Phonon dispersion relation - Revision history

Yanmingw: /* Background knowledge */

2012-06-19T17:46:05Z

Background knowledge

← Older revision		Revision as of 17:46, 19 June 2012
Line 9:		Line 9:
	==Background knowledge==		==Background knowledge==

	The following links may be helpful as a ~~starting~~ ~~point~~.		The following links may be helpful to read at first.

	[http://en.wikipedia.org/wiki/Phonon What is phonon?]		[http://en.wikipedia.org/wiki/Phonon What is phonon?]

Yanmingw at 01:51, 19 June 2012

2012-06-19T01:51:49Z

← Older revision		Revision as of 01:51, 19 June 2012
Line 1:		Line 1:
	<P ALIGN="CENTER">		<P ALIGN="CENTER">
	<FONT SIZE="+3" color="darkred"><STRONG>		<FONT SIZE="+3" color="darkred"><STRONG>
	Phonon Dispersion Relation</STRONG></font></P>		Phonon Dispersion Relation of graphene</STRONG></font></P>
	<DIV>		<DIV>
	<P ALIGN="CENTER"><STRONG>Yanming Wang, Saad Bhamla and Wei Cai</STRONG></P>		<P ALIGN="CENTER"><STRONG>Yanming Wang, Saad Bhamla and Wei Cai</STRONG></P>
Line 7:		Line 7:
	<BR><HR>		<BR><HR>

			==Background knowledge==
	==METHODS==

⚫	===Small displacement method for calculating the phonon dispersion relation===
			The following links may be helpful as a starting point.

			[http://en.wikipedia.org/wiki/Phonon What is phonon?]

			[http://en.wikipedia.org/wiki/Phonon#Dispersion_relation What is phonon dispersion relation?]

			[http://en.wikipedia.org/wiki/Planewave What is plane wave?]

		⚫	==Small displacement method for calculating the phonon dispersion relation==
	For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1		For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1

Yanmingw at 00:46, 19 June 2012

2012-06-19T00:46:53Z

← Older revision		Revision as of 00:46, 19 June 2012
Line 82:		Line 82:

	We observe that the REBO potential captures the acoustic modes quite well, but there is a slight deviation in the optical modes. This has also been reported in other literatures<ref>Zhang, H.; Lee, G.; Cho, K. Physical Review B 2011, 84,1-5</ref>. Since it is known that the thermal conductivity is mainly dependent on the acoustic modes, this strategy is satisfactory.		We observe that the REBO potential captures the acoustic modes quite well, but there is a slight deviation in the optical modes. This has also been reported in other literatures<ref>Zhang, H.; Lee, G.; Cho, K. Physical Review B 2011, 84,1-5</ref>. Since it is known that the thermal conductivity is mainly dependent on the acoustic modes, this strategy is satisfactory.

			<references/>

Yanmingw: /* TEST CASE */

2012-06-19T00:45:45Z

TEST CASE

← Older revision		Revision as of 00:45, 19 June 2012
Line 80:		Line 80:

	The three phonon dispersion branches, which originate from the <math>\Gamma</math>-point of the Brilliouin zone correspond to acoustic modes: an out-of plane mode (ZA), an in-plane transverse mode (TA), and in-plane longitudinal (LA), listed in order of increasing energy. The remaining three branches correspond to optical modes: one out-of plane mode (ZO), and two in-plane modes (TO) and (LO).		The three phonon dispersion branches, which originate from the <math>\Gamma</math>-point of the Brilliouin zone correspond to acoustic modes: an out-of plane mode (ZA), an in-plane transverse mode (TA), and in-plane longitudinal (LA), listed in order of increasing energy. The remaining three branches correspond to optical modes: one out-of plane mode (ZO), and two in-plane modes (TO) and (LO).

			We observe that the REBO potential captures the acoustic modes quite well, but there is a slight deviation in the optical modes. This has also been reported in other literatures<ref>Zhang, H.; Lee, G.; Cho, K. Physical Review B 2011, 84,1-5</ref>. Since it is known that the thermal conductivity is mainly dependent on the acoustic modes, this strategy is satisfactory.

Yanmingw at 00:41, 19 June 2012

2012-06-19T00:41:03Z

← Older revision		Revision as of 00:41, 19 June 2012
Line 78:		Line 78:
	You may want to compare this result with DFT calculation. Here is the plot. The red curves are from this small displacement method using REBO potential and the black dots are the data from DFT calculation.		You may want to compare this result with DFT calculation. Here is the plot. The red curves are from this small displacement method using REBO potential and the black dots are the data from DFT calculation.
	[[Image:Dispersion_curve.png\|500px]]		[[Image:Dispersion_curve.png\|500px]]

			The three phonon dispersion branches, which originate from the <math>\Gamma</math>-point of the Brilliouin zone correspond to acoustic modes: an out-of plane mode (ZA), an in-plane transverse mode (TA), and in-plane longitudinal (LA), listed in order of increasing energy. The remaining three branches correspond to optical modes: one out-of plane mode (ZO), and two in-plane modes (TO) and (LO).

Yanmingw at 00:35, 19 June 2012

2012-06-19T00:35:25Z

← Older revision		Revision as of 00:35, 19 June 2012
Line 74:		Line 74:

	In the end, you will have the same figure shown below,		In the end, you will have the same figure shown below,
	[[Image:W_k.jpg]]		[[Image:W_k.jpg\|500px]]

	You may want to compare this result with DFT calculation. Here is the plot. The red curves are from this small displacement method using REBO potential and the black dots are the data from DFT calculation.		You may want to compare this result with DFT calculation. Here is the plot. The red curves are from this small displacement method using REBO potential and the black dots are the data from DFT calculation.
	[[Image:Dispersion_curve.png]]		[[Image:Dispersion_curve.png\|500px]]

Yanmingw at 00:31, 19 June 2012

2012-06-19T00:31:22Z

← Older revision		Revision as of 00:31, 19 June 2012
Line 1:		Line 1:
	<H1 ALIGN="CENTER"~~>Phonon Dispersion Relation</H1~~>		<P ALIGN="CENTER">
			<FONT SIZE="+3" color="darkred"><STRONG>
			Phonon Dispersion Relation</STRONG></font></P>
	<DIV>		<DIV>
	<P ALIGN="CENTER"><STRONG>Yanming Wang, Saad Bhamla and Wei Cai</STRONG></P>		<P ALIGN="CENTER"><STRONG>Yanming Wang, Saad Bhamla and Wei Cai</STRONG></P>
		⚫	<P ALIGN="CENTER">June, 2012</P>
	</DIV>
⚫	<P ALIGN="CENTER">~~May~~, 2012</P>
	<BR><HR>		<BR><HR>

	=METHODS=		==METHODS==
	==Small displacement method for calculating the phonon dispersion relation==		===Small displacement method for calculating the phonon dispersion relation===
	For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1		For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1

Line 59:		Line 60:
	Equation 6 is an eigenfunction where <math>\xi</math> represents the eigenvalues, and <math>\vec{A}</math> represents the eigenvectors of the dynamical matrix <math>Q(\vec{k})</math>. We may notice each given <math>\vec{k}</math> coressponds to a unique <math>Q(\vec{k)}</math> and by solving the eigenvalues of <math>Q(\vec{k)}</math>, finally we can obtain the relation between <math>\vec{k}</math> and <math> \omega </math>, which is the '''phonon dispersion relation'''.		Equation 6 is an eigenfunction where <math>\xi</math> represents the eigenvalues, and <math>\vec{A}</math> represents the eigenvectors of the dynamical matrix <math>Q(\vec{k})</math>. We may notice each given <math>\vec{k}</math> coressponds to a unique <math>Q(\vec{k)}</math> and by solving the eigenvalues of <math>Q(\vec{k)}</math>, finally we can obtain the relation between <math>\vec{k}</math> and <math> \omega </math>, which is the '''phonon dispersion relation'''.

	=TEST CASE=		==TEST CASE==
	The '''tcl''' script [[media:Graphene_md.tcl.txt‎ \| Graphene_md.tcl]] can provide the atoms' configurations and the '''Hessian Matrix''' for graphene dispersion curve calculation. First, you need to compile the REBO potential in MD++ using the command like,		The '''tcl''' script [[media:Graphene_md.tcl.txt‎ \| Graphene_md.tcl]] can provide the atoms' configurations and the '''Hessian Matrix''' for graphene dispersion curve calculation. First, you need to compile the REBO potential in MD++ using the command like,

Yanmingw at 00:26, 19 June 2012

2012-06-19T00:26:19Z

← Older revision		Revision as of 00:26, 19 June 2012
Line 6:		Line 6:
	<BR><HR>		<BR><HR>

	==METHODS==		=METHODS=
	===Small displacement method for calculating the phonon dispersion relation===		==Small displacement method for calculating the phonon dispersion relation==
	For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1		For a crystal lattice composed of a number of atoms bound by a specific potential, an equilibrium or minimum energy state can be reached by relaxing the structure. This is achieved using the conjugate gradient relaxation method in MD++. Once the local minimum is reached, a Taylor expansion is used around this state in terms of the atomic displacements which gives Equation 1

Line 59:		Line 59:
	Equation 6 is an eigenfunction where <math>\xi</math> represents the eigenvalues, and <math>\vec{A}</math> represents the eigenvectors of the dynamical matrix <math>Q(\vec{k})</math>. We may notice each given <math>\vec{k}</math> coressponds to a unique <math>Q(\vec{k)}</math> and by solving the eigenvalues of <math>Q(\vec{k)}</math>, finally we can obtain the relation between <math>\vec{k}</math> and <math> \omega </math>, which is the '''phonon dispersion relation'''.		Equation 6 is an eigenfunction where <math>\xi</math> represents the eigenvalues, and <math>\vec{A}</math> represents the eigenvectors of the dynamical matrix <math>Q(\vec{k})</math>. We may notice each given <math>\vec{k}</math> coressponds to a unique <math>Q(\vec{k)}</math> and by solving the eigenvalues of <math>Q(\vec{k)}</math>, finally we can obtain the relation between <math>\vec{k}</math> and <math> \omega </math>, which is the '''phonon dispersion relation'''.

	==TEST CASE==		=TEST CASE=
	The '''tcl''' script [[media:Graphene_md.tcl.txt‎ \| Graphene_md.tcl]] can provide the atoms' configurations and the '''Hessian Matrix''' for graphene dispersion curve calculation. First, you need to compile the REBO potential in MD++ using the command like,		The '''tcl''' script [[media:Graphene_md.tcl.txt‎ \| Graphene_md.tcl]] can provide the atoms' configurations and the '''Hessian Matrix''' for graphene dispersion curve calculation. First, you need to compile the REBO potential in MD++ using the command like,

Yanmingw: /* TEST CASE */

2012-06-15T06:38:21Z

TEST CASE

← Older revision		Revision as of 06:38, 15 June 2012
Line 71:		Line 71:

	After the simulation, you can find two files named as '''perf.cn''' and '''hessian.out'''. The matlab script [[media:Graphene_phonon.m.txt‎ \| Graphene_phonon.m]] can extract data from the files and draw the phonon dispersion curve for graphene.		After the simulation, you can find two files named as '''perf.cn''' and '''hessian.out'''. The matlab script [[media:Graphene_phonon.m.txt‎ \| Graphene_phonon.m]] can extract data from the files and draw the phonon dispersion curve for graphene.

			In the end, you will have the same figure shown below,
			[[Image:W_k.jpg]]

			You may want to compare this result with DFT calculation. Here is the plot. The red curves are from this small displacement method using REBO potential and the black dots are the data from DFT calculation.
			[[Image:Dispersion_curve.png]]

Yanmingw: /* TEST CASE */

2012-06-15T06:29:27Z

TEST CASE

← Older revision		Revision as of 06:29, 15 June 2012
Line 60:		Line 60:

	==TEST CASE==		==TEST CASE==
			The '''tcl''' script [[media:Graphene_md.tcl.txt‎ \| Graphene_md.tcl]] can provide the atoms' configurations and the '''Hessian Matrix''' for graphene dispersion curve calculation. First, you need to compile the REBO potential in MD++ using the command like,
	[[media:Graphene_phonon.m.txt‎ \| Graphene_phonon.m]] [[media:Graphene_md.tcl.txt‎ \| Graphene_md.tcl]].

			make rebo build=R SYS=mac

			Please refer the details for compiling to [[Media: http://micro.stanford.edu/wiki/M01_Introduction_to_MD%2B%2B \| Introduction to MD++]].

			Then the following command can be run to generate the data for the calculation.

			bin/rebo_mac scripts/ME346/graphene_md.tcl 0 0

			After the simulation, you can find two files named as '''perf.cn''' and '''hessian.out'''. The matlab script [[media:Graphene_phonon.m.txt‎ \| Graphene_phonon.m]] can extract data from the files and draw the phonon dispersion curve for graphene.