Tight-binding model of graphene
This tutorial aims at introducing the tight-binding functionalities available in PyBigDFT. The TB module currently enables to:
define the geometry of a given system
extract the corresponding matrix elements from a linear-scaling calculation
compute the subsequent band structure
In a nutshell, a comparison is performed between a sub-system (sys_cs) and a supercell (sys_ls) on which a LS run was carried out. This process depends on the local electronic environment, thus enabling to probe the variation due to defects in a solid-state system.
[1]:
from BigDFT import Systems, Fragments, Logfiles, TB
from BigDFT.PostProcessing import BigDFTool
from BigDFT.Spillage import MatrixMetadata
[2]:
from matplotlib import cm
from matplotlib import pyplot as plt
from matplotlib.colors import LogNorm, Normalize
[3]:
import numpy as np
from scipy.constants import value
sq3 = np.sqrt(3)
Ha2eV = value('Hartree energy in eV') # (A)
/opt/intel/oneapi/intelpython/latest/lib/python3.9/site-packages/scipy/__init__.py:146: UserWarning: A NumPy version >=1.16.5 and <1.23.0 is required for this version of SciPy (detected version 1.23.5
warnings.warn(f"A NumPy version >={np_minversion} and <{np_maxversion}"
Geometry definition
Importantly, consistency is required between both systems lattice constants, so that geometrical correspondence can be established.
[4]:
a0 = 2.5 # lattice parameter
name = 'gr_scell' # supercell label
Definition of the primitive and conventional cells of an hexagonal lattice. The primitive cell is used to browse the TB workflow, while the conventional one is employed for the recap at the end of this tutorial.
[5]:
coord = {'prim': np.array([[0,0,0],
[a0/sq3,0,0]]),
'conv': np.array([[0,0,0],
[a0/sq3,0,0],
[a0*sq3/2,0,a0/2],
[a0*5/sq3/2,0,a0/2]])}
cell = {'prim': np.array([[a0*sq3/2,0,a0/2],
[0,float("inf"),0],
[a0*sq3/2,0,-a0/2]]),
'conv': np.array([[a0*sq3,0,0],
[0,float("inf"),0],
[0,0,a0]])}
From the definitions of atomic coordinates and the unit cell, the sub-system is constructed. Remember that Systems are collections of Fragments in BigDFT. In the following, the solid-state system is defined as one fragment.
[6]:
positions = [{'C':list(j)} for j in coord['prim']]
posinp = {'positions': positions, 'units': 'angstroem'}
frag = Fragments.Fragment(posinp=posinp)
sys_cs = Systems.System()
sys_cs["FRA:1"] = frag
Since non-orthorombic cells are not yet handled in BigDFT, the following procedure is required when defining the unit cell
[7]:
sys_cs.cell.cell = cell['prim']
Tight-binding parameters
The TB parametrization relies on definition on interaction region, where electronic contributions are added to the Hamiltonian. This approach is similar to the localization constraint employed in linear scaling. In addition to a Systems.System, a TB object need an interaction radius \(d\).
[8]:
d = 5 # 1nn or 3nn are advised for graphene
tb = TB.TightBinding(sys_cs, d=d)
Mapping between sites and Bravais vectors
From the previous parametrization, a mapping is established between atoms in the unit cell and their periodic images within a distance \(d\). The Bravais vectors connecting those sites are denoted \(R_{ij}\). The TB object therefore contains a dictionary \(R_{sh}\) defined such as:
\(R_{sh}\) = {…, {(\(i\),\(j\),(\(n_{ij}\))): \(R_{ij}\)}, …}
where \(i\), \(j\) are atoms sites, \(n_{ij}\) are the Bravais indices.
For example, the nearest-neighbors (\(1nn\)) model in graphene is defined as a dictionary of 8 entries, i.e. 2 on-site terms and 3 pairs of 1\(nn\) terms.
[9]:
tb._sites_in_shell(d=1.5)
[9]:
{(0, 0, (0, 0, 0)): [0.0, 0.0, 0.0],
(0, 1, (-1, 0, 0)): [-0.7216878364870318, 0.0, -1.25],
(0, 1, (0, 0, -1)): [-0.7216878364870318, 0.0, 1.25],
(0, 1, (0, 0, 0)): [1.4433756729740645, 0.0, 0.0],
(1, 0, (0, 0, 0)): [-1.4433756729740645, 0.0, 0.0],
(1, 0, (0, 0, 1)): [0.7216878364870318, 0.0, -1.25],
(1, 0, (1, 0, 0)): [0.7216878364870318, 0.0, 1.25],
(1, 1, (0, 0, 0)): [0.0, 0.0, 0.0]}
It can often be insightful to visualize the selected atoms of the TB model
[10]:
fig = plt.figure(figsize=(4,4))
ax = plt.gca()
x0,_,z0 = coord['prim'][0]
circle = plt.Circle((x0,z0), d, color='g', fill=False)
ax.add_patch(circle)
for (i,j,idk),r_ij in tb.R_sh.items():
if i>j:
continue
xi,_,zi = coord['prim'][i]
xj,_,zj = coord['prim'][i]+r_ij
plt.plot([xi,xj], [zi,zj], 'bo')
plt.plot([0,cell['prim'][0,0]], [0,cell['prim'][0,2]], 'r-')
plt.plot([0,cell['prim'][2,0]], [0,cell['prim'][2,2]], 'r-')
plt.axis('equal'); plt.axis('off'); plt.show()
Structure wrapping with supercell
Once the TB model is parametrized, a geometrical matching is established between the supercell and the sub-system
[11]:
if 'log' not in locals():
log = Logfiles.Logfile(f'log-{name}.yaml')
From the logfile, a Systems.System is loaded
[12]:
posinp = Logfiles.Logfile(f'{name}.yaml').astruct #['posinp']
sys_ls = Systems.system_from_dict_positions(posinp['positions'])
The primitive cell is then connected to the supercell through an atoms mapping. The supercell indices establish the correspondence between the two systems, with the associated MSE given to assess the matching quality.
[13]:
m_idx = tb.matching_index(sys_ls);
idx = m_idx['idx']; print(m_idx)
{'idx': [626, 209], 'mse': 8.845044099689063e-18}
Hamiltonian extraction
Now, the matrices elements of the associated atoms need to be extracted.
First, the hamiltonian and overlap matrices are loaded, as well as the metadata. We copy these matrices from the resource repository of BigDFT
[14]:
from futile.Utils import data_path, untar_archive
archive=f'data-{name}.tar.xz'
data_path(archive)
untar_archive(archive)
Executing: wget https://raw.githubusercontent.com/BigDFT-group/resources/main/datalake/data-gr_scell.tar.xz -O lfs.info
--2024-04-12 14:06:44-- https://raw.githubusercontent.com/BigDFT-group/resources/main/datalake/data-gr_scell.tar.xz
Resolving raw.githubusercontent.com (raw.githubusercontent.com)... 2606:50c0:8003::154, 2606:50c0:8002::154, 2606:50c0:8000::154, ...
Connecting to raw.githubusercontent.com (raw.githubusercontent.com)|2606:50c0:8003::154|:443... connected.
HTTP request sent, awaiting response... 200 OK
Length: 133 [text/plain]
Saving to: ‘lfs.info’
0K 100% 1.89M=0s
2024-04-12 14:06:44 (1.89 MB/s) - ‘lfs.info’ saved [133/133]
% Total % Received % Xferd Average Speed Time Time Time Current
Dload Upload Total Spent Left Speed
100 904 100 748 100 156 1897 395 --:--:-- --:--:-- --:--:-- 2294
Executing: wget https://github-cloud.githubusercontent.com/alambic/media/547350918/80/c0/80c09b6c1f74f890e4608c5780432295ff23f6a27f29b240f0bb37b9607b84de?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA5BA2674WPWWEFGQ5%2F20240412%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240412T140644Z&X-Amz-Expires=3600&X-Amz-Signature=58323c4ed9ca636868156d39a68977f56dcad8351dab538fe4b3d989f290e5e0&X-Amz-SignedHeaders=host&actor_id=0&key_id=0&repo_id=596478275&token=1 -O ./data-gr_scell.tar.xz
--2024-04-12 14:06:44-- https://github-cloud.githubusercontent.com/alambic/media/547350918/80/c0/80c09b6c1f74f890e4608c5780432295ff23f6a27f29b240f0bb37b9607b84de?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA5BA2674WPWWEFGQ5%2F20240412%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240412T140644Z&X-Amz-Expires=3600&X-Amz-Signature=58323c4ed9ca636868156d39a68977f56dcad8351dab538fe4b3d989f290e5e0&X-Amz-SignedHeaders=host&actor_id=0&key_id=0&repo_id=596478275&token=1
Resolving github-cloud.githubusercontent.com (github-cloud.githubusercontent.com)... 185.199.111.154, 185.199.110.154, 185.199.109.154, ...
Connecting to github-cloud.githubusercontent.com (github-cloud.githubusercontent.com)|185.199.111.154|:443... connected.
HTTP request sent, awaiting response... 200 OK
Length: 12322968 (12M) [application/octet-stream]
Saving to: ‘./data-gr_scell.tar.xz’
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9200K .......... .......... .......... .......... .......... 76% 69.6M 0s
9250K .......... .......... .......... .......... .......... 77% 3.94M 0s
9300K .......... .......... .......... .......... .......... 77% 3.89M 0s
9350K .......... .......... .......... .......... .......... 78% 60.7M 0s
9400K .......... .......... .......... .......... .......... 78% 4.03M 0s
9450K .......... .......... .......... .......... .......... 78% 54.1M 0s
9500K .......... .......... .......... .......... .......... 79% 3.98M 0s
9550K .......... .......... .......... .......... .......... 79% 4.08M 0s
9600K .......... .......... .......... .......... .......... 80% 44.6M 0s
9650K .......... .......... .......... .......... .......... 80% 4.01M 0s
9700K .......... .......... .......... .......... .......... 81% 44.0M 0s
9750K .......... .......... .......... .......... .......... 81% 4.01M 0s
9800K .......... .......... .......... .......... .......... 81% 62.2M 0s
9850K .......... .......... .......... .......... .......... 82% 3.92M 0s
9900K .......... .......... .......... .......... .......... 82% 3.83M 0s
9950K .......... .......... .......... .......... .......... 83% 11.2M 0s
10000K .......... .......... .......... .......... .......... 83% 5.45M 0s
10050K .......... .......... .......... .......... .......... 83% 4.02M 0s
10100K .......... .......... .......... .......... .......... 84% 13.2M 0s
10150K .......... .......... .......... .......... .......... 84% 5.45M 0s
10200K .......... .......... .......... .......... .......... 85% 11.7M 0s
10250K .......... .......... .......... .......... .......... 85% 3.82M 0s
10300K .......... .......... .......... .......... .......... 86% 5.74M 0s
10350K .......... .......... .......... .......... .......... 86% 11.0M 0s
10400K .......... .......... .......... .......... .......... 86% 5.03M 0s
10450K .......... .......... .......... .......... .......... 87% 3.83M 0s
10500K .......... .......... .......... .......... .......... 87% 105M 0s
10550K .......... .......... .......... .......... .......... 88% 3.79M 0s
10600K .......... .......... .......... .......... .......... 88% 21.3M 0s
10650K .......... .......... .......... .......... .......... 88% 4.41M 0s
10700K .......... .......... .......... .......... .......... 89% 21.3M 0s
10750K .......... .......... .......... .......... .......... 89% 4.54M 0s
10800K .......... .......... .......... .......... .......... 90% 3.97M 0s
10850K .......... .......... .......... .......... .......... 90% 29.5M 0s
10900K .......... .......... .......... .......... .......... 90% 4.28M 0s
10950K .......... .......... .......... .......... .......... 91% 22.8M 0s
11000K .......... .......... .......... .......... .......... 91% 4.32M 0s
11050K .......... .......... .......... .......... .......... 92% 28.9M 0s
11100K .......... .......... .......... .......... .......... 92% 3.28M 0s
11150K .......... .......... .......... .......... .......... 93% 5.53M 0s
11200K .......... .......... .......... .......... .......... 93% 11.8M 0s
11250K .......... .......... .......... .......... .......... 93% 5.54M 0s
11300K .......... .......... .......... .......... .......... 94% 10.7M 0s
11350K .......... .......... .......... .......... .......... 94% 6.05M 0s
11400K .......... .......... .......... .......... .......... 95% 4.05M 0s
11450K .......... .......... .......... .......... .......... 95% 8.94M 0s
11500K .......... .......... .......... .......... .......... 95% 6.20M 0s
11550K .......... .......... .......... .......... .......... 96% 9.69M 0s
11600K .......... .......... .......... .......... .......... 96% 6.11M 0s
11650K .......... .......... .......... .......... .......... 97% 3.95M 0s
11700K .......... .......... .......... .......... .......... 97% 9.91M 0s
11750K .......... .......... .......... .......... .......... 98% 5.64M 0s
11800K .......... .......... .......... .......... .......... 98% 3.95M 0s
11850K .......... .......... .......... .......... .......... 98% 12.8M 0s
11900K .......... .......... .......... .......... .......... 99% 5.49M 0s
11950K .......... .......... .......... .......... .......... 99% 3.91M 0s
12000K .......... .......... .......... .... 100% 107M=2.0s
2024-04-12 14:06:47 (5.93 MB/s) - ‘./data-gr_scell.tar.xz’ saved [12322968/12322968]
[14]:
['data-gr_scell',
'data-gr_scell/sparsematrix_metadata.dat',
'data-gr_scell/overlap_sparse.mtx',
'data-gr_scell/hamiltonian_sparse.mtx']
[15]:
if 'h' and 's' not in locals():
tool = BigDFTool()
h = tool.get_matrix_h(log)
s = tool.get_matrix_s(log)
metadatafile = f'data-{name}/sparsematrix_metadata.dat'
metadata = MatrixMetadata(metadatafile)
The matrices elements are extracted using the shell_matrix function, returning \(H\) and \(S\) inside a dictionary that also contains information on the basis set.
[16]:
m_sh = tb.shell_matrix(sys_ls, [h,s], metadata)
list(m_sh.keys())
[16]:
['id', 'h', 's']
The basis set is defined by the atoms mapping and their orbitals. In this case, the 626\(^{th}\) and 209\(^{th}\) atoms (corresponding to the \(A\) and \(B\) lattices) have 4 orbitals each, starting from the 0\(^{th}\) and the 4\(^{th}\) index, respectively. The last entry in the orbitals definition therefore give the total number of orbitals.
[17]:
m_sh['id']
[17]:
{'atoms': [626, 209], 'orbs': [0, 4, 8]}
The Hamiltonian \(H_{ij}\) between sites \(i\) and \(j\) is defined in a similar way to \(R_{ij}\). Visualizing the different matrix elements is therefore straightforwards. In this example, the on-site and \(1nn\) terms are displayed.
[18]:
fig,axs = plt.subplots(1,2,figsize=(10,4))
axs[0].imshow(abs(m_sh['h'][(0,0,(0,0,0))]),
norm=LogNorm(vmin=1e-3,vmax=1e0),
# norm=Normalize(vmin=0,vmax=.4),
cmap=cm.Blues,aspect='equal')
axs[1].imshow(abs(m_sh['h'][(0,1,(0,0,0))]),
norm=LogNorm(vmin=1e-3,vmax=1e0),
# norm=Normalize(vmin=0,vmax=.4),
cmap=cm.Blues,aspect='equal')
plt.show()
Band structure computation
Once the Hamiltonian extraction is completed, the band structure is obtained in a few steps.
First, the \(k\)-path is defined
[19]:
hsp = {'G1': [0., 0., 0.],
'M': [0.5, 0., 0.5],
'K': [0.666666, 0., 0.333333],
'G2': [0., 0., 0.]}
k = tb.k_path(hsp)
The eigenvalues are then computed by solving the Schrodinger’s equation. The k_matrix functions returns the k-resolved matrices and their corresponding eigenvalues.
[20]:
Hk,Sk,Ek = tb.k_matrix(k, m_sh)
Finally, the band structure is plotted
[21]:
Ef = log.fermi_level
TB.plot_bs(k, Ek-Ef)
plt.ylim([-14,14])
plt.show()
In a nutshell
To summarize the tight-binding workflow, the conventional cell is studied.
First, the sub-system is updated with the proper geometry
[22]:
positions = [{'C':list(j)} for j in coord['conv']]
posinp = {'positions': positions, 'units': 'angstroem'}
frag = Fragments.Fragment(posinp=posinp)
sys_cs = Systems.System()
sys_cs["FRA:1"] = frag
sys_cs.cell.cell = cell['conv']
The corresponding TB object is defined
[23]:
tb = TB.TightBinding(sys_cs, d=5)
The matrix description is extracted
[24]:
m_sh = tb.shell_matrix(sys_ls, [h,s], metadata)
The band structure is then computed for the associated k-points
[25]:
hsp = {'G': [0., 0., 0.],
'Y': [0., 0., 0.5]}
k = tb.k_path(hsp)
[26]:
Hk,Sk,Ek = tb.k_matrix(k, m_sh)
[27]:
Ef = log.fermi_level
TB.plot_bs(k, Ek-Ef)
plt.ylim([-14,10])
plt.show()
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