Skip to content

dIdV

dIdV

Calculating and visualizing differential conductence (dI/dV) spectra.

This module provides routines to compute differential conductance spectra using second-order perturbation theory.

Functions:

Name Description
calc_IETS_2nd

Compute dI/dV using a second-order tunneling model and thermally populated states.
tunnelingBroadenedStepFunction

Compute the tunneling broadened step function used in the IETS calculations.
calc_IETS_2ndR

Compute dI/dV using a second-order tunneling model with voltage-dependent rate equations.
calc_derivative

Numerically differentiate current with respect to voltage.
plot_IETS

Plot IETS spectra for one or all spins using Plotly.
Dependencies
  • Uses spinfinity.SpinSys.SpinSys class to represent the spin system and its properties.
  • Uses spinfinity.SpinHamiltonian for eigenvalue and eigenvector calculations.
  • Uses functions from spinfinity.Bath for population and rate calculations.
References

calc_IETS_2nd(spinsys, V_array=None, norm=False)

Calculate the derivative of the tunneling current (dI/dV) in 2nd Order.

This functions calculates the dI/dV signal as a function of voltage using a second-order tunneling model with thermally populated states.

Parameters:

Name Type Description Default
spinsys SpinSys

The spin system object containing eigenvalues, eigenvectors, temperature, and other relevant parameters.

 required
V_array ndarray

An array of voltage values (in mV) at which to calculate dI/dV. If None, a default range from -30 mV to 30 mV with 200 points will be used.

 None
norm bool

If True, normalize the output dI/dV array to the range [0, 1]. Default is False.

 False

Returns:

Name Type Description
V_array ndarray

Array of voltage values over the specified range.
dIdV_array ndarray

Calculated dI/dV values corresponding to each voltage in V_array.
Source code in spinfinity/dIdV.py 
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
def calc_IETS_2nd(
    spinsys: SpinSys,
    V_array: np.ndarray = None,
    norm: bool = False,
):
    """
    Calculate the derivative of the tunneling current (dI/dV) in 2nd Order.

    This functions calculates the dI/dV signal as a function of voltage using
    a second-order tunneling model with thermally populated states.

    Parameters
    ----------
    spinsys : SpinSys
        The spin system object containing eigenvalues, eigenvectors, temperature, 
        and other relevant parameters.
    V_array : ndarray, optional
        An array of voltage values (in mV) at which to calculate dI/dV. 
        If None, a default range from -30 mV to 30 mV with 200 points will be used.
    norm : bool, optional
        If True, normalize the output dI/dV array to the range [0, 1]. Default is False.

    Returns
    -------
    V_array : ndarray
        Array of voltage values over the specified range.
    dIdV_array : ndarray
        Calculated dI/dV values corresponding to each voltage in V_array.
    """
    # Initialize the arrays
    # Default range from -30 mV to 30 mV with 200 points
    if V_array is None:
        V_array = np.linspace(-30, 30, 200)  
    N = len(V_array)
    dIdV_array = np.zeros(N, dtype=float)

    # Calculate the Eignevalues and Eignevectors if not already done
    if not hasattr(spinsys, 'eigVectors') or spinsys.eigVectors is None:
        hamil.calc_EigEnergies(spinsys)

    # Calculate the thermal population (Boltzmann distribution)
    bath.calc_Populations(spinsys, AllowPumping=False)

    # Calculate the Rate Matrix     
    RateMatrix = bath.calc_TunnelingMatrixElements(spinsys, "st")[0]

    for i in range(spinsys.dimensionOfMatrix):
        if spinsys.pop_0[i] > 10**(-6):
            for j in range(spinsys.dimensionOfMatrix):
                ep = tunnelingBroadenedStepFunction(
                    np.array(
                        (spinsys.E_All[j] - spinsys.E_All[i] - V_array)
                        / (spinsys.T * spinsys.kB)
                    )
                )
                en = tunnelingBroadenedStepFunction(
                    np.array(
                        (spinsys.E_All[j] - spinsys.E_All[i] + V_array)
                        / (spinsys.T * spinsys.kB)
                    )
                )
                ytemp = (spinsys.pop_0[i] * (RateMatrix[j, i] 
                                             * ep + RateMatrix[i, j] * en))
                dIdV_array = dIdV_array + ytemp

    # Normalize with Rate_00 to get correct units
    Rate_0 = RateMatrix[0][0]  # Transition matrix element of the ground state
    dIdV_array = dIdV_array * spinsys.G_st / Rate_0

    # Normalize dI/dV if required
    if norm:
        dIdV_array = dIdV_array / dIdV_array[0]  # Normalize to the first value
    return V_array, dIdV_array

calc_IETS_2ndR(spinsys, V_array=None, AllowPumping=True, norm=True)

Calculate the IETS spectrum using the rate equation approach.

This function calculates the dI/dV spectrum as a function of voltage using a second-order tunneling model combined with voltage-dependent rate equations that redistribute the spin state populations.

Parameters:

Name Type Description Default
spinsys SpinSys

The spin system object containing all relevant parameters and matrices.

 required
V_array ndarray

An array of voltage values (in mV) at which to calculate dI/dV. If None, a default range from -30 mV to 30 mV with 200 points will be used.

 None
AllowPumping bool

If True, allows for non-thermal population of spin states (pumping). If False, uses thermal (Boltzmann) population. Default is True.

 True
norm bool

If True, normalizes the resulting dI/dV spectrum to the range [0, 1]. Default is True.

 True

Returns:

Name Type Description
V_array ndarray

Array of voltage values (in mV) at which the spectrum is calculated.
dIdV ndarray

The normalized differential conductance (dI/dV) spectrum.
P ndarray of shape (dimensionOfMatrix, N)

Matrix of state populations for each voltage point.
Notes

The calculation is based on the rate equation approach and supports both thermal and non-thermal population distributions. For AllowPumping=False the function should output the same result as calc_IETS_2nd. The function updates the following attributes of spinsys:

  • spinsys.pop_0: The thermal population distribution (Boltzmann distribution) for the spin states.
  • spinsys.MatrixST: The tunneling matrix elements for tip-sample scattering.
  • spinsys.MatrixTS: The tunneling matrix elements for sample-tip scattering.
  • spinsys.MatrixSS: The tunneling matrix elements for sample-sample scattering.
  • spinsys.MatrixTT: The tunneling matrix elements for tip-tip scattering.
Source code in spinfinity/dIdV.py 
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
def calc_IETS_2ndR(
    spinsys: SpinSys,
    V_array: np.ndarray = None,
    AllowPumping: bool = True,
    norm: bool = True
):
    """
    Calculate the IETS spectrum using the rate equation approach.

    This function calculates the dI/dV spectrum as a function of voltage using
    a second-order tunneling model combined with voltage-dependent rate equations that 
    redistribute the spin state populations. 

    Parameters
    ----------
    spinsys : SpinSys
        The spin system object containing all relevant parameters and matrices.
    V_array : ndarray, optional
        An array of voltage values (in mV) at which to calculate dI/dV.
        If None, a default range from -30 mV to 30 mV with 200 points will be used.
    AllowPumping : bool, optional
        If True, allows for non-thermal population of spin states (pumping). 
        If False, uses thermal (Boltzmann) population. Default is True.
    norm : bool, optional
        If True, normalizes the resulting dI/dV spectrum to the range [0, 1]. 
        Default is True.

    Returns
    -------
    V_array : ndarray
        Array of voltage values (in mV) at which the spectrum is calculated.
    dIdV : ndarray
        The normalized differential conductance (dI/dV) spectrum.
    P : ndarray of shape (dimensionOfMatrix, N)
        Matrix of state populations for each voltage point.

    Notes
    -----
    The calculation is based on the rate equation approach and supports both thermal
    and non-thermal population distributions. `For AllowPumping=False` the function
    should output the same result as `calc_IETS_2nd`.
    The function updates the following attributes of `spinsys`:

    - `spinsys.pop_0`: The thermal population distribution (Boltzmann distribution)
      for the spin states.
    - `spinsys.MatrixST`: The tunneling matrix elements for tip-sample scattering.
    - `spinsys.MatrixTS`: The tunneling matrix elements for sample-tip scattering.
    - `spinsys.MatrixSS`: The tunneling matrix elements for sample-sample scattering.
    - `spinsys.MatrixTT`: The tunneling matrix elements for tip-tip scattering.
    """
    # Initialize the arrays
    # Default range from -30 mV to 30 mV with 200 points
    if V_array is None:
        V_array = np.linspace(-30, 30, 200)  
    N = len(V_array)
    Current = np.zeros((N), dtype=float)
    P = np.zeros((spinsys.dimensionOfMatrix, N), dtype=float)
    Rates = np.zeros((4, spinsys.dimensionOfMatrix, spinsys.dimensionOfMatrix))

    # Calculate the Eignevalues and Eignevectors if not already done
    if not hasattr(spinsys, 'eigVectors') or spinsys.eigVectors is None:
        hamil.calc_EigEnergies(spinsys)

    # Getting the relevant matrix elements
    spinsys.MatrixST = bath.calc_TunnelingMatrixElements(spinsys, "st")[0]
    spinsys.MatrixTS = spinsys.MatrixST.T
    spinsys.MatrixSS = bath.calc_TunnelingMatrixElements(spinsys, "ss")
    spinsys.MatrixTT = bath.calc_TunnelingMatrixElements(spinsys, "tt")[0]

    MatrixSS_summed = np.zeros((spinsys.dimensionOfMatrix, spinsys.dimensionOfMatrix))
    for i in range(spinsys.NSpins):
        MatrixSS_summed += spinsys.MatrixSS[i, :, :] * spinsys.G_ss[i]

    Rate_0 = spinsys.MatrixST[0][0]  # Transition matrix element of the ground state

    # Rate-Factors using Loth et al. (2010), https://doi.org/10.1038/nphys1616
    spinsys.G_st = (1 - spinsys.b0) * spinsys.G_st
    RateFactor_ts = spinsys.G_st / (const.e * Rate_0) / 1e3
    RateFactor_st = spinsys.G_st / (const.e * Rate_0) / 1e3
    RateFactor_ss = 1 / (const.e * Rate_0) / 1e3
    # Gtt is calculated from Gss and Gst, the Gtt value 0,1 here allows it or not
    RateFactor_tt = (
    spinsys.G_st**2
    / spinsys.G_ss[spinsys.ReadoutSpin]
    / (const.e * Rate_0)
    / 1e3 * spinsys.G_tt
    )
    # Calculate the rates that stem from Tip tip and sample sample scattering,
    #  since they are not affected by the voltage
    RateIntegrals_ss = bath.calc_RateIntegrals(spinsys, 'ss')
    RateIntegrals_tt = RateIntegrals_ss
    Rates[2, :, :] = RateFactor_ss * MatrixSS_summed * RateIntegrals_ss
    Rates[3, :, :] = RateFactor_tt * spinsys.MatrixTT * RateIntegrals_tt * spinsys.G_tt

    # This part happens now as often as we need it
    for I_index in range(N):
        V_DC = V_array[I_index]
        spinsys.V_DC = V_DC

        # Calculate changing rate integrals for the current voltage
        RateIntegrals_ts = bath.calc_RateIntegrals(spinsys, 'ts')
        RateIntegrals_st = bath.calc_RateIntegrals(spinsys, 'st')

        # Calculate rates based on rate factors and integrals
        Rates[0, :, :] = RateFactor_ts * spinsys.MatrixTS * RateIntegrals_ts
        Rates[1, :, :] = RateFactor_st * spinsys.MatrixST * RateIntegrals_st

        # Sum the rates
        Rates_Summed = (
            Rates[0, :, :] + Rates[1, :, :] +
            Rates[2, :, :] + Rates[3, :, :] 
        )

        # Build the Rate-Equation matrix
        K = np.zeros((spinsys.dimensionOfMatrix, spinsys.dimensionOfMatrix))
        for i in range(spinsys.dimensionOfMatrix):
            for j in range(spinsys.dimensionOfMatrix):
                if i != j:
                    K[i, i] -= Rates_Summed[i, j]  # Rates leaving state i
                    K[i, j] = Rates_Summed[j, i]   # Rates leading to state i

        # Calculate the thermal population (Boltzmann distribution)
        p0 = np.exp(-spinsys.E_All / (spinsys.kB * spinsys.T))
        pTot = np.sum(p0)
        spinsys.pop_0 = p0 / pTot  # Normalized thermal population

        # Solve the Rate-Equation if non-thermal population is desired
        if AllowPumping:
            try:
                p = null_space(K)
            except Exception as e:
                print("Warning: null_space failed, using thermal population. Error:", e)
                p = np.empty((0, 0))
            if p.size == 0:  # If no valid solution, use thermal distribution
                Populations = spinsys.pop_0
                print("Warning: No valid solution for rate equations found, "
                "using thermal population instead.")
            else:
                Populations = p[:, 0] / np.sum(p[:, 0])
        else:
            Populations = spinsys.pop_0
            print("Using thermal population (Boltzmann distribution).")

        # Deriving the tunneling current using Equation 41 from Ternes (2015),
        # https://dx.doi.org/10.1088/1367-2630/17/6/063016
        I_value = 0
        for i in range(spinsys.dimensionOfMatrix):
            for j in range(spinsys.dimensionOfMatrix):
                I_temp = const.e * Populations[i] * (Rates[0, i, j] - Rates[1, i, j])
                I_value += I_temp

        # Adjust current calculation
        I_value += spinsys.b0 * spinsys.G_st * spinsys.V_DC

        Current[I_index] = I_value
        P[:, I_index] = Populations    

    # Calculate the derivative (dI/dV)
    dIdV = calc_derivative(Current, (V_array * 1e-3))  # Convert V_array to volts

    # Normalize dI/dV if required
    if norm:
        dIdV = (dIdV / dIdV[0])  # Normalize to the first value

    return V_array, dIdV, P

calc_derivative(Current, Voltage)

Calculate the numerical derivative dI/dV of current (I) with respect to voltage (V).

Parameters:

Name Type Description Default
Current array_like

Array of current values.

 required
Voltage array_like

Array of voltage values. Must be the same length as Current.

 required

Returns:

Name Type Description
dIdV ndarray

Array of numerical derivatives dI/dV, same length as input arrays.
Source code in spinfinity/dIdV.py 
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
def calc_derivative(Current: list, Voltage: list):
    """
    Calculate the numerical derivative dI/dV of current (I) with respect to voltage (V).

    Parameters
    ----------
    Current : array_like
        Array of current values.
    Voltage : array_like
        Array of voltage values. Must be the same length as `Current`.

    Returns
    -------
    dIdV : ndarray
        Array of numerical derivatives dI/dV, same length as input arrays.
    """
    n = len(Current)  # Length of the array
    dIdV = np.zeros(n)  # Initialize the output array for derivative

    # Forward difference for the first point
    dIdV[0] = (Current[1] - Current[0]) / (Voltage[1] - Voltage[0])

    # Central difference for the middle points
    for i in range(1, n - 1):
        dIdV[i] = (Current[i + 1] - Current[i - 1]) / (Voltage[i + 1] - Voltage[i - 1])

    # Backward difference for the last point
    dIdV[n - 1] = (Current[n - 1] - Current[n - 2]) / (Voltage[n - 1] - Voltage[n - 2])

    return dIdV

plot_IETS(spinsys, V_array=None, norm=False, order='2nd', plotAllSpins=False)

Plot the Inelastic Electron Tunneling Spectroscopy (IETS) spectrum.

The function calculates the IETS spectrum for the given spin system and visualizes it using Plotly. It supports both 2nd order and 2nd order with rate equations calculations.

Parameters:

Name Type Description Default
spinsys SpinSys

The spin system object for which the IETS spectrum is to be calculated.

 required
V_array ndarray

An array of voltage values (in mV) at which to calculate the spectrum. If None, a default range from -30 mV to 30 mV with 200 points will be used.

 None
norm bool

If True, normalize the calculated spectrum (default is False).

 False
order str

The order of the IETS calculation. Supported orders are '2nd' and '2ndR'.

 '2nd'

Returns:

Name Type Description
return_dict dict

A dictionary containing: - 'fig': plotly.graph_objects.Figure The plotly figure object containing the plot. - 'V_array': numpy.ndarray The voltage values used in the spectrum. - 'dIdVsignal': numpy.ndarray The calculated dI/dV signal values.

Raises:

Type Description
ValueError

If an unsupported order is specified.
Notes
  • the dimension of the dIdVsignal changes when plotAllSpins=True
Source code in spinfinity/dIdV.py 
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
def plot_IETS(
    spinsys: SpinSys,
    V_array: np.ndarray = None,
    norm: bool = False,
    order: str = '2nd',
    plotAllSpins: bool = False,
):
    """
    Plot the Inelastic Electron Tunneling Spectroscopy (IETS) spectrum.

    The function calculates the IETS spectrum for the given spin system 
    and visualizes it using Plotly. It supports both 2nd order and 2nd order
    with rate equations calculations.

    Parameters
    ----------
    spinsys : SpinSys
        The spin system object for which the IETS spectrum is to be calculated.
    V_array : numpy.ndarray, optional
        An array of voltage values (in mV) at which to calculate the spectrum.
        If None, a default range from -30 mV to 30 mV with 200 points will be used.
    norm : bool, optional
        If True, normalize the calculated spectrum (default is False).
    order : str, optional
        The order of the IETS calculation. Supported orders are '2nd' and '2ndR'.

    Returns
    -------
    return_dict : dict
        A dictionary containing:
        - 'fig': plotly.graph_objects.Figure
            The plotly figure object containing the plot.
        - 'V_array': numpy.ndarray
            The voltage values used in the spectrum.
        - 'dIdVsignal': numpy.ndarray
            The calculated dI/dV signal values.

    Raises
    ------
    ValueError
        If an unsupported order is specified.

    Notes
    -----
    - the dimension of the dIdVsignal changes when plotAllSpins=True
    """
    # Define y-axis label according to normalization
    y_label = 'Normalized dI/dV (a.u.)' if norm else 'dI/dV (S)'

    # Default to 200 points if V_array is None
    N = len(V_array) if V_array is not None else 200  

    if not plotAllSpins:

        # Initialize x and y arrays
        x = np.zeros(N)  # Voltage 
        y = np.zeros(N)  # dI/dV Signal
        # Calculate the IETS spectrum for the specified order
        if order == '2nd':
            x, y = calc_IETS_2nd(spinsys, V_array=V_array, norm=norm)
        elif order == '2ndR':
            x, y, _ = calc_IETS_2ndR(spinsys, V_array=V_array, norm=norm)
        else:
            raise ValueError("Unsupported order. Currently only '2nd'"
            "and '2ndR' order is implemented.")

        # Plot the IETS spectrum using Plotly with white background and black border
        y_max = np.max(y)
        if y_max == 0:
            y_max = 1.0
        x0, x1 = float(np.min(x)), float(np.max(x))
        y0, y1 = 0.0, float(y_max * 1.1)

        fig = go.Figure()
        fig.add_trace(go.Scatter(
            x=x,
            y=y,
            mode='lines',
            name='IETS Spectrum',
            line=dict(color='black', width=2)
        ))

        fig.update_layout(
            title=f'IETS Spectrum ({order} Order)',
            xaxis_title='Bias Voltage (mV)',
            yaxis_title=y_label,
            plot_bgcolor='white',
            paper_bgcolor='white',
            width=800,
            height=500,
            margin=dict(l=60, r=20, t=60, b=60)
        )

        fig.update_xaxes(range=[x0, x1], showgrid=True,
                          gridcolor='lightgray', zeroline=False)
        fig.update_yaxes(range=[y0, y1], showgrid=True, 
                         gridcolor='lightgray', zeroline=False)

        fig.update_layout(
            shapes=[dict(
            type="rect",
            xref="x",
            yref="y",
            x0=x0,
            x1=x1,
            y0=y0,
            y1=y1,
            line=dict(color="black", width=2),
            fillcolor="rgba(0,0,0,0)"
            )]
        )

        fig.show()

    elif plotAllSpins:

        # Initialize x and y arrays
        x = np.zeros(N)
        y = np.zeros((len(spinsys.Spins), N))

        for i in range(len(spinsys.Spins)):
            spinsys.ReadoutSpin = i

            if order == '2nd':
                x, y[i, :] = calc_IETS_2nd(spinsys, V_array=V_array, norm=norm)
            elif order == '2ndR':
                x, y[i, :], _ = calc_IETS_2ndR(spinsys, V_array=V_array, norm=norm)
            else:
                raise ValueError("Unsupported order. Currently only '2nd' "
                "and '2ndR' order are implemented.")

            # Plot all IETS spectra in the same figure on top of each other
            # Only create the figure once, before the loop
            # if i == 0:
        # After computing x and y for all spins, build a Plotly figure
        color_list = plotly.colors.qualitative.Plotly
        fig = go.Figure()

        # Add one trace per spin (overlayed)
        nspins = y.shape[0]
        for i in range(nspins):
            fig.add_trace(go.Scatter(
            x=x,
            y=y[i, :],
            mode='lines',
            line=dict(color=color_list[i % len(color_list)], width=2),
            name=f'Spin {i}'
            ))

        # Determine axis ranges similar to single-spin branch
        y_max = np.max(y) if np.max(y) != 0 else 1.0
        x0, x1 = float(np.min(x)), float(np.max(x))
        y0, y1 = 0.0, float(y_max * 1.1)

        fig.update_layout(
            title=f'IETS Spectrum ({order} Order) for All Spins',
            xaxis_title='Bias Voltage (mV)',
            yaxis_title=y_label,
            plot_bgcolor='white',
            paper_bgcolor='white',
            width=800,
            height=500,
            margin=dict(l=60, r=20, t=60, b=60)
        )

        fig.update_xaxes(range=[x0, x1], showgrid=True, 
                         gridcolor='lightgray', zeroline=False)
        fig.update_yaxes(range=[y0, y1], showgrid=True, 
                         gridcolor='lightgray', zeroline=False)

        # add a black rectangle border around the plotting area
        fig.update_layout(
            shapes=[dict(
            type="rect",
            xref="x",
            yref="y",
            x0=x0,
            x1=x1,
            y0=y0,
            y1=y1,
            line=dict(color="black", width=2),
            fillcolor="rgba(0,0,0,0)"
            )]
        )

        fig.show()

    return_dict = {
        'fig': fig,
        'V_array': x,
        'dIdVsignal': y}      

    return return_dict

tunnelingBroadenedStepFunction(x_V)

Compute the tunneling broadened step function.

This function calculates a broadened step function used in tunneling spectroscopy.

Parameters:

Name Type Description Default
x_V array_like or float

Input value(s) at which to evaluate the broadened step function.

 required

Returns:

Name Type Description
y ndarray or float

The value(s) of the tunneling broadened step function at x_V. If the computation results in NaN values, they are replaced with 0.0.
Source code in spinfinity/dIdV.py 
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
def tunnelingBroadenedStepFunction(x_V):
    """
    Compute the tunneling broadened step function.

    This function calculates a broadened step function used in tunneling spectroscopy.

    Parameters
    ----------
    x_V : array_like or float
        Input value(s) at which to evaluate the broadened step function.

    Returns
    -------
    y : ndarray or float
        The value(s) of the tunneling broadened step function at `x_V`. 
        If the computation results in NaN values, they are replaced with 0.0.
    """
    z = np.exp(x_V)
    y = (1 + (x_V - 1) * z) / ((z - 1) ** 2)
    y = np.nan_to_num(y, nan=0.0)
    return y