1import numpy as np 

2from ase.io.jsonio import read_json, write_json 

3 

4 

5class STM: 

6 def __init__(self, atoms, symmetries=None, use_density=False): 

7 """Scanning tunneling microscope. 

8 

9 atoms: Atoms object or filename 

10 Atoms to scan or name of file to read LDOS from. 

11 symmetries: list of int 

12 List of integers 0, 1, and/or 2 indicating which surface 

13 symmetries have been used to reduce the number of k-points 

14 for the DFT calculation. The three integers correspond to 

15 the following three symmetry operations:: 

16 

17 [-1 0] [ 1 0] [ 0 1] 

18 [ 0 1] [ 0 -1] [ 1 0] 

19 

20 use_density: bool 

21 Use the electron density instead of the LDOS. 

22 """ 

23 

24 self.use_density = use_density 

25 

26 if isinstance(atoms, str): 

27 with open(atoms, 'r') as fd: 

28 self.ldos, self.bias, self.cell = read_json(fd, 

29 always_array=False) 

30 self.atoms = None 

31 else: 

32 self.atoms = atoms 

33 self.cell = atoms.cell 

34 self.bias = None 

35 self.ldos = None 

36 assert not self.cell[2, :2].any() and not self.cell[:2, 2].any() 

37 

38 self.symmetries = symmetries or [] 

39 

40 def calculate_ldos(self, bias): 

41 """Calculate local density of states for given bias.""" 

42 if self.ldos is not None and bias == self.bias: 

43 return 

44 

45 self.bias = bias 

46 

47 calc = self.atoms.calc 

48 

49 if self.use_density: 

50 self.ldos = calc.get_pseudo_density() 

51 return 

52 

53 if bias < 0: 

54 emin = bias 

55 emax = 0.0 

56 else: 

57 emin = 0 

58 emax = bias 

59 

60 nbands = calc.get_number_of_bands() 

61 weights = calc.get_k_point_weights() 

62 nkpts = len(weights) 

63 nspins = calc.get_number_of_spins() 

64 eigs = np.array([[calc.get_eigenvalues(k, s) 

65 for k in range(nkpts)] 

66 for s in range(nspins)]) 

67 eigs -= calc.get_fermi_level() 

68 ldos = np.zeros(calc.get_pseudo_wave_function(0, 0, 0).shape) 

69 

70 for s in range(nspins): 

71 for k in range(nkpts): 

72 for n in range(nbands): 

73 e = eigs[s, k, n] 

74 if emin < e < emax: 

75 psi = calc.get_pseudo_wave_function(n, k, s) 

76 ldos += weights[k] * (psi * np.conj(psi)).real 

77 

78 if 0 in self.symmetries: 

79 # (x,y) -> (-x,y) 

80 ldos[1:] += ldos[:0:-1].copy() 

81 ldos[1:] *= 0.5 

82 

83 if 1 in self.symmetries: 

84 # (x,y) -> (x,-y) 

85 ldos[:, 1:] += ldos[:, :0:-1].copy() 

86 ldos[:, 1:] *= 0.5 

87 

88 if 2 in self.symmetries: 

89 # (x,y) -> (y,x) 

90 ldos += ldos.transpose((1, 0, 2)).copy() 

91 ldos *= 0.5 

92 

93 self.ldos = ldos 

94 

95 def write(self, filename): 

96 """Write local density of states to JSON file.""" 

97 write_json(filename, (self.ldos, self.bias, self.cell)) 

98 

99 def get_averaged_current(self, bias, z): 

100 """Calculate avarage current at height z (in Angstrom). 

101 

102 Use this to get an idea of what current to use when scanning.""" 

103 

104 self.calculate_ldos(bias) 

105 nz = self.ldos.shape[2] 

106 

107 # Find grid point: 

108 n = z / self.cell[2, 2] * nz 

109 dn = n - np.floor(n) 

110 n = int(n) % nz 

111 

112 # Average and do linear interpolation: 

113 return ((1 - dn) * self.ldos[:, :, n].mean() + 

114 dn * self.ldos[:, :, (n + 1) % nz].mean()) 

115 

116 def scan(self, bias, current, z0=None, repeat=(1, 1)): 

117 """Constant current 2-d scan. 

118 

119 Returns three 2-d arrays (x, y, z) containing x-coordinates, 

120 y-coordinates and heights. These three arrays can be passed to 

121 matplotlibs contourf() function like this: 

122 

123 >>> import matplotlib.pyplot as plt 

124 >>> plt.contourf(x, y, z) 

125 >>> plt.show() 

126 

127 """ 

128 

129 self.calculate_ldos(bias) 

130 

131 L = self.cell[2, 2] 

132 nz = self.ldos.shape[2] 

133 h = L / nz 

134 

135 ldos = self.ldos.reshape((-1, nz)) 

136 

137 heights = np.empty(ldos.shape[0]) 

138 for i, a in enumerate(ldos): 

139 heights[i] = find_height(a, current, h, z0) 

140 

141 s0 = heights.shape = self.ldos.shape[:2] 

142 heights = np.tile(heights, repeat) 

143 s = heights.shape 

144 

145 ij = np.indices(s, dtype=float).reshape((2, -1)).T 

146 x, y = np.dot(ij / s0, self.cell[:2, :2]).T.reshape((2,) + s) 

147 

148 return x, y, heights 

149 

150 def scan2(self, bias, z, repeat=(1, 1)): 

151 """Constant height 2-d scan. 

152 

153 Returns three 2-d arrays (x, y, I) containing x-coordinates, 

154 y-coordinates and currents. These three arrays can be passed to 

155 matplotlibs contourf() function like this: 

156 

157 >>> import matplotlib.pyplot as plt 

158 >>> plt.contourf(x, y, I) 

159 >>> plt.show() 

160 

161 """ 

162 

163 self.calculate_ldos(bias) 

164 

165 nz = self.ldos.shape[2] 

166 ldos = self.ldos.reshape((-1, nz)) 

167 

168 I = np.empty(ldos.shape[0]) 

169 

170 zp = z / self.cell[2, 2] * nz 

171 zp = int(zp) % nz 

172 

173 for i, a in enumerate(ldos): 

174 I[i] = self.find_current(a, zp) 

175 

176 s0 = I.shape = self.ldos.shape[:2] 

177 I = np.tile(I, repeat) 

178 s = I.shape 

179 

180 ij = np.indices(s, dtype=float).reshape((2, -1)).T 

181 x, y = np.dot(ij / s0, self.cell[:2, :2]).T.reshape((2,) + s) 

182 

183 # Returing scan with axes in Angstrom. 

184 return x, y, I 

185 

186 def linescan(self, bias, current, p1, p2, npoints=50, z0=None): 

187 """Constant current line scan. 

188 

189 Example:: 

190 

191 stm = STM(...) 

192 z = ... # tip position 

193 c = stm.get_averaged_current(-1.0, z) 

194 stm.linescan(-1.0, c, (1.2, 0.0), (1.2, 3.0)) 

195 """ 

196 

197 heights = self.scan(bias, current, z0)[2] 

198 

199 p1 = np.asarray(p1, float) 

200 p2 = np.asarray(p2, float) 

201 d = p2 - p1 

202 s = np.dot(d, d)**0.5 

203 

204 cell = self.cell[:2, :2] 

205 shape = np.array(heights.shape, float) 

206 M = np.linalg.inv(cell) 

207 line = np.empty(npoints) 

208 for i in range(npoints): 

209 p = p1 + i * d / (npoints - 1) 

210 q = np.dot(p, M) * shape 

211 line[i] = interpolate(q, heights) 

212 return np.linspace(0, s, npoints), line 

213 

214 def pointcurrent(self, bias, x, y, z): 

215 """Current for a single x, y, z position for a given bias.""" 

216 

217 self.calculate_ldos(bias) 

218 

219 nx = self.ldos.shape[0] 

220 ny = self.ldos.shape[1] 

221 nz = self.ldos.shape[2] 

222 

223 # Find grid point: 

224 xp = x / np.linalg.norm(self.cell[0]) * nx 

225 dx = xp - np.floor(xp) 

226 xp = int(xp) % nx 

227 

228 yp = y / np.linalg.norm(self.cell[1]) * ny 

229 dy = yp - np.floor(yp) 

230 yp = int(yp) % ny 

231 

232 zp = z / np.linalg.norm(self.cell[2]) * nz 

233 dz = zp - np.floor(zp) 

234 zp = int(zp) % nz 

235 

236 # 3D interpolation of the LDOS at point (x,y,z) at given bias. 

237 xyzldos = (((1 - dx) + (1 - dy) + (1 - dz)) * self.ldos[xp, yp, zp] + 

238 dx * self.ldos[(xp + 1) % nx, yp, zp] + 

239 dy * self.ldos[xp, (yp + 1) % ny, zp] + 

240 dz * self.ldos[xp, yp, (zp + 1) % nz]) 

241 

242 return dos2current(bias, xyzldos) 

243 

244 def sts(self, x, y, z, bias0, bias1, biasstep): 

245 """Returns the dI/dV curve for position x, y at height z (in Angstrom), 

246 for bias from bias0 to bias1 with step biasstep.""" 

247 

248 biases = np.arange(bias0, bias1+biasstep, biasstep) 

249 I = np.zeros(biases.shape) 

250 

251 for b in np.arange(len(biases)): 

252 print(b, biases[b]) 

253 I[b] = self.pointcurrent(biases[b], x, y, z) 

254 

255 dIdV = np.gradient(I, biasstep) 

256 

257 return biases, I, dIdV 

258 

259 def find_current(self, ldos, z): 

260 """ Finds current for given LDOS at height z.""" 

261 nz = self.ldos.shape[2] 

262 

263 zp = z / self.cell[2, 2] * nz 

264 dz = zp - np.floor(zp) 

265 zp = int(zp) % nz 

266 

267 ldosz = (1 - dz) * ldos[zp] + dz * ldos[(zp + 1) % nz] 

268 

269 return dos2current(self.bias, ldosz) 

270 

271 

272def dos2current(bias, dos): 

273 # Borrowed from gpaw/analyse/simple_stm.py: 

274 # The connection between density n and current I 

275 # n [e/Angstrom^3] = 0.0002 sqrt(I [nA]) 

276 # as given in Hofer et al., RevModPhys 75 (2003) 1287 

277 return 5000. * dos**2 * (1 if bias > 0 else -1) 

278 

279 

280def interpolate(q, heights): 

281 qi = q.astype(int) 

282 f = q - qi 

283 g = 1 - f 

284 qi %= heights.shape 

285 n0, m0 = qi 

286 n1, m1 = (qi + 1) % heights.shape 

287 z = (g[0] * g[1] * heights[n0, m0] + 

288 f[0] * g[1] * heights[n1, m0] + 

289 g[0] * f[1] * heights[n0, m1] + 

290 f[0] * f[1] * heights[n1, m1]) 

291 return z 

292 

293 

294def find_height(ldos, current, h, z0=None): 

295 if z0 is None: 

296 n = len(ldos) - 2 

297 else: 

298 n = int(z0 / h) 

299 while n >= 0: 

300 if ldos[n] > current: 

301 break 

302 n -= 1 

303 else: 

304 return 0.0 

305 

306 c2, c1 = ldos[n:n + 2] 

307 return (n + 1 - (current - c1) / (c2 - c1)) * h 

308 

309 

310def delta(biases, bias, width): 

311 """Return a delta-function centered at 'bias'""" 

312 x = -((biases - bias) / width)**2 

313 return np.exp(x) / (np.sqrt(np.pi) * width)