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

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

As a result, when electrically characterizing thesecomponents, the test signals need to be kept small to prevent componentbreakdown or other damage.

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

If the device has a low resistance, the resulting voltage will bevery small. Thus, great care is needed to reduce offset voltage andnoise, which can normally be ignored when measuring higher signallevels.

Even if the resistance is far from zero, the voltage to be measuredis often very small due to the need to source only a small current andavoid damaging the device. This power limitation often makescharacterizing the resistance of modern devices and materials verychallenging.

Low-level measurements

There are many factors that make low-voltage measurements dif- ficult.For instance, various noise sources can hinder resolving the actualvoltage, and thermoelectric voltages (thermoelectricEMFs ) can causeerror offsets and drift in voltage readings.

In the past, one could simply increase the test current until theDUT's response voltage was muchlarger 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 thedevice's resistance, or even the destruction of the device. The key toobtaining accurate, consistent measurements is eliminating the error.

Figure1: (a) The schematic shows a standard DC resistance measurement setup;(b) Changing the standard measurement by using four leads eliminateserrors. |

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

Resistance is calculated using Ohm'sLaw – i.e.the DC voltage measured across the device dividedbythe DC stimulus current yields the resistance. Voltage readings will bethe sum of the induced voltage across the device (VR), lead and contactresistance (Vl_{ead} res), other 1/f noise contributions (V1/fnoise),white noise (V_{white noise} ) and thermoelectric voltages (V_{t} ).

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

Thermoelectric voltages normally have a 1/f characteristic. Thismeans 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 voltageacross the DUT induced by the applied current. It's possible to reducethermoelectric voltages using techniques such as all-copper circuitconstruction, thermal isolation, precise temperature control andfrequent contact cleaning.

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

Delta method

One way to eliminate a constant thermoelectric voltage is to use adelta method in which voltage measurements are made first at a positivethen at a negative test current. A modified technique can be used tocompensate for changing thermoelectric voltages.

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

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

For the linearapproximation to be valid, the current source mustalternate quickly and the voltmeter must make accurate voltagemeasurements within a short interval. If these conditions are met, thethree-step delta technique yields an accurate voltage reading of theintended signal unaffected by thermoelectric offsets and drifts.

An analysis of the mathematics for one three-step delta cycle willdemonstrate how the technique compensates for temperature differencesin the circuit, thereby reducing measurement error.

Figure2: The graph depicts an alternating, three-point delta method ofmeasuring voltage with no thermoelectric voltage error; (b) A linearlyincreasing temperature generates a changing thermoelectric voltageerror, which is eliminated by the three-point delta method. |

Consider the example in Figure 2aabove where: Test current = ±5 nanoamperes and device =500 ohm resistance. Ignoring thermoelectric voltage errors, thevoltages 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 shortterm in such a way that it produces a voltage profile like that shownin Figure 2b above , where Vtis climbing 100nV with each successive reading.

As Figure 2b shows, thevoltages now measured by the voltmeter include error due to theincreasing thermoelectric voltage in the circuit and are no longer ofequal magnitude.

However, the absolute difference between the measurements is inerror by a constant 100nV, so it's possible to cancel this term. Thefirst 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 currentstep 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 apositive error term in the calculation of Vb. When the thermal drift islinear, these error terms are equal in magnitude. Thus, we can cancelthe 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 thermoelectricvoltages. Therefore, the voltmeter measurement is the voltage inducedby the stimulus current alone.

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

Figure3: The graph compares the results of applying a two- and three-pointdelta method and shows significant noise reduction using thethree-point method. |

Figure 3 above compares1,000 measurements of a 100 ohms resistor made with a 10 nanoAmperestest current taken over approximately 100s. In this example, the rateof change in thermoelectric voltage is no more than 7microvolts/second.

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

Equipment requirements

The success of the three-step delta method depends on the linearapproximation of the thermal drift viewed over a short interval. Thisapproximation requires that the measurement cycle time be faster thanthe thermal time constant of the test system.

This imposes certain requirements on the current source andvoltmeter used. The current source must alternate quickly in evenlytimed intervals so that the thermoelectric voltage changes at equalamounts between measurements.

Figure4: The I-V curve method involves differentiating the signal, whichamplifies noise. |

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

Synchronization favors hardware handshaking between instruments sothat the voltmeter can makevoltage measurements only after the currentsource has settled; the current source doesn't switch polarity untilafter the voltage measurement has been completed.

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

For reliable resistance measurements, the voltmeter must maintainthis speed without sacrificing low-noise characteristics. In low-powerapplications, the current source must be capable of outputting lowvalues of current so as not to exceed the maximum power rating of thedevice. This ability is particularly important for moderately highandhigh-impedance devices.

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

This derivative is called the differential conductance, dG =dI/dV(or its inverse, the differentialresistance, dR = dV/dI).Thefundamental reason that differential conductance is interesting is thatthe 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 electronenergyspectroscopy, tunnelingspectroscopy or density of states .

Differential conductancemeasurements

Typically, researchers perform differential conductance measurementsusing one of two methods: obtaining an I-V curve with a calculatedderivative or using an AC technique (Figure5, below ).

Figure5: The AC technique can use as many as six components, making it a farmore complex setup than the I-V curve method. |

The I-V curve method requiresonly one source and one measurementinstrument, which makes it relatively easy to coordinate and control.

A current-voltage sweep is made and the mathematical derivative isfound. However, taking the mathematical derivative amplifies anymeasurement noise, so tests must be run multiple times and the resultsaveraged to smooth the curve before the derivative is calculated. Thisleads to long test times.

Figure6: Differential conductance measurements can be made using just twoinstruments that incorporate all of the instruments used in the ACtechnique. |

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

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

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

Figure7: The waveform used in the new technique is a linear staircasefunction that combines an alternating current with a staircase current. |

The waveform can be broken down into an alternating current and astaircase current. Using the exact same calculations as in the deltamethod, 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 driftingoffsets, it is also immune to the effects of a linearly changingstaircase. In addition, the nanovoltmeter used in this method has lowernoise than lock-in amplifiers at the alternation frequency.

There are several benefits to this method. One is that in the areasof highest conductance, more data points are taken by sourcing thesweep in equal current steps. These areas are of greatest interest toresearchers and give detailed data. In addition, using just oneinstrument to source current and measure voltage greatly simplifiesequipment setup. Lastly, reduced noise can lower test times from anhour to only 5mins.

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

Thismeans it's no longer necessary to take extreme care to minimizethermally-induced voltage noise in the wiring of resistance measuringsystems. Applying the same technique to differential conductancemeasurements considerably reduces noise and test complexity.

Adam Daire is Product Marketer at Keithley Instruments Inc.