Using three-point current reversal to reduce error in low resistance/power measurements

Adam Daire, Keithley Instruments Inc.

May 25, 2007

Adam Daire, Keithley Instruments Inc.

Electronic components continue to shrink as consumers demand faster, more feature-rich products in ever-smaller form factors. Because of their small sizes, these components usually have limited power-handling capability.

As a result, when electrically characterizing these components, the test signals need to be kept small to prevent component breakdown or other damage.

Testing these devices and materials often requires low voltage measurements. This involves sourcing a known current, measuring the resulting voltage and calculating resistance.

If the device has a low resistance, the resulting voltage will be very small. Thus, great care is needed to reduce offset voltage and noise, which can normally be ignored when measuring higher signal levels.

Even if the resistance is far from zero, the voltage to be measured is often very small due to the need to source only a small current and avoid damaging the device. This power limitation often makes characterizing the resistance of modern devices and materials very challenging.

Low-level measurements
There are many factors that make low-voltage measurements dif- ficult. For instance, various noise sources can hinder resolving the actual voltage, and thermoelectric voltages (thermoelectric EMFs) can cause error offsets and drift in voltage readings.

In the past, one could simply increase the test current until the DUT's response voltage was much larger than these errors.

But with today's smaller devices, this is no longer an option. Increased test current can result in device heating, changes in the device's resistance, or even the destruction of the device. The key to obtaining accurate, consistent measurements is eliminating the error.

Figure 1: (a) The schematic shows a standard DC resistance measurement setup; (b) Changing the standard measurement by using four leads eliminates errors.

For low-voltage measurement applications, error is composed largely of white noise (random noise across all frequencies) and 1/f noise. Thermoelectric voltages typically having 1/f distribution are generated from temperature differences in the circuit.

Resistance is calculated using Ohm's Law - i.e.the DC voltage measured across the device divided by the DC stimulus current yields the resistance. Voltage readings will be the sum of the induced voltage across the device (VR), lead and contact resistance (Vl_ead res), other 1/f noise contributions (V1/f noise), white noise (V_{white noise}) and thermoelectric voltages (V_t).

Using four separate leads to connect the voltmeter and current source to the device eliminates lead resistance because the voltmeter won't measure the voltage drop across the source leads. Implementing filtering may reduce white noise, but will not reduce 1/f noise significantly, which often sets the measurement noise floor.

Thermoelectric voltages normally have a 1/f characteristic. This means there can be significant offset - the more measurements taken, the more drift there will be.

Taken together, the offset and drift may even exceed VR, the voltage across the DUT induced by the applied current. It's possible to reduce thermoelectric voltages using techniques such as all-copper circuit construction, thermal isolation, precise temperature control and frequent contact cleaning.

No matter what steps are taken to minimize thermoelectric voltages, it's impossible to eliminate them. It would be preferable to use a method that would allow accurate resistance measurements even in the presence of large thermoelectric voltages, instead of working to minimize them.

Delta method
One way to eliminate a constant thermoelectric voltage is to use a delta method in which voltage measurements are made first at a positive then at a negative test current. A modified technique can be used to compensate for changing thermoelectric voltages.

Over the short term, thermoelectric drift can be approximated as a linear function. The difference between consecutive voltage readings is the slope or the rate of change in thermoelectric voltage.

This slope is constant, so it may be canceled by alternating the current source three times to make two delta measurements - one at a negative-going step and one at a positive-going step.

For the linear approximation to be valid, the current source must alternate quickly and the voltmeter must make accurate voltage measurements within a short interval. If these conditions are met, the three-step delta technique yields an accurate voltage reading of the intended signal unaffected by thermoelectric offsets and drifts.

An analysis of the mathematics for one three-step delta cycle will demonstrate how the technique compensates for temperature differences in the circuit, thereby reducing measurement error.

Figure 2: The graph depicts an alternating, three-point delta method of measuring voltage with no thermoelectric voltage error; (b) A linearly increasing temperature generates a changing thermoelectric voltage error, which is eliminated by the three-point delta method.

Consider the example in Figure 2a above where: Test current = ±5 nanoamperes and device = 500 ohm resistance. Ignoring thermoelectric voltage errors, the voltages measured at each of the steps are:

V1 = 2.5 microvolts
V2 = "2.5 microvolts
V3 = 2.5 microvolts

Let's assume the temperature is linearly increasing over the short term in such a way that it produces a voltage profile like that shown in Figure 2b above, where Vt is climbing 100nV with each successive reading.

As Figure 2b shows, the voltages now measured by the voltmeter include error due to the increasing thermoelectric voltage in the circuit and are no longer of equal magnitude.

However, the absolute difference between the measurements is in error by a constant 100nV, so it's possible to cancel this term. The first step is to calculate the delta voltages. The first delta voltage (Va) is equal to:

Va = negative-going step = (V1 " V2)/2 = 2.45 microvolts

The second delta voltage (Vb) is made at the positive-going current step and is equal to:

Vb = positive-going step = (V3 " V2)/2 = 2.55 microvolts

The thermoelectric voltage adds a negative error term in Va and a positive error term in the calculation of Vb. When the thermal drift is linear, these error terms are equal in magnitude. Thus, we can cancel the error by taking the average of Va and Vb:

Vf = final voltage reading = (Va + Vb)/2 = ½[(V1 " V2)/2 + (V3 " V2)/2] = 2.5 microvolts

The delta technique eliminates error due to changing thermoelectric voltages. Therefore, the voltmeter measurement is the voltage induced by the stimulus current alone.

As alternation continues, every successive reading is the average of the three most recent A/D conversions. The three-step delta technique is the best choice for high-accuracy resistance measurements.

Figure 3: The graph compares the results of applying a two- and three-point delta method and shows significant noise reduction using the three-point method.

Figure 3 above compares 1,000 measurements of a 100 ohms resistor made with a 10 nanoAmperes test current taken over approximately 100s. In this example, the rate of change in thermoelectric voltage is no more than 7 microvolts/second.

The two step delta technique fluctuates 30 percent as the thermoelectric error voltage drifts. In contrast, the three-step delta technique has much lower noise - the measurement is unaffected by the thermoelectric variations in the test circuit.

Equipment requirements
The success of the three-step delta method depends on the linear approximation of the thermal drift viewed over a short interval. This approximation requires that the measurement cycle time be faster than the thermal time constant of the test system.

This imposes certain requirements on the current source and voltmeter used. The current source must alternate quickly in evenly timed intervals so that the thermoelectric voltage changes at equal amounts between measurements.

Figure 4: The I-V curve method involves differentiating the signal, which amplifies noise.

The voltmeter must be tightly synchronized with the current source and capable of making accurate measurements over short intervals.

Synchronization favors hardware handshaking between instruments so that the voltmeter can make voltage measurements only after the current source has settled; the current source doesn't switch polarity until after the voltage measurement has been completed.

The measurement speed of the voltmeter is critical in determining total cycle time; faster voltage measurements mean shorter cycle times.

For reliable resistance measurements, the voltmeter must maintain this speed without sacrificing low-noise characteristics. In low-power applications, the current source must be capable of outputting low values of current so as not to exceed the maximum power rating of the device. This ability is particularly important for moderately highand high-impedance devices.

Another important measurement technique for characterizing solid-state and nanoscale devices is differential conductance. For these materials, things are rarely simplified to Ohm's Law. With these nonlinear devices, the resistance is no longer a constant, so a detailed measurement of the slope of that I-V curve at every point is needed to study them (Figure 4, above).

This derivative is called the differential conductance, dG = dI/dV (or its inverse, the differential resistance, dR = dV/dI). The fundamental reason that differential conductance is interesting is that the conductance reaches a maximum at voltages or electron energies (eV) at which the electrons are most active.

In different fields, this measurement may be called electron energy spectroscopy, tunneling spectroscopy or density of states.

Differential conductance measurements
Typically, researchers perform differential conductance measurements using one of two methods: obtaining an I-V curve with a calculated derivative or using an AC technique (Figure 5, below).

Figure 5: The AC technique can use as many as six components, making it a far more complex setup than the I-V curve method.

The I-V curve method requires only one source and one measurement instrument, which makes it relatively easy to coordinate and control.

A current-voltage sweep is made and the mathematical derivative is found. However, taking the mathematical derivative amplifies any measurement noise, so tests must be run multiple times and the results averaged to smooth the curve before the derivative is calculated. This leads to long test times.

Figure 6: Differential conductance measurements can be made using just two instruments that incorporate all of the instruments used in the AC technique.

The AC technique (Figure 6, above) reduces noise and test times. It superimposes a low amplitude AC sine wave on a swept DC bias. This involves many pieces of equipment and is hard to control and coordinate. Assembling such a system is time-consuming and requires extensive knowledge of electrical circuitry. So while the AC technique produces marginally lower noise, it is much more complex.

There is, however, another way to obtain differential conductance measurements. This simple and low-noise technique involves a current source that combines the DC and AC components into one instrument.

There is no need to do a secondary measure of the current because the instrument is a true current source. Figure 7 below shows the current sourced in a differential conductance measurement

Figure 7: The waveform used in the new technique is a linear staircase function that combines an alternating current with a staircase current.

The waveform can be broken down into an alternating current and a staircase current. Using the exact same calculations as in the delta method, accurate resistance or conductance measurements can be made, with measurements at each point of the staircase.

Three-step delta benefits
Because the three-step delta technique eliminates linearly drifting offsets, it is also immune to the effects of a linearly changing staircase. In addition, the nanovoltmeter used in this method has lower noise than lock-in amplifiers at the alternation frequency.

There are several benefits to this method. One is that in the areas of highest conductance, more data points are taken by sourcing the sweep in equal current steps. These areas are of greatest interest to researchers and give detailed data. In addition, using just one instrument to source current and measure voltage greatly simplifies equipment setup. Lastly, reduced noise can lower test times from an hour to only 5mins.

Thermoelectric EMFs are often the dominant source of error in low resistance/low power resistance measurements. This error can be almost completely removed using a three-point current reversal technique.

This means it's no longer necessary to take extreme care to minimize thermally-induced voltage noise in the wiring of resistance measuring systems. Applying the same technique to differential conductance measurements considerably reduces noise and test complexity.

Adam Daire is Product Marketer at Keithley Instruments Inc.