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""" This sub-module contains DC I-V curve models.
These models range from simple (e.g., 'perfect' and 'polynomial') to
complex (e.g., 'exponential' and 'expanded').
"""
import numpy as np
# I-V curve models -----------------------------------------------------------
def perfect(voltage):
"""Generate a perfect I-V curve.
This is the ideal I-V curve. The current is equal to 0 when the normalized
bias voltage is less than 1, equal to 0.5 when the normalized bias voltage
is equal to 1, and equal to the normalized bias voltage otherwise.
The current and voltage are normalized to Igap and Vgap, respectively,
where Igap=Vgap/Rn is the gap current, Vgap is the gap voltage, and Rn is
the normal resistance.
Args:
voltage (ndarray): normalized bias voltage
Returns:
ndarray: normalized current
"""
if isinstance(voltage, np.ndarray):
current = np.copy(voltage)
current[voltage == 1.] = 0.5
current[voltage == -1.] = -0.5
current[np.abs(voltage) < 1] = 0
return current
elif isinstance(voltage, float) or isinstance(voltage, int):
if np.abs(voltage) < 1:
return 0
if np.abs(voltage) > 1:
return voltage
if voltage == 1.:
return 0.5
if voltage == -1.:
return -0.5
def perfect_kk(voltage, max_kk=100.):
"""Generate Kramers-Kronig transform of the perfect I-V curve.
This is an analytic solution.
Args:
voltage (ndarray): normalized bias voltage
max_kk (float, optional): if output is NaN, use this max value
instead, default is 100
Returns:
ndarray: kk-transform of perfect i-v curve
"""
if isinstance(voltage, np.ndarray):
kk = np.empty_like(voltage, dtype=float)
mask = np.abs(voltage) != 1.
kk[mask] = (-1 / np.pi * (2 + voltage[mask] * (np.log(np.abs((voltage[mask] - 1) / (voltage[mask] + 1))))))
# np.log(v-1) will cause result=Nan at v=-1 and v=1
# replace with max value instead
kk[np.invert(mask)] = max_kk
return kk
elif isinstance(voltage, float) or isinstance(voltage, int):
if abs(voltage) == 1.:
return max_kk
else:
return -1 / np.pi * (2 + voltage * (np.log(abs((voltage - 1) / (voltage + 1)))))
def polynomial(voltage, order=50):
"""Generate the polynomial I-V curve model.
From Kennedy (1999) [see full references in online docs].
Args:
voltage (ndarray): normalized bias voltage
order (float): order of polynomial (usually between 30 and 50)
Returns:
ndarray: normalized current
"""
current = voltage ** (2 * order + 1) / (1 + voltage ** (2 * order))
return current
def exponential(voltage, vgap=2.8e-3, rn=14., rsg=300., agap=4e4, model='fixed'):
"""The exponential I-V curve model that is used in some papers from
Chalmers.
From Rashid et al. (2016) [see full references in online docs].
Note:
- The equation from this paper will result in an I-V curve that has a
subgap resistance that is half the value that it is supposed to be.
- The normal resistance will also be slightly lower than it is
supposed to be.
- I fixed this model. This model can be selected by setting
``model='fixed'``. The original model can be selected by
setting ``model='original'``.
Args:
voltage (ndarray): normalized bias voltage
vgap (float, optional): gap voltage, in units [V], default is 2.8e-3
rn (float, optional): normal resistance, in units [ohms], default
is 14
rsg (float, optional): sub-gap resistance, in units [ohms], default
is 300
agap (float, optional): gap linearity coefficient (typically around
4e4), default is 4e4
model (str, optional): model to used (either 'fixed' or 'original'),
default is "fixed"
Returns:
ndarray: normalized current
"""
igap = vgap / rn
v_v = voltage * vgap # voltage in units [V]
if model.lower() == 'fixed' or model.lower() == 'corrected':
np.seterr(over='ignore')
i_a = (
# Sub-gap resistance
v_v / (rsg * 2) * (1 / (1 + np.exp(-agap * (v_v + vgap)))) -
v_v / (rsg * 2) * (1 / (1 + np.exp(agap * (v_v + vgap)))) -
v_v / (rsg * 2) * (1 / (1 + np.exp(-agap * (v_v - vgap)))) +
v_v / (rsg * 2) * (1 / (1 + np.exp(agap * (v_v - vgap)))) +
# Normal resistance
v_v / rn * (1 / (1 + np.exp(agap * (v_v + vgap)))) +
v_v / rn * (1 / (1 + np.exp(-agap * (v_v - vgap)))))
return i_a / igap
elif model.lower() == 'original':
np.seterr(over='ignore')
i_a = (
# Sub-gap resistance
v_v / rsg * (1 / (1 + np.exp(-agap * (v_v + vgap)))) +
v_v / rsg * (1 / (1 + np.exp(agap * (v_v - vgap)))) +
# Normal resistance
v_v / rn * (1 / (1 + np.exp(agap * (v_v + vgap)))) +
v_v / rn * (1 / (1 + np.exp(-agap * (v_v - vgap)))))
return i_a / igap
else:
raise ValueError("Model not recognized.")
def expanded(voltage, vgap=2.8e-3, rn=14., rsg=5e2, agap=4e4, a0=1e4,
ileak=5e-6, vnot=2.85e-3, inot=1e-5, anot=2e4, ioff=1e-5):
"""My "expanded" I-V curve model.
This model is based on the exponential I-V curve model, but I have added
the ability to include leakage current, the proximity effect, the onset of
thermal tunnelling, and the reduced current amplitude often seen
above the gap. It is able to recreate experimental data very well, but it
is very complex.
Args:
voltage (ndarray): normalized bias voltage
vgap (float, optional): gap voltage, in units [V], default is 2.8e-3
rn (float, optional): normal resistance, in units [ohms], default
is 14
rsg (float, optional): sub-gap resistance, in units [ohms], default
is 5e2
agap (float, optional): gap linearity coefficient, default is 4e4
a0 (float, optional): linearity coefficient at the origin, default
is 1e4
ileak (float, optional): amplitude of leakage current, default is 5e-6
vnot (float, optional): notch location, in units [V], default
is 2.85e-3
inot (float, optional): notch current amplitude, in units [A], default
is 1e-5
anot (float, optional): linearity of notch, default is 2e4
ioff (float, optional): current offset, default is 1e-5
Returns:
ndarray: normalized current
"""
v_v = voltage * vgap
igap = vgap / rn
np.seterr(over='ignore')
i_a = (
# Leakage current
ileak * 2 * (1 / (1 + np.exp(-a0 * v_v))) -
ileak * np.ones_like(voltage) -
ileak * (1 / (1 + np.exp(-agap * (v_v - vgap)))) +
ileak * (1 / (1 + np.exp(agap * (v_v + vgap)))) +
# Sub-gap resistance
v_v / (rsg * 2) * (1 / (1 + np.exp(-agap * (v_v + vgap)))) -
v_v / (rsg * 2) * (1 / (1 + np.exp(agap * (v_v + vgap)))) -
v_v / (rsg * 2) * (1 / (1 + np.exp(-agap * (v_v - vgap)))) +
v_v / (rsg * 2) * (1 / (1 + np.exp(agap * (v_v - vgap)))) +
# Transition and normal resistance
v_v / rn * (1 / (1 + np.exp(agap * (v_v + vgap)))) +
v_v / rn * (1 / (1 + np.exp(-agap * (v_v - vgap)))) +
# Notch above gap (proximity effect)
inot / (1 + np.exp(anot * (v_v - vnot))) +
inot / (1 + np.exp(anot * (v_v + vnot))) - inot +
# Current offset seen above the gap
ioff / (1 + np.exp(agap * (v_v - vgap))) +
ioff / (1 + np.exp(agap * (v_v + vgap))) - ioff +
0)
return i_a / igap
|
