import numpy as np
from ase.io.jsonio import read_json, write_json
[docs]class STM:
def __init__(self, atoms, symmetries=None, use_density=False):
"""Scanning tunneling microscope.
atoms: Atoms object or filename
Atoms to scan or name of file to read LDOS from.
symmetries: list of int
List of integers 0, 1, and/or 2 indicating which surface
symmetries have been used to reduce the number of k-points
for the DFT calculation. The three integers correspond to
the following three symmetry operations::
[-1 0] [ 1 0] [ 0 1]
[ 0 1] [ 0 -1] [ 1 0]
use_density: bool
Use the electron density instead of the LDOS.
"""
self.use_density = use_density
if isinstance(atoms, str):
with open(atoms) as fd:
self.ldos, self.bias, self.cell = read_json(fd,
always_array=False)
self.atoms = None
else:
self.atoms = atoms
self.cell = atoms.cell
self.bias = None
self.ldos = None
assert not self.cell[2, :2].any() and not self.cell[:2, 2].any()
self.symmetries = symmetries or []
[docs] def calculate_ldos(self, bias):
"""Calculate local density of states for given bias."""
if self.ldos is not None and bias == self.bias:
return
self.bias = bias
calc = self.atoms.calc
if self.use_density:
self.ldos = calc.get_pseudo_density()
return
if bias < 0:
emin = bias
emax = 0.0
else:
emin = 0
emax = bias
nbands = calc.get_number_of_bands()
weights = calc.get_k_point_weights()
nkpts = len(weights)
nspins = calc.get_number_of_spins()
eigs = np.array([[calc.get_eigenvalues(k, s)
for k in range(nkpts)]
for s in range(nspins)])
eigs -= calc.get_fermi_level()
ldos = np.zeros(calc.get_pseudo_wave_function(0, 0, 0).shape)
for s in range(nspins):
for k in range(nkpts):
for n in range(nbands):
e = eigs[s, k, n]
if emin < e < emax:
psi = calc.get_pseudo_wave_function(n, k, s)
ldos += weights[k] * (psi * np.conj(psi)).real
if 0 in self.symmetries:
# (x,y) -> (-x,y)
ldos[1:] += ldos[:0:-1].copy()
ldos[1:] *= 0.5
if 1 in self.symmetries:
# (x,y) -> (x,-y)
ldos[:, 1:] += ldos[:, :0:-1].copy()
ldos[:, 1:] *= 0.5
if 2 in self.symmetries:
# (x,y) -> (y,x)
ldos += ldos.transpose((1, 0, 2)).copy()
ldos *= 0.5
self.ldos = ldos
[docs] def write(self, filename):
"""Write local density of states to JSON file."""
write_json(filename, (self.ldos, self.bias, self.cell))
[docs] def get_averaged_current(self, bias, z):
"""Calculate avarage current at height z (in Angstrom).
Use this to get an idea of what current to use when scanning."""
self.calculate_ldos(bias)
nz = self.ldos.shape[2]
# Find grid point:
n = z / self.cell[2, 2] * nz
dn = n - np.floor(n)
n = int(n) % nz
# Average and do linear interpolation:
return ((1 - dn) * self.ldos[:, :, n].mean() +
dn * self.ldos[:, :, (n + 1) % nz].mean())
[docs] def scan(self, bias, current, z0=None, repeat=(1, 1)):
"""Constant current 2-d scan.
Returns three 2-d arrays (x, y, z) containing x-coordinates,
y-coordinates and heights. These three arrays can be passed to
matplotlibs contourf() function like this:
>>> import matplotlib.pyplot as plt
>>> plt.contourf(x, y, z)
>>> plt.show()
"""
self.calculate_ldos(bias)
L = self.cell[2, 2]
nz = self.ldos.shape[2]
h = L / nz
ldos = self.ldos.reshape((-1, nz))
heights = np.empty(ldos.shape[0])
for i, a in enumerate(ldos):
heights[i] = find_height(a, current, h, z0)
s0 = heights.shape = self.ldos.shape[:2]
heights = np.tile(heights, repeat)
s = heights.shape
ij = np.indices(s, dtype=float).reshape((2, -1)).T
x, y = np.dot(ij / s0, self.cell[:2, :2]).T.reshape((2,) + s)
return x, y, heights
[docs] def scan2(self, bias, z, repeat=(1, 1)):
"""Constant height 2-d scan.
Returns three 2-d arrays (x, y, I) containing x-coordinates,
y-coordinates and currents. These three arrays can be passed to
matplotlibs contourf() function like this:
>>> import matplotlib.pyplot as plt
>>> plt.contourf(x, y, I)
>>> plt.show()
"""
self.calculate_ldos(bias)
nz = self.ldos.shape[2]
ldos = self.ldos.reshape((-1, nz))
current = np.empty(ldos.shape[0])
zp = z / self.cell[2, 2] * nz
zp = int(zp) % nz
for i, a in enumerate(ldos):
current[i] = self.find_current(a, zp)
s0 = current.shape = self.ldos.shape[:2]
current = np.tile(current, repeat)
s = current.shape
ij = np.indices(s, dtype=float).reshape((2, -1)).T
x, y = np.dot(ij / s0, self.cell[:2, :2]).T.reshape((2,) + s)
# Returing scan with axes in Angstrom.
return x, y, current
[docs] def linescan(self, bias, current, p1, p2, npoints=50, z0=None):
"""Constant current line scan.
Example::
stm = STM(...)
z = ... # tip position
c = stm.get_averaged_current(-1.0, z)
stm.linescan(-1.0, c, (1.2, 0.0), (1.2, 3.0))
"""
heights = self.scan(bias, current, z0)[2]
p1 = np.asarray(p1, float)
p2 = np.asarray(p2, float)
d = p2 - p1
s = np.dot(d, d)**0.5
cell = self.cell[:2, :2]
shape = np.array(heights.shape, float)
M = np.linalg.inv(cell)
line = np.empty(npoints)
for i in range(npoints):
p = p1 + i * d / (npoints - 1)
q = np.dot(p, M) * shape
line[i] = interpolate(q, heights)
return np.linspace(0, s, npoints), line
[docs] def pointcurrent(self, bias, x, y, z):
"""Current for a single x, y, z position for a given bias."""
self.calculate_ldos(bias)
nx = self.ldos.shape[0]
ny = self.ldos.shape[1]
nz = self.ldos.shape[2]
# Find grid point:
xp = x / np.linalg.norm(self.cell[0]) * nx
dx = xp - np.floor(xp)
xp = int(xp) % nx
yp = y / np.linalg.norm(self.cell[1]) * ny
dy = yp - np.floor(yp)
yp = int(yp) % ny
zp = z / np.linalg.norm(self.cell[2]) * nz
dz = zp - np.floor(zp)
zp = int(zp) % nz
# 3D interpolation of the LDOS at point (x,y,z) at given bias.
xyzldos = (((1 - dx) + (1 - dy) + (1 - dz)) * self.ldos[xp, yp, zp] +
dx * self.ldos[(xp + 1) % nx, yp, zp] +
dy * self.ldos[xp, (yp + 1) % ny, zp] +
dz * self.ldos[xp, yp, (zp + 1) % nz])
return dos2current(bias, xyzldos)
[docs] def sts(self, x, y, z, bias0, bias1, biasstep):
"""Returns the dI/dV curve for position x, y at height z (in Angstrom),
for bias from bias0 to bias1 with step biasstep."""
biases = np.arange(bias0, bias1 + biasstep, biasstep)
current = np.zeros(biases.shape)
for b in np.arange(len(biases)):
print(b, biases[b])
current[b] = self.pointcurrent(biases[b], x, y, z)
dIdV = np.gradient(current, biasstep)
return biases, current, dIdV
[docs] def find_current(self, ldos, z):
""" Finds current for given LDOS at height z."""
nz = self.ldos.shape[2]
zp = z / self.cell[2, 2] * nz
dz = zp - np.floor(zp)
zp = int(zp) % nz
ldosz = (1 - dz) * ldos[zp] + dz * ldos[(zp + 1) % nz]
return dos2current(self.bias, ldosz)
def dos2current(bias, dos):
# Borrowed from gpaw/analyse/simple_stm.py:
# The connection between density n and current I
# n [e/Angstrom^3] = 0.0002 sqrt(I [nA])
# as given in Hofer et al., RevModPhys 75 (2003) 1287
return 5000. * dos**2 * (1 if bias > 0 else -1)
def interpolate(q, heights):
qi = q.astype(int)
f = q - qi
g = 1 - f
qi %= heights.shape
n0, m0 = qi
n1, m1 = (qi + 1) % heights.shape
z = (g[0] * g[1] * heights[n0, m0] +
f[0] * g[1] * heights[n1, m0] +
g[0] * f[1] * heights[n0, m1] +
f[0] * f[1] * heights[n1, m1])
return z
def find_height(ldos, current, h, z0=None):
if z0 is None:
n = len(ldos) - 2
else:
n = int(z0 / h)
while n >= 0:
if ldos[n] > current:
break
n -= 1
else:
return 0.0
c2, c1 = ldos[n:n + 2]
return (n + 1 - (current - c1) / (c2 - c1)) * h
def delta(biases, bias, width):
"""Return a delta-function centered at 'bias'"""
x = -((biases - bias) / width)**2
return np.exp(x) / (np.sqrt(np.pi) * width)