Phonon Dispersion Relation and Density of States of a Simple Cubic Lattice

Phonon dispersion relation and density of states of a simple cubic lattice

Student project for the course Molecular and Solid State Physics by Eva Meisterhofer Contents

1 The linear spring model 3 1.1 Nearest neighbours ...... 3 1.2 Nearest and next nearest neighbours ...... 4

2 Numerical calculation in Matlab 6 2.1 Matlab Files ...... 6 2.2 Figures ...... 9

2 1 The linear spring model

A simple model for describing lattice vibrations in a crystal is to assume that the atoms are masses connected by linear springs. With this approximation we want to calculate the phonon dispersion relation and density of states for a simple cubic lattice.

1.1 Nearest neighbours When considering only the six nearest neighbours, the equation of motion in the x- direction (Newton’s law) is:

d2ux m lmn = C[(ux − ux ) + (ux − ux )] dt2 l+1mn lmn l−1mn lmn The solutions of this diﬀerential equation are eigenfunctions of the translation operator:

ux = Ax exp [i(l~k~a + m~k~a + n~k~a )] = Ax exp [i(lk a + mk a + nk a)] lmn ~k 1 2 3 ~k x y z The expressions for the y- and z-directions are similar. Since the three equations of motion are independent, the three-dimensional problem decouples into three one- dimensional problems. By substituting the eigenfunction solutions into Newton’s law, the diﬀerential equations become algebraic equations which can be used to calculate the dispersion relation.

2 mω ~ulmn = M~ulmn

 2 akx  4C sin ( 2 ) 0 0    ak  M =  0 4C sin2 ( y ) 0   2    2 akz 0 0 4C sin ( 2 ) The phonon dispersion relation can be obtained by calculating the eigenvalues λ of the Matrix M: r λ ω = m Since the eigenvalues of a diagonal matrix are the diagonal elements, the dispersion relation for a simple cubic lattice considering only the nearest neighbours is: r r r 4C akx 4C aky 4C akz ω1 = sin ( ) , ω2 = sin ( ) , ω3 = sin ( ) m 2 m 2 m 2

3 1.2 Nearest and next nearest neighbours When considering the 6 nearest and the 12 next nearest neighbours, the equation of motion in the x-direction is:

2 x d ulmn x x x x m dt2 = C1[(ul+1mn − ulmn) + (ul−1mn − ulmn)] C2 x x x x x x x x + 2 [(ul+1m+1n − ulmn) + (ul+1m−1n − ulmn) + (ul−1m+1n − ulmn) + (ul−1m−1n − ulmn) x x x x x x x x + (ul+1mn+1 − ulmn) + (ul+1mn−1 − ulmn) + (ul−1mn+1 − ulmn) + (ul−1mn−1 − ulmn) y y y y y y y y + (ul+1m+1n − ulmn) − (ul+1m−1n − ulmn) − (ul−1m+1n − ulmn) + (ul−1m−1n − ulmn) z z z z z z z z + (ul+1mn+1 − ulmn) − (ul+1mn−1 − ulmn) − (ul−1mn+1 − ulmn) + (ul−1mn−1 − ulmn)]

C1 is the spring constant for the nearest and C2 the spring constant for the next nearest neighbours. The expressions for the y- and z-directions are similar. By substituting the eigenfunction solutions into the equations of motion, the three coupled diﬀerential equations become algebraic equations:

2 mω ~ulmn = M~ulmn The phonon dispersion relation can be obtained by calculating the eigenvalues λ of the Matrix M: r λ ω = m

4  2C1[1 − cos (akx)] 2C2 sin (akx) sin (aky ) 2C2 sin (akx) sin (akz )  +2C2[2 − cos(akx)cos(aky ) − cos(akx)cos(akz )]         2C2 sin (akx) sin (aky ) 2C1[1 − cos (aky )] 2C2 sin (aky ) sin (akz )  M =    +2C2[2 − cos(akx)cos(aky ) − cos(aky )cos(akz )]       2C2 sin (akx) sin (akz ) 2C2 sin (aky ) sin (akz ) 2C1[1 − cos (akz )]  +2C2[2 − cos(akx)cos(akz ) − cos(aky )cos(akz )] 5 2 Numerical calculation in Matlab

2.1 Matlab Files The following Matlab program calculates and plots the phonon dispersion relation and density of states for simple cubic considering the nearest and next nearest neighbours. The calculation is performed for a set of diﬀerent quotients of the two spring constants C1 . C2

1 % Phonon dispersion relation and density of states for a simple cubic 2 % lattice using the linear spring model 3 4 %−−−−−p a r a m e t e r s −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 5 % dimensions 6 d = 3 ; 7 8 % lattice constant in real space in meters 9 a = 1 ; 10 11 % quotient of the spring constants for nearest & next nearest neighbours 12 % c q u o t = c1 / c2 13 c quot = [inf ,6,3,2,1]; %( g u e s s ) 14 15 % number of points in k−s p a c e 16 n = 1 0 0 0 ; % for plotting the dispersion relationship 17 n dos = 10000000; % for calculating the DoS 18 19 % number of histogram bins for the DoS 20 w bin = 1 5 0 ; 21 22 % brilloin zone symmetry points 23 g = [ 0 ; 0 ; 0 ] ; 24 x = [ 0 ; pi /a ; 0 ] ; 25 m = [ pi /a ; pi /a ; 0 ] ; 26 r = [ pi /a ; pi /a ; pi /a ] ; 27 28 % dispersion relation plot order (symmetry points) 29 po = [g,x,m,g,r]; 30 p o l a b e l = { ’G’ , ’X’ , ’M’ , ’G’ , ’R’ } ; 31 32 %−−−−−phonon dispersion relation −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 33 % creating n linearly spaced k vectors between each pair of symmetry 34 % p o i n t s : 35 n po = s i z e ( po , 2 ) −1; 36 k = nan ( d , ( n−1)∗ n po +1) ; % kx ; ky ; k z 37 kk = zeros ( 1 , s i z e ( k , 2 ) ) ; % length of path in k−space for plot axis 38 k k s = zeros ( 1 , n po +1) ; % values of kk at the symmetry points 39 40 k(:,1) = po(:,1); 41 kk ( 1 ) = k k s ( 1 ) ; 42 for ind1 = 1 : n po 43 44 i n d 1 start = (ind1 −1) ∗( n−1)+2; 45 i n d 1 e n d = ind1 ∗( n−1)+1; 46 47 k k s(ind1+1) = kk s ( ind1 )+norm(po(: ,ind1+1)−po(: ,ind1)); 48 temp kk = linspace ( k k s ( ind1 ) , k k s(ind1+1),n); 49 kk ( 1 , i n d 1 s t a r t : i n d 1 e n d ) = temp kk ( 2 : end ); 50 51 for ind2 = 1 : d 52 temp k = linspace (po(ind2 ,ind1) ,po(ind2 ,ind1+1),n); 53 k(ind2 , ind1 s t a r t : i n d 1 e n d ) = temp k ( 2 : end ); 54 end 55 end 56 57 58 % calculating the frequencies omega/(sqrt(C/m)) for each k from the dispersion relation: 59 % nearest neighbours 60 w1 = f u n disp1(k,a);

6 61 62 % nearest & next nearest neighbours 63 n u m e l c quot = numel(c q u o t ) ; 64 w2 = nan ( s i z e ( k , 1 ) , s i z e (k,2) ,numel c q u o t ) ; 65 for ind1 = 1:numel c q u o t 66 w2(:,:,ind1) = fun disp2(k,a,c quot(ind1)); 67 end 68 69 70 71 % plotting the dispersion relation: 72 % nearest neighbours 73 ymax1 = round (11∗max(w1(:)))/10; 74 f i gu r e ( 1 ) 75 subplot (2 ,1+ numel c quot , 1 ) 76 plot ( kk , w1 , ’ b− ’) 77 for ind1 = 1 : ( n po +1) 78 l i n e ( [ k k s ( ind1 ) k k s(ind1)],[0 ymax1], ’Color’,’r’) 79 end 80 ylim([0 ymax1]) 81 xlim([kk(1) kk( end )]) 82 t i t l e (’Phonon dispersion relation of sc, nearest neighbours’) 83 ylabel (’ \ omega {norm} = \omega / sqrt(C/m) ’) 84 set ( gca , ’XTick’ ,kk s ) 85 set ( gca , ’XTickLabel ’ ,po l a b e l ) 86 87 % nearest and next neighbours 88 for ind1 = 1:numel c q u o t 89 ymax2 = round (11∗max(max(w2) ) ) / 1 0 ; 90 subplot (2 ,1+ numel c quot ,1+ind1) 91 plot (kk,w2(: ,: ,ind1),’b− ’) 92 for ind2 = 1 : ( n po +1) 93 l i n e ( [ k k s ( ind2 ) k k s(ind2)],[0 ymax2(ind1)], ’Color’,’r’) 94 end 95 ylim([0 ymax2(ind1)]) 96 xlim([kk(1) kk( end )]) 97 t i t l e ({ ’Phonon dispersion relation of sc,’; ’nearest and next nearest neighbours’;[ ’C1/C2 = ’, num2str ( c quot(ind1))] } ) 98 ylabel (’ \ omega {norm} = \omega / sqrt(C 1 /m) ] ’ ) 99 set ( gca , ’XTick’ ,kk s ) 100 set ( gca , ’XTickLabel ’ ,po l a b e l ) 101 end 102 103 104 %−−−−−density of states −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− 105 % choosing random k vectors in the first brilloin zone 106 k rand = 2∗ pi /a ∗( rand ( d , n do s ) −0.5) ; 107 108 % calculating the frequencies from the dispersion relation: 109 % nearest neighbours 110 w1 rand = f u n d i s p 1 ( k rand , a ) ; 111 112 % nearest & next nearest neighbours 113 w2 rand = nan ( s i z e ( k rand , 1 ) , s i z e ( k rand ,2) ,numel c q u o t ) ; 114 for ind1 = 1:numel c q u o t 115 w2 rand(:,:,ind1) = fun d i s p 2 ( k rand , a , c quot(ind1)); 116 end 117 118 119 120 % h i s t o g r a m 121 % nearest neighbours 122 [ dw1 , wn1 ] = h i s t ( w1 rand ( : ) , w bin ) ; 123 dos1 = dw1/ n do s ∗ w bin /(max( wn1 )−min( wn1 ) ) ∗ 1/( a ˆ3) ; %normalisation 124 125 subplot (2 ,1+ numel c quot ,2+numel c q u o t ) 126 plot ( wn1 , dos1 ) 127 xlim([0 wn1( end )]) 128 t i t l e (’Phonon density of states of sc, nearest neighbours’) 129 xlabel (’ \ omega {norm} = \omega / sqrt(C/m) ’) 130 ylabel (’D(\ omega {norm })’) 131 132 %DoS text file 133 f i d = fopen ( ’ s c d o s onlynearest.txt ’,’w’); 134 f p r i n t f ( f i d , ’%s \n’,’Phonon density of states of sc, only nearest neighbours’);

7 135 f p r i n t f ( f i d , ’%s \ t%s \n’ , ’omega/sqrt(C/m) ’ , ’D(omega/sqrt(C/m)) ’); 136 f p r i n t f ( f i d , ’%f \ t%f \n’, [wn1; dos1]); 137 f c l o s e ( f i d ) ; 138 139 140 % nearest & next nearest neighbours 141 dw2 = nan(numel c quot , w bin ) ; 142 wn2 = nan(numel c quot , w bin ) ; 143 dos2 = nan(numel c quot , w bin ) ; 144 for ind1 = 1:numel c q u o t 145 w2 rand ind 1 = w2 rand(: ,: ,ind1); 146 [dw2(ind1 ,:) ,wn2(ind1 ,:) ] = h i s t ( w2 ran d ind 1 ( : ) , w bin ) ; 147 dos2(ind1 ,:) = dw2(ind1 ,:)/n do s ∗ w bin /(max(wn2(ind1 ,:) )−min(wn2(ind1 ,:) )) ∗ 1/( a ˆ3) ; % normalisation 148 149 subplot (2 ,1+ numel c quot ,2+numel c q u o t+ind1 ) 150 plot (wn2(ind1 ,:) ,dos2(ind1 ,:) ) 151 xlim([0 wn2(ind1 , end )]) 152 t i t l e ({ ’Phonon density of states of sc,’; ’nearest and next nearest neighbours’;[ ’C2/C1 = ’, num2str ( c quot(ind1))] } ) 153 xlabel (’ \ omega {norm} = \omega / sqrt(C 1 /m) ’ ) 154 ylabel (’D(\ omega {norm })’) 155 156 %DoS text file 157 f i d = fopen ( [ ’ s c d o s ’, num2str ( c quot(ind1)),’.txt’], ’w’); 158 f p r i n t f ( f i d , ’%s \n’,’Phonon density of states of sc, nearest and next nearest neighbours’); 159 f p r i n t f ( f i d , ’%s%s \n’,’C2/C1 = ’, num2str ( c quot(ind1))); 160 f p r i n t f ( f i d , ’%s \ t%s \n’, ’omega/sqrt(C 1/m) ’ , ’D(omega/sqrt(C 1 /m) ) ’ ) ; 161 f p r i n t f ( f i d , ’%f \ t%f \n’, [wn2(ind1 ,:); dos2(ind1 ,:) ]); 162 f c l o s e ( f i d ) ; 163 end

1 function [w] = f u n d i s p 1 ( k , a ) 2 % Phonon dispersion relation for a simple cubic lattice 3 % in the linear spring model considering only the nearest neighbours. 4 % Input: matrix k with wavenumber column vectors [kx;ky;kz], 5 % lattice constant a 6 % Output: frequencies w = [w1;w2;w3] 7 8 w = 2∗abs ( sin ( k∗a /2) ) ; 9 10 end

1 function [w] = f u n disp2(k,a,c q u o t ) 2 % Phonon dispersion relation for a simple cubic lattice in the linear 3 % spring model considering the nearest and next nearest neighbours. 4 % Input: matrix k with wavenumber column vectors [kx;ky;kz], 5 % lattice constant a, spring constant quotient c q u o t 6 % Output: frequencies w = [w1;w2;w3] 7 8 w = nan ( s i z e ( k ) ) ; 9 M = nan ( 3 ) ; 10 11 for ind1 = 1 : s i z e ( k , 2 ) 12 % Berechnung der Matrix M 13 M( 1 , 1 ) = 2∗(1− cos ( a∗k(1,ind1)))+2/c q u o t ∗(2−cos ( a∗k ( 1 , ind1 ) ) ∗ cos ( a∗k ( 2 , ind1 ) )−cos ( a∗k ( 1 , ind1 ) ) ∗ cos ( a∗k(3,ind1))); 14 M( 1 , 2 ) = 2/ c q u o t ∗ sin ( a∗k ( 1 , ind1 ) ) ∗ sin ( a∗k(2,ind1)); 15 M( 1 , 3 ) = 2/ c q u o t ∗ sin ( a∗k ( 1 , ind1 ) ) ∗ sin ( a∗k(3,ind1)); 16 M(2,1) = M(1,2); 17 M( 2 , 2 ) = 2∗(1− cos ( a∗k(2,ind1)))+2/c q u o t ∗(2−cos ( a∗k ( 1 , ind1 ) ) ∗ cos ( a∗k ( 2 , ind1 ) )−cos ( a∗k ( 2 , ind1 ) ) ∗ cos ( a∗k(3,ind1))); 18 M( 2 , 3 ) = 2/ c q u o t ∗ sin ( a∗k ( 2 , ind1 ) ) ∗ sin ( a∗k(3,ind1)); 19 M(3,1) = M(1,3); 20 M(3,2) = M(2,3); 21 M( 3 , 3 ) = 2∗(1− cos ( a∗k(3,ind1)))+2/c q u o t ∗(2−cos ( a∗k ( 1 , ind1 ) ) ∗ cos ( a∗k ( 3 , ind1 ) )−cos ( a∗k ( 2 , ind1 ) ) ∗ cos ( a∗k(3,ind1))); 22 23 %Berechnung von omega/sqrt(C1/m) aus den Eigenwerten von M: 24 w ( : , ind1 ) = sqrt ( eig (M) ) ; 25 end 26 27 end

8 2.2 Figures

Figure 1: Phonon dispersion relation for simple cubic, only nearest neighbours

Figure 2: Phonon density of states for simple cubic, only nearest neighbours

9 Figure 3: Phonon dispersion relation for simple cubic, nearest and next nearest neighbours, C1 = ∞ C2

Figure 4: Phonon density of states for simple cubic, nearest and next nearest neighbours, C1 = ∞ C2

10 Figure 5: Phonon dispersion relation for simple cubic, nearest and next nearest neighbours, C1 = 6 C2

Figure 6: Phonon density of states for simple cubic, nearest and next nearest neighbours, C1 = 6 C2

11 Figure 7: Phonon dispersion relation for simple cubic, nearest and next nearest neighbours, C1 = 3 C2

Figure 8: Phonon density of states for simple cubic, nearest and next nearest neighbours, C1 = 3 C2

12 Figure 9: Phonon dispersion relation for simple cubic, nearest and next nearest neighbours, C1 = 2 C2

Figure 10: Phonon density of states for simple cubic, nearest and next nearest neighbours, C1 = 2 C2

13 Figure 11: Phonon dispersion relation for simple cubic, nearest and next nearest neighbours, C1 = 1 C2

Figure 12: Phonon density of states for simple cubic, nearest and next nearest neighbours, C1 = 1 C2