Density of States

Introduction

In this example, we calculate the phonon (vibrational) density of states (DOS) of graphene at 300 K and zero pressure. The method is based on the velocity auto-correlation (VAC) function. The DOS is calculated as the Fourier transform of the VAC [Dickey 1969].

Importing Relevant Functions

The inputs/outputs for GPUMD are processed using the Atomic Simulation Environment (ASE).

[6]:

from pylab import *
from ase.build import graphene_nanoribbon
from ase.io import write

Preparing the Inputs

We use a sheet of graphene consisting of 8640 carbon atoms and use a Tersoff potential [Tersoff 1989] parameterized by Lindsay et al. [Lindsay 2010].

Generate the model.xyz file:

[2]:

gnr = graphene_nanoribbon(60, 36, type='armchair', sheet=True, vacuum=3.35/2, C_C=1.44)
gnr.euler_rotate(theta=90)
l = gnr.cell.lengths()
gnr.cell = gnr.cell.new((l[0], l[2], l[1]))
l = l[2]
gnr.center()
gnr.pbc = [True, True, False]
gnr

[2]:

Atoms(symbols='C8640', pbc=[True, True, False], cell=[149.64918977395098, 155.52, 3.35])

[3]:

write("model.xyz", gnr)

The first few lines of the model.xyz file are:

8640
Lattice="149.64918977395098 0.0 0.0 0.0 155.52 0.0 0.0 0.0 3.35" Properties=species:S:1:pos:R:3 pbc="T T F"
C      147.77857490       0.72000000       1.67500000
C      149.02565148       1.44000000       1.67500000
C      149.02565148       2.88000000       1.67500000
C      147.77857490       3.60000000       1.67500000

Explanation for the first line:

  • The first number states that the number of particles is 8640.

  • Explanation for the second line:

    • This line consists of a number of keyword=value pairs separated by spaces. Spaces before and after = are allowed. All the characters are case-insensitive.

    • lattice="ax ay az bx by bz cx cy cz" gives the box vectors.

    • properties=property_name:data_type:number_of_columns. We only read the following items:    -species:S:1atom type (Mandatory)    -pos:R:3position vector (Mandatory)    -mass:R:1mass (Optional: default mass values will be used when this is missing), not included here.    -vel:R:3velocity vector (Optional), not included here.    -group:I:number_of_grouping_methods` grouping methods (Optional), not included here.

The run.in file:

The run.in input file is given below:

potential   ../../../potentials/tersoff/Graphene_Lindsay_2010_modified.txt
velocity    300

ensemble    npt_ber 300 300 100 0 0 0 53.4059 53.4059 53.4059 2000
time_step   1
dump_thermo 100
run         200000

ensemble    nve
compute_dos 5 200 400.0
run         200000

The first line uses the potential keyword to define the potential to be used, which is specified in the file Graphene_Lindsay_2010_modified.txt.

The second line uses the velocity keyword and sets the velocities to be initialized with a temperature of 300 K.

There are two runs:

  • The first run serves as the equilibration stage, where the NPT ensemble (the Berendsen method) is used. The temperature is 300 K and the pressures are zero in all the directions. The coupling constants are 100 and 2000 time steps for the thermostat and the barostat (The elastic constant, or inverse compressibility parameter needed in the barostat is estimated to be 53.4059 GPa; this only needs to be correct up to the order of magnitude.), respectively. The time_step for integration is 1 fs. There are \(2\times 10^5\) steps (200 ps) for this run and the thermodynamic quantities will be output every 1000 steps.

  • The second run serves as the production run, where the NVE ensemble is used. The line with compute_dos means that velocities will be recorded every 5 steps (5 fs) and 200 VAC data (the maximum correlation time is then about 1 ps) will be calculated. The last parameter in this line is the maximum angular frequency considered, \(\omega_{\rm max} = 2\pi\nu_{\rm max} =400\) THz, which is large enough for graphene. The production run lasts 200 ps.

Results and Discussion

Computation Time

Using a GeForce RTX 2080 Ti GPU, the NEMD simulation takes about 1.5 minutes. The speed of this simulation is about \(3.3\times 10^7\) atom x step / second.

Figure Properties

[4]:

aw = 2
fs = 16
font = {'size'   : fs}
matplotlib.rc('font', **font)
matplotlib.rc('axes' , linewidth=aw)

def set_fig_properties(ax_list):
    tl = 8
    tw = 2
    tlm = 4

    for ax in ax_list:
        ax.tick_params(which='major', length=tl, width=tw)
        ax.tick_params(which='minor', length=tlm, width=tw)
        ax.tick_params(which='both', axis='both', direction='in', right=True, top=True)

Plot DOS and VAC

The dos.out and mvac.out output files are loaded and processed to create the following figure.

We transform the angular frequency \(\omega\) to regular frequency \(\nu\) using \(\nu\) = \(\frac{\omega}{2\pi}\).

[23]:

dos_array = np.loadtxt("dos.out")
dos = {}
dos["omega"], dos['DOSx'], dos['DOSy'], dos['DOSz'] = dos_array[:,0], dos_array[:,1], dos_array[:,2], dos_array[:,3]
dos['DOSxyz'] = dos['DOSx']+dos['DOSy']+dos['DOSz']
dos["nu"] = dos["omega"] / (2 * np.pi)

vac_array = np.loadtxt("mvac.out")
vac = {}
vac["t"], vac['VACx'], vac['VACy'], vac['VACz'] = vac_array[:,0], vac_array[:,1], vac_array[:,2], vac_array[:,3]
vac['VACxyz'] = vac['VACx']+vac['VACy']+vac['VACz']
vac['VACxyz'] /= vac['VACxyz'].max()

print('DOS:', list(dos.keys()))
print('VAC:', list(vac.keys()))

DOS: ['omega', 'DOSx', 'DOSy', 'DOSz', 'DOSxyz', 'nu']
VAC: ['t', 'VACx', 'VACy', 'VACz', 'VACxyz']

[24]:

figure(figsize=(12,10))
subplot(2,2,1)
set_fig_properties([gca()])
plot(vac['t'], vac['VACx']/vac['VACx'].max(), color='C3',linewidth=3)
plot(vac['t'], vac['VACy']/vac['VACy'].max(), color='C0', linestyle='--',linewidth=3)
plot(vac['t'], vac['VACz']/vac['VACz'].max(), color='C2', linestyle='-.',zorder=100,linewidth=3)
xlim([0,1])
gca().set_xticks([0,0.5,1])
ylim([-0.5, 1])
gca().set_yticks([-0.5,0,0.5,1])
ylabel('VAC (Normalized)')
xlabel('Correlation Time (ps)')
legend(['x','y', 'z'])
title('(a)')

subplot(2,2,2)
set_fig_properties([gca()])
plot(dos['nu'], dos['DOSx'], color='C3',linewidth=3)
plot(dos['nu'], dos['DOSy'], color='C0', linestyle='--',linewidth=3)
plot(dos['nu'], dos['DOSz'], color='C2', linestyle='-.',zorder=100, linewidth=3)
xlim([0, 60])
gca().set_xticks(range(0,61,20))
ylim([0, 1500])
gca().set_yticks(np.arange(0,1501,500))
ylabel('PDOS (1/THz)')
xlabel(r'$\nu$ (THz)')
legend(['x','y', 'z'])
title('(b)')

subplot(2,2,3)
set_fig_properties([gca()])
plot(vac['t'], vac['VACxyz'], color='k',linewidth=3)
xlim([0,1])
gca().set_xticks([0,0.5,1])
ylim([-0.5, 1])
gca().set_yticks([-0.5,0,0.5,1])
ylabel('VAC (Normalized)')
xlabel('Correlation Time (ps)')
title('(c)')

subplot(2,2,4)
set_fig_properties([gca()])
plot(dos['nu'], dos['DOSxyz'], color='k',linewidth=3)
xlim([0, 60])
gca().set_xticks(range(0,61,20))
ylim([0, 2500])
gca().set_yticks(np.arange(0,2501,500))
ylabel('PDOS (1/THz)')
xlabel(r'$\nu$ (THz)')
title('(d)')

tight_layout()
show()
../_images/tutorials_density_of_states_13_0.png

(a) Normalized VAC for individual directions. (b) DOS for individual directions. (c) Total Normalized VAC. (d) Total DOS.

For 3D isotropic systems, the results along different directions are equivalent and can be averaged, but for 2D materials like graphene, it is natural to consider the in-plane part (the \(x\) and \(y\) directions in the simulation) and the out-of-plane part (the \(z\) direction) separately. It can be seen that the two components behave very differently. We can see that the cutoff frequency for the out-of-plane component (about 40 THz) is smaller than that for the in-plane component (about 52 THz).

Plot Quantum-Corrected Heat Capacity

In classical MD simulations, the heat capacity per atom is almost \(k_{\rm B}\) even at temperatures that are much lower than the Debye temperature. With the DOS available, one can obtaine the following quantum heat capacity per atom in direction \(\alpha\):

\[C_{\alpha}(T) = \int_0^{\infty} \frac{d\omega}{2\pi} \rho_{\alpha}(\omega) \frac{x^2 \exp(x)}{(\exp(x)-1)^2},\]

where

\[x=\frac{\hbar\omega}{k_{\rm B} T}\]

and \(\rho_{\alpha}(\omega)\) is the density of states in direction \(\alpha\) normalized to 1:

\[\int_0^{\infty} \frac{d\omega}{2\pi} \rho_{\alpha}(\omega) = 1.\]
[25]:

temperatures = np.arange(100,5001,100)  # [K]
num_atoms = len(gnr)
Cx, Cy, Cz = list(), list(), list()  # [k_B/atom] Heat capacity per atom
hnu = 6.63e-34*dos['nu']*1.e12  # [J]

for temperature in temperatures:
    kBT = 1.38e-23*temperature  # [J]
    x = hnu/kBT
    expr = np.square(x)*np.exp(x)/(np.square(np.expm1(x)))
    Cx.append(np.trapz(dos['DOSx']*expr, dos['nu'])/num_atoms)
    Cy.append(np.trapz(dos['DOSy']*expr, dos['nu'])/num_atoms)
    Cz.append(np.trapz(dos['DOSz']*expr, dos['nu'])/num_atoms)

[26]:

figure(figsize=(8,6))
set_fig_properties([gca()])
mew, ms, mfc, lw = 1, 8, 'none', 2.5
plot(temperatures, Cx, lw=lw,marker='d',mfc=mfc,ms=ms,mew=mew)
plot(temperatures, Cy, lw=lw,marker='s',mfc=mfc,ms=ms,mew=mew)
plot(temperatures, Cz, lw=lw,marker='o',mfc=mfc,ms=ms,mew=mew)
xlim([0,5100])
gca().set_xticks(range(0,5001,1000))
ylim([0, 1.1])
gca().set_yticks(np.linspace(0,1,6))
ylabel(r'Heat Capacity (k$_B$/atom)')
xlabel('Temperature (K)')
legend(['x','y','z'])

tight_layout()
show()
../_images/tutorials_density_of_states_18_0.png

Quantum heat capacity per atom as a function of temperature.

The above figure shows the calculated per-atom quantum heat capacity in different directions.

Again, the in-plane (\(x\) and \(y\) directions) and out-of-plane (\(z\) direction) phonons behave differently.

For every direction, the quantum heat capacity increases from 0 to \(k_{\rm B}\) with increasing temperature.

One can also calculate the Debye temperature as a function of temperature \(\Theta(T)\), but we leave it to the reader.

References