"AC�a.U”Define the 3 3 www.quantumchemistry.net strain”�Ü©5½Â§ AC�ÌÝI3§S?È�§$1?È����¬žÑ\"defvect.f� SNXeµ C%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C >this simple program to get the primitive vectors after C $\delta$ strain, in order to calculate the independent C elastic constants of solids. C usage: C!!!!! Please first prepare the undeformed POSCAR in OLDPOS C >defvector.x C >type defvector.x > create new POSCAR in file fort.3 C%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% program defvector real*8 privect,strvect,delta,strten,strain,pos, alat dimension privect(3,3),strvect(3,3),strten(3,3),strain(6) dimension pos(50,3) character*10 bravlat, title, direct integer i,j,k,ntype, natomi, nn dimension natomi(10) C%%%%%%%%% Read the undeformed primitive vector and atomic postion %%%%%%% open(7,file=’OLDPOS’) C%% In first line of OLDPOS, please add the number C%% of the type of atoms after the title read(7,*) title, ntype read(7,*) alat do i=1,3 read(7,*) (privect(i,j),j=1,3) write(*,*) (privect(i,j),j=1,3) enddo read(7,*) (natomi(i),i=1,ntype) nn=0 do i =1, ntype nn=nn+natomi(i) enddo read(7,*) direct do i=1, nn read(7,*) (pos(i,j),j=1,3) enddo C%%%%%%%%% Read the amti of strain %%%%%%%%%%%%%%% read(*,*) delta C%%%%%%%%% Define the strain %%%%%%%%%%%%%% strain(1)=delta strain(2)=0.0 strain(3)=0.0 strain(4)=0.0 strain(5)=0.0 strain(6)=0.0 C%%%%%%%%% Define the strain tensor %%%%%%%%%%%%%%%%%%%%%%%% strten(1,1)=strain(1)+1.0 4 3 www.quantumchemistry.net strten(1,2)=0.5*strain(6) strten(1,3)=0.5*strain(5) strten(2,1)=0.5*strain(6) strten(2,2)=strain(2)+1.0 strten(2,3)=0.5*strain(4) strten(3,1)=0.5*strain(5) strten(3,2)=0.5*strain(4) strten(3,3)=strain(3)+1.0 C%%%%%%%%% Transform the primitive vector to the new vector under strain%%%%% C strvect(i,j)=privect(i,j)*(I+strten(i,j)) do k=1,3 do i=1,3 strvect(i,k)=0.0 do j=1,3 strvect(i,k)=strvect(i,k)+privect(i,j)*strten(j,k) enddo enddo enddo C%%%%%%%% Write the new vector under strain%%%%%%%%%%%% do i=1,3 write(*,100)(strvect(i,j),j=1,3) enddo 100 format(3f20.15) C%%%%%%%%% Create the POSCAR for total energy calculation %%%%%%%%%%%%%%5 write(3,’(A10)’) title write(3,’(f15.10)’) alat do i=1,3 write(3,100)(strvect(i,j),j=1,3) enddo write(3,’(10I4)’) (natomi(i), i=1,ntype) write(3,’(A6)’) Direct do i=1, nn write(3,100) (pos(i,j),j=1,3) enddo C%%%%%%% end 3defvector.f¥§�f«a�ê8dCþntype5½Â�§ŒǑ10§XJ�f�«aê8� Œ§IgC�ÄNŒê|natomi(10)±9Ñў” write(3,’(10I4)’) (natomi(i), i=1,ntype)”� ‚ª”10I4”" 3defvector.f¥�˜� A½�AC�§ÒŒ±?Èdefvector.f£�^g77 -o defector.x defector.f)���¬defvector.x" • O��VASPOŽ�Ñ\©‡KPOINTSÚPOTCAR"±9?1�f ˜¶þOŽ�INCAR.relax§ §�SNXeµ SYSTEM = AlN ENCUT = 400 ISTART = 0 ICHARG = 2 5 3 www.quantumchemistry.net ISMEAR = 0; SIGMA = 0.2 NSW = 60; IBRION = 2 EDIFF = 1E-5 EDIFFG = -1E-2 ISIF = 2 POTIM = 0.2 PREC = Accurate LWAVE = .FALSE. , 3O�é`z���(�?1oU?1�INCAR.static§§�SNXeµ SYSTEM = AlN ENCUT = 400 ISTART = 0 ICHARG = 2 ISMEAR = -5 EDIFF = 1E-5 PREC = Accurate LWAVE = .FALSE. o�`5§Ò´kéAC��POSCAR?1�½Ä¥§é�f ˜�`z§2 é`z���(�?1·�oUOŽ��ACNX��oUEtot (δ)"Ù¥AC��� ©POSCARÏLdefvector.x5��" • é˜X�ÌÝδ�A½AC?1þ˜Ú�OŽ"���˜|Etot(δ)−Etot(0) V0 ∼ δêâ", �é§?1�g¼ê[Ü���g‘�Xê" 5¿µEtot(0)´™AC´NX�oU§V0´™ACNX�NÈ"3VASPOŽ¥§‚ �êŠü ´eVÚA˚3"1 eV/A˚3 = 160.2 GPa" ¡��f ˜`zÚoUOŽ�Œ±ÏL˜‡bash��5?1§Xµ #!/bin/sh for i in -0.018 -0.015 -0.012 -0.09 -0.06 -0.03 0.00 \ 0.03 0.06 0.09 0.012 0.015 0.018 do echo $i | defvector.x cp fort.3 POSCAR #### cat > INCAR < INCAR <>SUMMARY done • éÙ�A½�AC§Uþ¡12Ú3Ú2‰˜X��OŽ��ƒA�˜|Etot (δ)−Etot(0) V0 ∼ δê⧱9[Ü" éÙ�¬X('X��¬N)��5ACU!ACÚ�5~ê�'X§Œ±ë©z [1, 4]" ë©z [1] P. Ravindran, L. Fast, P. A. Korzhavyi, B. Johansson, and J. Wills, J. Appl. Phys. 84, 4891 (1998). [2] G. Grimvall, Thermophysical properties of materials, Elsevier/North-Holland, Amsterdam, 1999. [3] S. Q. Wang and H. Q. Ye, J. Phys. Condens. Matter. 15, 5307 (2003). [4] Z. F. Hou, cond-mat/0601216, unpublished. (2006) 7