This page describes how to set up a four-wire resistance measurement of a probe of indium tin oxide ITO.

Before the measurement in the climate chamber started, the resistance at room temperature was measured to identify in which value range the resistance of the probe is for a 2-wire measurement, a 2-wire measurement with constant voltage and a 4-wire measurement and to check if the measurement has an influence on the probe. After connecting the wires with the SMU and the probe, the python code below was used to measure the resistance.

from KeithleyV15 import SMU26xx
import datetime
import csv
import numpy as np
import matplotlib.pyplot as plt
import time
import probscale
import atexit
"""
Example Four-terminal sensing (Vierleitermessung) with one channel

First 2-wire measurement with Channel A -> R_I + R_M are measured
Than 4-wire measurement with Channel A Sense-wire R_M is measured

-------------------------------
     |                             |
    |-|              --------------�
    | | R_I          |             |
    |-|              |            |-|
     |               |            | | R_M
  ------- SMU_A   ------- SMU_A   |-|
  |  A  | voltage | A_S | Sense    |
  ------- source  ------- wires    |
     |               |             |
     ----------------�--------------

A...   SMU Channel A voltage source
A_S... SMU Channel A sense wires for the measurement of R_M only used in 4-wire mode
R_M... Resistor that you want to measure
R_I... Resistor that interferes with the direct measurement of R_M in 2-wire mode

In this example the R_I is simulating possible lead and contact resistances, that are interfering with a direct
measurement of R_M. Using the Four-terminal sensing (Vierleitermessung) with the 4-wire mode the lead and contact
resistances are ignored and only the R_M is measured. This setup is used for electrical resistance measurements in
which the resistor that is measured has a resistance in the range of the lead and contact resistances or when a very
precise measurement is necessary.

The 2-wire measurement will measure the series circuit of R_M + R_I (+ lead and contact resistances)
The 4-wire measurement will measure the R_M

"""

# Connect to the Sourcemeter
try:
    sm = SMU26xx("TCPIP0::129.27.158.13::inst0::INSTR")
except:
    sm = SMU26xx("TCPIP0::129.27.158.13::inst0::INSTR")

atexit.register(lambda: sm.disconnect() )

# Select the channel that is connected
#smu = sm.get_channel(sm.CHANNEL_A)
smu = sm.get_channel(sm.CHANNEL_B)

""" Define the current for both measurements. The current limit for all measurements is 10*current."""
""" The current has to be so low that the measured voltage is less than 20 V!!!!"""

current = 1e-5

#---------------------------------------------------------------------------
''' setup files for saving data '''
pathname = './'
filename_base = '4point_measure'
# Create unique filenames for saving the data
time_for_name = datetime.datetime.now().strftime("%Y_%m_%d_%H%M%S")
filename_csv = pathname + filename_base + time_for_name +'.csv'
filename2_csv = pathname + filename_base + '_summary_' + time_for_name +'.csv'
# optional: consider also graphical representation
filename_pdf = pathname + filename_base + time_for_name +'.pdf'
filenameprob_pdf = pathname + filename_base + time_for_name + '_prob' +'.pdf'

# Header for csv
with open(filename_csv, 'a') as csvfile:
        writer = csv.writer(csvfile, delimiter=';',  lineterminator='\n')
        writer.writerow(["R2wire / Ohms", "V2wire /V", "I2wire /A",
                         "R2wire_constV / Ohms", "V2wire_constV /V", "I2wire_constV /A",
                         "R4wire/Ohms", "V4wire /V", "I4wire /A",])

with open(filename2_csv, 'a') as csvfile2:
        writer2 = csv.writer(csvfile2, delimiter=';',  lineterminator='\n')
        #writer2.writerow(["mean", "mean", "mean"])
        #writer2.writerow(["R2wire / Ohms", "R2wire_constV / Ohms", "R4wire/Ohms"])
        writer2.writerow(["measurement", "mean / Ohms", "standard deviation / Ohms"])

#---------------------------------------------------------------------------

R2wire=[]
R2wire_cvoltage=[]
R4wire=[]
n     = 50;

for i in range(n):
    """ Configure Channel B for a 2-wire measurement """

# reset to default settings
    smu.reset()

# set the sense mode to local (2-wire) - this is not necessary if you reset the channel previously
    smu.set_sense_2wire()

# setup the operation mode and what will be shown at the display
    smu.set_mode_current_source()
    smu.display_resistance()

# define the initial parameters for Channel A
    smu.set_voltage_range(1)
    smu.set_voltage_limit(1)
    smu.set_voltage(0)
    smu.set_current_range(current*10)
    smu.set_current_limit(current*10)
    smu.set_current(0)

""" Do the 2-wire measurement """

# enable the outputs
    smu.enable_output()

# set the current
    smu.set_current(current)

# measure the current and the voltage
    current2wire = smu.measure_current()
    voltage2wire = smu.measure_voltage()

# disable the outputs
    smu.disable_output()

#------------------------------------------------------------------------------------

""" Configure Channel B for a 2-wire measurement with constant voltage """

# reset to default settings
    smu.reset()

# set the sense mode to local (2-wire) - this is not necessary if you reset the channel previously
    #smu.set_sense_2wire()

# setup the operation mode and what will be shown at the display
    smu.set_mode_voltage_source()
    smu.display_resistance()

# define the initial parameters for Channel B
    smu.set_voltage_range(1)
    smu.set_voltage_limit(1)
    smu.set_voltage(0)
    smu.set_current_range(current*10)
    smu.set_current_limit(current*10)
    smu.set_current(0)

""" Do the 2-wire measurement by applying constant voltage """

# enable the outputs
    smu.enable_output()

# set the voltage
    smu.set_voltage(0.1)

# measure the current and the voltage
    current2wire_cvoltage = smu.measure_current()
    voltage2wire_cvoltage = smu.measure_voltage()

# disable the outputs
    smu.disable_output()

#-----------------------------------------------------------------------------------

""" Configure Channel A for a 4-wire measurement """
    # reset to default settings
    smu.reset()

# setup the operation mode and what will be shown at the display
    smu.set_mode_current_source()
    smu.display_resistance()

# set the sense mode to remote (4-wire)
    smu.set_sense_4wire()

""" Do the 4-wire measurement """

# enable the outputs
    smu.enable_output()

# set the current
    #smu.set_current(current*(i%2))
    smu.set_current(current)
    # measure the current and the voltage
    current4wire = smu.measure_current()
    voltage4wire = smu.measure_voltage()

# disable the outputs
    smu.disable_output()

""" Calculate and display the Measurement """

R2wire.append(voltage2wire / current2wire)
    R2wire_cvoltage.append(voltage2wire_cvoltage / current2wire_cvoltage)
    R4wire.append(voltage4wire / current4wire)

print("Cycle: " + str(i))
    print("Voltage 2-Wire = " + str(voltage2wire) + ". Current 2-Wire = " + str(current2wire))
    print("Voltage 2-Wire Const. Voltage = " + str(voltage2wire_cvoltage) + ". Current 2-Wire = " + str(current2wire_cvoltage))
    print("Voltage 4-Wire = " + str(voltage4wire) + ". Current 4-Wire = " + str(current4wire))

print("The resistance measured with the 2-wire mode is " + str(R2wire[i]/1e6) + "MOhm")
    print("The resistance measured with the 2-wire mode with constant voltage is " + str(R2wire_cvoltage[i]/1e6) + "MOhm")
    print("The resistance measured with the 4-wire mode is " + str(R4wire[i]/1e6) + "MOhm")

# write data to csv
    with open(filename_csv, 'a') as csvfile:
         writer = csv.writer(csvfile, delimiter=';',  lineterminator='\n')
         writer.writerow([R2wire[i], voltage2wire,current2wire,
                          R2wire_cvoltage[i],voltage2wire_cvoltage,current2wire_cvoltage,
                          R4wire[i], voltage4wire, current4wire])

time.sleep(1)

print("\n\n\nResistance 2-wire mode ="+str(np.mean(R2wire)/1e6)+"+/-"+str(np.std(R2wire)/1e6)+"MOhm")
print("Resistance 2-wire mode with constant voltage ="+str(np.mean(R2wire_cvoltage)/1e6)+"+/-"+str(np.std(R2wire_cvoltage)/1e6)+"MOhm")
print("Resistance 4-wire mode ="+str(np.mean(R4wire)/1e6)+"+/-"+str(np.std(R4wire)/1e6)+"MOhm")

with open(filename2_csv, 'a') as csvfile2:
    writer2 = csv.writer(csvfile2, delimiter=';',  lineterminator='\n')
    #writer2.writerow([np.mean(R2wire), np.mean(R2wire_cvoltage),np.mean(R4wire)])
    #writer2.writerow(["std", "std", "std"])
    #writer2.writerow([np.std(R2wire), np.std(R2wire_cvoltage),np.std(R4wire)])
    #writer2.writerow(["R2wire / Ohms", "R2wire_constV / Ohms", "R4wire/Ohms"])
    writer2.writerow(["R2wire", np.mean(R2wire), np.std(R2wire)])
    writer2.writerow(["R2wire_constV", np.mean(R2wire_cvoltage),np.std(R2wire_cvoltage)])
    writer2.writerow(["R4wire/Ohms", np.mean(R4wire),np.std(R4wire)])

""" Disconnect from the SMU """

# reset the SMU
smu.reset()

# disconnect from the SMU
sm.disconnect()

# plot?
fig, [ax1, ax2] = plt.subplots(2,1, sharex=True)
acquistion_cycle = np.arange(1, n+1)
#print(acquistion_cycle.shape)
#print(R4wire.shape)
ax1.plot(acquistion_cycle, R4wire,'x-', color = 'red', linewidth=2, label="R4wire")
ax2.plot(acquistion_cycle, R2wire,'x-', color = 'blue', linewidth=2, label="R2wire")
ax2.plot(acquistion_cycle, R2wire_cvoltage,'x-', color = 'green', linewidth=2, label="R2wire_constV")
#plt.bar(acquistion_cycle, R2wire)
#plt.bar(acquistion_cycle, R2wire_cvoltage)
#plt.bar(acquistion_cycle, R4wire)

ax2.set_xlabel('cycle')
ax1.set_ylabel('Resistance / Ohms')
ax2.set_ylabel('Resistance / Ohms')
plt.suptitle('Comparison of 2wire and 4wire resistances ')
# ax2.set_tick_params(labelsize = 14)
ax2.legend(loc='lower right')
plt.savefig(filename_pdf)
plt.show()

"""_, [ax1,ax2,ax3] = plt.subplots(1,3)
probscale.probplot(
    R2wire,
    ax=ax1,
    probax='y', # flip the plot
    bestfit=True, # draw a best-fit line
    estimate_ci=True,
    datalabel='2W CC Measurement Points',  # labels and markers...
    problabel='Non-exceedance probability',
    scatter_kws=dict(marker='d', zorder=2, mew=1.25, mec='w', markersize=10),
    line_kws=dict(color='0.17', linewidth=2.5, zorder=0, alpha=0.75),
)
probscale.probplot(
    R2wire_cvoltage,
    ax=ax2,
    probax='y', # flip the plot
    bestfit=True, # draw a best-fit line
    estimate_ci=True,
    datalabel='2W CV Measurement Points',  # labels and markers...
    problabel='Non-exceedance probability',
    scatter_kws=dict(marker='d', zorder=2, mew=1.25, mec='w', markersize=10),
    line_kws=dict(color='0.17', linewidth=2.5, zorder=0, alpha=0.75),
)
probscale.probplot(
    R4wire,
    ax=ax3,
    probax='y', # flip the plot
    bestfit=True, # draw a best-fit line
    estimate_ci=True,
    datalabel='4W Measurement Points',  # labels and markers...
    problabel='Non-exceedance probability',
    scatter_kws=dict(marker='d', zorder=2, mew=1.25, mec='w', markersize=10),
    line_kws=dict(color='0.17', linewidth=2.5, zorder=0, alpha=0.75),
)
plt.savefig(filenameprob_pdf)
plt.show()"""

four-terminal_sensing_SenseMode_modified.py

As can be seen in the figure below, the fluctuation of the values for the resistance during the measurements are smaller than 0.5%, that means that the measurement does not have a big influence on the probe.

To figure out the temperature depence of the resistivity and the voltage, the measurements were then performed in a climate chamber, which switched the temperature in 2 K steps from T = (10,0 ± 0,2)°C to T = (40,0 ± 0,2)°C, using the python code below.

from KeithleyV15 import SMU26xx
from voetsch import VT4002
import datetime
import csv
import numpy as np
import matplotlib.pyplot as plt
import time
import probscale

"""
Example Four-terminal sensing (Vierleitermessung) with one channel

First 2-wire measurement with Channel A -> R_I + R_M are measured
Then 4-wire measurement with Channel A Sense-wire R_M is measured

The 2-wire measurement will measure the series circuit of R_M + R_I (+ lead and contact resistances)
The 4-wire measurement will measure the R_M

"""

# Connect to the Sourcemeter
#sm = SMU26xx("TCPIP0::129.27.158.13::inst0::INSTR")

# Select the channel that is connected
#smu = sm.get_channel(sm.CHANNEL_A)
#smu = sm.get_channel(sm.CHANNEL_B)

""" Define the current for both measurements. The current limit for all measurements is 10*current."""
""" The current has to be so low that the measured voltage is less than 20 V!!!!"""

# define connection parameters
ip_address = "129.27.158.42"
username = "simpacuser"   # fill in the username on the front of the climate chamber
password = "u1s2e3r4"   # fill in the password on the front of the climate chamber

climate_chamber = VT4002(ip_address, username, password)

climate_chamber.enable()

""" ******* Connect to the Sourcemeter ******** """

# initialize the Sourcemeter and connect to it
# you may need to change the IP address depending on which sourcemeter you are using
sm = SMU26xx("TCPIP0::129.27.158.13::inst0::INSTR")

# get one channel of the Sourcemeter (we only need one for this measurement)
smu = sm.get_channel(sm.CHANNEL_B)

""" ******* Configure the SMU Channel A ******** """

current = 1e-5

# number measurements
n = 150

# define sweep parameters
sweep_start = 10.
sweep_end = 40.1
sweep_step = 2.

# enable the output
smu.enable_output()

""" ******* Sweep the temperature ******** """

def do_single_meas(meth='cc2w', current=1e-9, voltage=0.2):
    # reset to default settings
    smu.reset()

# set the sense mode to local (2-wire) - this is not necessary if you reset the channel previously
    if meth == 'cc2w':
        smu.set_sense_2wire()
    elif meth == 'cv2w':
        smu.set_mode_voltage_source()
    elif meth == '4w':
        smu.set_sense_4wire()
    else:
        raise RuntimeError()

# setup the operation mode and what will be shown at the display
    smu.set_mode_current_source()
    smu.display_resistance()

# define the initial parameters for Channel A
    smu.set_voltage_range(voltage)
    smu.set_voltage_limit(voltage)
    smu.set_voltage(0)
    smu.set_current_range(current*10)
    smu.set_current_limit(current*10)
    smu.set_current(0)

""" Do the 2-wire measurement """

# enable the outputs
    smu.enable_output()

# set the current
    smu.set_current(current)

# measure the current and the voltage
    current2wire = smu.measure_current()
    voltage2wire = smu.measure_voltage()

# disable the outputs
    smu.disable_output()

return current2wire, voltage2wire

def do_measurements(T=20.):
    voltage2wire = np.zeros(n)
    voltage2wire_cvoltage = np.zeros(n)
    voltage4wire = np.zeros(n)

current2wire = np.zeros(n)
    current2wire_cvoltage = np.zeros(n)
    current4wire = np.zeros(n)

R2wire = np.zeros(n)
    R2wire_cvoltage = np.zeros(n)
    R4wire = np.zeros(n)

for i in range(0, n):
        """ Configure Channel B for a 2-wire measurement """
        current2wire[i], voltage2wire[i] = do_single_meas(meth='cc2w', current=current, voltage=1)

""" Configure Channel B for a 2-wire measurement with constant voltage """
        current2wire_cvoltage[i], voltage2wire_cvoltage[i] = do_single_meas(meth='cv2w', current=current, voltage=1)

""" Configure Channel A for a 4-wire measurement """
        current4wire[i], voltage4wire[i] = do_single_meas(meth='4w', current=current, voltage=1)
        print("{} / {} done".format(i+1, n))

""" Calculate and display the Measurement """

################################################################
    # begin saving

R2wire = voltage2wire / current2wire
    R2wire_cvoltage = voltage2wire_cvoltage / current2wire_cvoltage
    R4wire = voltage4wire / current4wire

fname = 'meas_T{:.2f}'.format(T)
    np.savez(fname + '.npz',
             voltage2wire=voltage2wire, current2wire=current2wire,
             voltage2wire_cvoltage=voltage2wire_cvoltage, current2wire_cvoltage=current2wire_cvoltage,
             voltage4wire=voltage4wire, current4wire=current4wire,
             R2wire=R2wire, R2wire_cvoltage=R2wire_cvoltage, R4wire=R4wire)

plt.figure()
    plt.plot(R2wire, ".")
    plt.plot(R2wire_cvoltage, ".")
    plt.plot(R4wire, ".")
    plt.savefig(fname + '_points.pdf')
    #plt.show(block=False)

plt.figure()
    probscale.probplot(
        voltage4wire * 1e3,
        probax='y', # flip the plot
        bestfit=True, # draw a best-fit line
        estimate_ci=True,
        datalabel='voltage 4W / mV',  # labels and markers...
        problabel='probability',
        scatter_kws=dict(marker='d', zorder=2, mew=1.25, mec='w', markersize=10),
        line_kws=dict(color='0.17', linewidth=2.5, zorder=0, alpha=0.75),
    )
    plt.savefig(fname + '_probV.pdf')
    #plt.show(block=False)

plt.figure()
    probscale.probplot(
        R4wire,
        probax='y', # flip the plot
        bestfit=True, # draw a best-fit line
        estimate_ci=True,
        datalabel='resistivity',  # labels and markers...
        problabel='probability',
        scatter_kws=dict(marker='d', zorder=2, mew=1.25, mec='w', markersize=10),
        line_kws=dict(color='0.17', linewidth=2.5, zorder=0, alpha=0.75),
    )
    plt.savefig(fname + '_probR.pdf')
    #plt.show(block=False)

mean_V = np.mean(voltage4wire)
    std_V  = np.std(voltage4wire) / np.sqrt(n)
    mean_R = np.mean(R4wire)
    std_R  = np.std(R4wire) / np.sqrt(n)

return mean_V, std_V, mean_R, std_R

Ts = np.arange(sweep_start, sweep_end, sweep_step)
#Ts = np.array([23, 24])
mean_V = 0.*Ts
std_V = 0.*Ts
mean_R = 0.*Ts
std_R =  0.*Ts

# define variables we store the measurement in
temp1 = 0.*Ts
temp2 = 0.*Ts

for i, T in enumerate(Ts):
    temp1[i] = climate_chamber.get_actual_temperature()
    print('Temperature 1 (°C): ' + str(temp1[i]))

climate_chamber.go_to_temperature(T, 0.2, 60)

temp1[i] = climate_chamber.get_actual_temperature()
    mean_V[i], std_V[i], mean_R[i], std_R[i] = do_measurements(T)

#Read out the current temperature and print the result to the console
    temp2[i] = climate_chamber.get_actual_temperature()
    print(' ')
    print('Temperature 2 (°C): ' + str(temp2[i]))

temp = (temp1 + temp2)/2

fig, [ax1, ax2] = plt.subplots(2,1, sharex=True)
ax1.plot(temp, mean_V)
ax1.fill_between(temp, mean_V - std_V, mean_V + std_V, zorder=0, alpha=0.3)
ax1.set_ylabel("V / V")
ax2.plot(temp, mean_R)
ax2.fill_between(temp, mean_R - std_R, mean_R + std_R, zorder=0, alpha=0.3)
ax2.set_ylabel("R / $\Omega$")
ax2.set_xlabel("T / °C")

plt.savefig("meas_over_T.pdf")

np.savez('meas_over_T.npz',
             temp=temp,
             mean_V=mean_V, std_V=std_V,
             mean_R=mean_R, std_R=std_R)

""" Disconnect from the SMU """

climate_chamber.go_to_temperature(23, 2, 60)

climate_chamber.disable()

# reset the SMU
smu.reset()

# disconnect from the SMU
sm.disconnect()

plt.show(block=True)

four-terminal_sensing_SenseMode.py

In the figures shown below, which are the measurements for T = 10°C, T = 20°C and T = 40°C, one can see that the values for the resistivity follow a gaussian distribution (line) around the mean, which are laying, depending on the temperature between 2770 Ohm, for high temperatures, and 2735 Ohm for low temperatures. One can see, that the higher the temperature, the lower the resistivity.

In the following figures, which are again the results of the measurements for T = 10°C, T = 20°C and T = 40°C, one can see, that the voltage at the four-point measurements, follows also a gaussian distribution and has a lower value, the higher the temperature is. The mean values are centered around 27 mV.

As a summary, the next figure shows the temperature depence of the resistivity and the voltage from 10°C to 40°C.

Semiconductors show another temperature behaviour than metals. In metals the resistivity increases with higher temperature. In semiconductors the conductivity increases with higher temperature, that means that the resistivity decreases. The higher the thermal energy, the more electrons overcome the band gap between the valence band and the conduction band. The conductivity in the valence band increases, which means that the resistivity decreases. In metals the conduction band and the valence band overlap, so a metal always has charge carriers. When the temperatur increases, the collision of the electrons and therefore the resistivity increases.

Four-wire resistance measurement