Probing thermopower on the microscale



Thermoelectric (TE) generators provide electrical energy from direct conversion of heat by means of the Seebeck effect; without moving parts, completely silent, and with negligible maintenance. As any other heat engine this conversion exploits only a fraction of the Carnot efficiency (Rowe (ed.), CRC Handbook of Thermoelectrics (CRC Press Inc., 1995), p. 19 1). The TE efficiency is linked to the thermoelectric figure of merit Z, which itself is given by basic material properties: Z = S2σ/κ. These are the electrical conductivity σ, the thermal conductivity κ and the Seebeck coefficient or thermopower S, which is known as the factor of proportionality between voltage output and applied temperature difference in a given TE sample. A distinct sensitivity to the carrier concentration and structural variations make the control and stabilisation of thermopower very challenging in complex material structures since degradation by diffusion, decomposition or evaporation can be observed in many cases during synthesis, operation and even in the process of characterisation of TE semiconductors; particularly at elevated temperatures. Investigating structural and compositional properties, stability, and performance of TE materials, and consequently aiming to understand their interaction, mainly methods like X-ray diffraction (XRD), energy-dispersive X-ray spectroscopy (EDX), scanning electron microscopy (SEM) or integral temperature dependent measurements of particular transport properties are used. Although TE materials research satisfies highest requirements on accuracy, the above mentioned techniques are not perfectly qualified to investigate promising material classes thoroughly. Against the background of usually complex material structures this article aims to show, that an efficient characterisation of TE materials becomes accessible for several questions by use of a spatially resolved determination of the thermopower.

1 Probing methods for transport properties

The thermoelectric (TE) figure of merit Z is the central material parameter in the field of TEs and significant efforts are undertaken to optimise TE materials by improving Z. Among the different material classes semiconductors have the best TE properties and are therefore usually employed. Their properties can be shifted by doping or structural adjustment but are interdependent by relations from solid-state physics. Thus, direct adjustment of a single property alone is usually prevented but can be achieved in complex material structures, establishing different transfer characteristics for phonons and charge carriers 2. This allows for an independent tuning of σ and κ to a certain extent, while S is permanently coupled to σ by the density of electronic states 3. Prominent development strategies for bulk materials comprise approaches, which reduce the lattice contribution to the thermal conductivity by phonon scattering in nano-structured materials 4–7. Such materials can be composed of highly ordered or randomly distributed phases with a desirably high mismatch in acoustic impedance. Other interesting materials are composed of a single homogeneous phase containing loosely bound guest atoms within relatively large interstitial voids of their covalent structures giving rise to efficient phonon scattering. However, an improvement of the TE figure of merit or the functional evidence of stable structures and compositions is ultimately relying on the experimental verification of the TE transport properties by means of appropriate measurement techniques. The persistent lack of ready-to-use, accurate and scale-fitted scanning measurement facilities for the functional properties of TE materials is still hampering efficient research on particular micro- or macrosystems and is thus preventing more rapid progress in the development of TE devices. Although a number of attempts have been reported, it has never been achieved yet to combine a complete set of TE properties – the Seebeck coefficient, the electrical and thermal conductivity – simultaneously in one reliable scanning measurement system.

In terms of spatially resolved measurements of transport properties, the majority of concepts are based on atomic force microscopes (AFM) or scanning tunnelling microscopes (STM). Since the invention of STM 8 in the early 1980s many scientific works have been performed in this field, and sophisticated methods arose like scanning thermal (SThM) and scanning thermoelectric microscopy (SThEM). First high resolution methods for measuring the topology 9, the chemical potential 10 and the thermal conductivity 11 were directly stimulated by the STM progress and partially accomplished on AFM assemblies.

In general, both STM- and AFM-based measurements provide atomic spatial resolution. However, differences between these techniques appear, considering the preconditions for a quantitative determination of transport properties. In terms of thermal characterisations by STM methods manifold setups have shown repeatedly highly resolved temperature distributions on various materials and systems. But since the thermal resistance of a tunnel barrier in UHV setups is dominating, a small variation of the local thermal conductivity of the sample is difficult to observe due to the small changes in heat flow 12. Thus a quantitative measurement of the thermal conductivity was solely demonstrated on a micro-thermocouple equipped STM-related setup 13, supported by a quantitative finite element model (FEM) simulation for interpretation of results.

Other approaches 14 used thermally driven tunnel currents to measure a local thermovoltage, which is modulated by the electronic band structure, as it is known from classic STM measurement principles. Based on the Mott formula 15, an approximation for the thermovoltage was introduced by Støvneng and Lipavský 16, which is valid for tunnelling junctions if the local density of states close to the Fermi level is a weakly varying function of energy. This formal relation primarily comprises the densities of electronic states, the average work function and the temperatures of the tunnelling tip and the sample. Since the thermal coupling between the tip and the sample is negligible in vacuum, the temperature difference remains constant 17 while scanning the sample surface. The contrast between different materials, which could be observed, was mainly assigned to variations of the local density of electronic states. This opens more or less wide room, with respect to the deduced absolute S value, for interpretation of the captured thermovoltage, which itself can be measured with high accuracy. Additionally, the density of states discretises with dimension reduction and depends on the material itself. Therefore this approach is qualified to be only conditionally operational in terms of a versatile and direct S measurement technique for various materials and systems, and for a meaningful characterisation of effective material performance.

A remarkable approach was demonstrated by Majumdar et al. 18 based on a thermocouple equipped AFM cantilever probe. Taking concurrent thermal and electrical measurements the ratio of the Seebeck coefficient by the thermal conductivity of the sample was initially determined against a calibration material. Due to the Peltier effect caused by an additional direct electric current, which is passed through the probe–sample contact, a balanced heat flow condition was established at the tip, which finally allows for an absolute determination of the desired quantities. Irrespective of the fact that the local electrical conductivity information cannot be separated from the contact resistance contribution, this approach allows for a direct measurement of the thermal contact resistance by perturbing the thermocouple at the apex of the tip by a small harmonic electrical current. Nevertheless, Majumdar mentioned spurious effects by conduction fluctuations and the need to average the thermovoltages over a long time in order to enhance the accuracy of the S value. Since Majumdar used ultra-sharp tips with sub-30 nm resolution, the contact dimensions became close to the phonon and electron mean free paths for the studied chalcogenide structures, which complicates the interpretation of measurement data.

In this context Lyeo et al. 19 found, that the spatial resolution for a quantitative S measurement is not only limited by the size of the contact but also depends on the size of the non-uniform temperature zone within the sample, which is defined by the phonon mean free path at nanometer scales. Since the non-uniform temperature zone needs to be substantially larger than the electron-phonon scattering mean free path in order to establish a local thermal equilibrium between electrons and phonons, Lyeo and coworkers experimentally recognised a broadening effect on the measured S profile at the length scale of about two or three times of the inter-dopant spacing. Based on his experiments Bian et al. 20 designed a three-dimensional (3D) model for the studied GaAs p/n-junction and stated, that a local TE voltage can only be defined and calculated within a uniform volume larger than the relaxation length of energy/charge carriers and the inter-dopant spacing due to its statistical nature. This spatial limit for local profiling of the Seebeck coefficient consequently amounts to a few upto above 10 nm for relevant TE semiconductors.

A quantitative determination of the local Seebeck coefficient on the microscale was introduced by Süßmann et al. 21 and later used by Dietrich 22. They demonstrated a lateral resolution of 10 µm on Czochralski-grown Bi0.5Sb1.5Te3 samples using a thermocouple-equipped micro-tip in contact mode for the investigation of the axial and radial homogeneity of the stoichiometric composition of a grown crystal. With a modified setup, Reinshaus et al. 23 later demonstrated that the position of an abrupt p/n-transition could be detected with a local resolution of about 1 µm, evaluating the transient signal immediately after setting the micro-tip onto the sample surface. A very recent setup for local S mapping was introduced by Iwanaga and Snyder 24, who used a comparable setup as Süßmann and Dietrich, but heated the sample instead of the probe tip, effectively inversing the heat flow. Iwanaga showed significant differences between the local distributions of the Seebeck coefficient of two PbTe bulk samples, although their nominal compositions were very similar and integral measurements could not give a proof of significant differences in the temperature dependence of the Seebeck coefficient.

Basically, many works employed constructional variants of known principles, using different probe designs or sensor concepts for mapping of transport properties; applied to different material systems while sometimes constructional adapted to either thin film 27–31 or bulk 24, 32, 33 samples and thus differently exposed to specific measurement errors. On the other hand several works addressed completely new spatially resolving approaches, which were inspired by classic bulk or thin film measurement techniques and partly enabled by new developments on miniaturised electronic drives and the availability of suitable commercial sensors or components thereof. Outstanding examples for such scanning devices can be found in the 3ω-based thermal conductivity probe 34, 35 and the ultra-compact scanning stage for the determination of the electrical conductivity 36 by means of the four point method, both imaging the properties on the nanometer scale.

Summarising scale relations in S mapping it can be assumed, that any discrepancy in measurements at nanometer or sub-nanometer scales can be attributed to the breakdown of the continuum assumptions, making a correlation to the macroscopic Seebeck coefficient difficult. As shown above, STM-related measuring principles, using non-contact approaches are exposed to restrictions which complicate or avoid a quantitative determination of macroscopically relevant transport properties in many cases. Furthermore, many quantitative contact-based measuring principles are suffering from missing information on thermal and electrical contact resistances or consideration of typical error sources of the employed thermopower measurement principles. Since some errors can lead to significant falsification of the deduced S values, the following sections regard to a basic methodical concept as well as to main difficulties, when thermopower has to be measured on the microscale.

1.1 Basic design concept of thermopower probing systems

Quantitative determination of spatially resolved thermopower was accomplished mainly by contact-based measuring principles. A probe tip has to be placed onto a sample surface in order to generate a temperature gradient either by cooling or heating of a small sample volume. The resulting temperature gradient generates a thermovoltage in the vicinity of the contact area. Thus, the basic setup involves a scanning stage with a tempered probe tip or sample. Secondly, suitable sensors have to be placed at the tip and sample for tapping of the thermovoltage and for measuring the excitatory temperature difference over the sample.

Several contact-based temperature sensors can be shortlisted such as thermistors, resistors, diodes and thermocouples. Although those probes all offer temperature resolutions of 1–10 mK 37, 38, thermocouples possess practical and methodical advantages and achieved more acceptance in reported system developments. This is mainly attributed to the stronger miniaturised while still punctual sensing characteristic of thermocouples, incorporating lower thermal mass and consequently facilitating measurements of fast transients of thermovoltages. Furthermore, thermocouples do not need additional electrical supply which would require space and might be a source of additional electrical noise.

Alternatives can also be taken into account with respect to the type of heat source or temperature management. Heat sources such as laser or resistive heaters can be included, while care must be taken to prevent electrical and thermal crosstalk to the sample and sensing leads by means of a suitable installation and shielding concept. Furthermore, isothermal conditions must be established within the sample, since spurious thermal gradients contribute to the voltage signal, which is generated close to the contacted sample spot. Resulting technical requirements on temperature control consequently become challenging, particularly for temperature dependent characterisation or larger samples being measured. Since some systematic error sources will show a distinct dependence on the acting temperature gradient, this should be kept constant and desirably on a minimum level for an entire scan.

Remaining differences among known devices mainly address constructive but not methodical variants and are related to differing probe designs and sensor installations such as cantilevers 18, multi-channel probe tip approaches 39, commercially available spring loaded probe tips 30, modified sheathed thermocouples 25, 26, coaxial single wire thermocouples 40 or custom made solutions employing carbide metal probe tips 41.

In fact, when thermocouples are considered for sensing of thermovoltages and temperature differences, the evaluation procedure for the determination of the local Seebeck coefficient simplifies to the following expressions, which are based on two voltage measurements across copper (Cu) and constantan (CuNi) circuits of two type T thermo-couples, as shown in Fig. 1.

equation image((1))
equation image((2))
equation image((3))

Combining the Cu[BOND]Cu and the CuNi[BOND]CuNi circuits of the thermocouples, voltages U1 and U2 are measured yielding the local Seebeck coefficient SS of the sample. SCu and SCuNi are the Seebeck coefficients of the Cu and CuNi thermocouple wires, respectively. The difference of Seebeck coefficient values in the bracket terms of Eqs. (1) and (2) is the differential thermopower of the particular material combination. In fact, this evaluation implies the assumption of a linear voltage response to the acting temperature difference 42, which is satisfied if the temperature dependence of the differential thermopower becomes insignificant within the considered temperature interval.

Figure 1.

(online color at: Measurement scheme for quantitative thermopower mapping: a heated tip operating in contact mode generates a temperature gradient. The resulting thermovoltage as well as the temperatures are measured.

2 Accuracy and spatial resolution of thermopower probing systems

Despite the fact, that each implemented system is subject to its individual level of measuring error, several systematic errors are inherent to all systems, irrespective of the particular design. The following subsections describe those factors, while summarising impact levels and possibilities to improve accuracy within the last subsection.

2.1 Measuring equipment and sensor characteristics

At room temperature (RT), absolute Seebeck coefficient values of relevant TE semiconductors are mostly in the range between 50 and 300 µV K−1, while typically revealing a non-linear curve/trend with temperature changes. Consequently small temperature differences lower than 10 K are desirable in order to comply with the implicit assumption of linearity in Eq. (3). This yields a thermovoltage of approximately 0.5–3 mV. Since measuring devices provide a finite voltage resolution, a direct impact on the accuracy of thermopower determination can be derived. Based on Eq. (3) a relative error for the thermopower ΔSS/SS due to a limited voltage resolution ΔU can be expressed as follows:

equation image((4))

Figure 2 shows the course of this error plotted against the Seebeck coefficient of the sample for a temperature difference ΔT = (T2 − T1) = 5 K and an assumed voltage resolution ΔU = 1 µV which is typically for daily laboratory practice.

Figure 2.

Calculated relative error in Seebeck coefficient measurement due to a limited voltage resolution of 1 µV of the measuring device. Curve plotted for a temperature difference of 5 K and type T thermocouples being used. The error is relevant for samples with a low Seebeck coefficient and shows an asymmetry with respect to the sign of thermopower.

Under these conditions the thermopower of typical TE semiconductors is subject to a relative error of approximately 1%. In principle this error scales linear with inaccuracy of the measurement devices and reciprocal to the acting temperature difference. Thus, when a nano-voltmeter with a typical resolution of 50 nV is used, this error reduces to 0.5‰ above an absolute thermopower of 50 µV K−1. However, a different situation arises for materials with low S, such as metals or high temperature TE materials, which possess relatively low S values at RT, such as p-type PbTe. This error may reach, for example, approximately 40% at S = 6 µV K−1. Even for a voltage resolution of 50 nV this error has still to be considered, as it amounts to 2% for the corresponding case. Furthermore, the relative error in Fig. 2 shows an asymmetry with respect to the sign of S of the sample. This is caused by the individual characteristics of the incorporated thermocouple leads, yielding a vanishing voltage response according to the Eqs. (1) and (2) at S values of the sample, which correspond to those of CuNi or Cu.

In this context a fundamental difficulty of S measurements can be addressed, which particularly becomes significant when miniaturised sensors are used or generally when low S values have to be measured. Irrespective which type of thermocouple is used for determination of S, the measurable thermovoltage is determined by the acting temperature difference and the differential thermopower of the couple, which forms from the material combination of the sample and the particular sensing lead. Whereas differential thermopower values (here type T: SCuNi − SCu) are well determined for particular material combinations, as given for the characteristics of standardised thermocouples, any uncertainty in the Seebeck coefficient of thermocouple materials adds an offset error to the calculation of S. Typical sources for materials thermopower can be found for instance in the Landolt-Börnstein 43 database but likewise in many other publications 44–46. A direct comparison for the temperature dependent Seebeck coefficient of different qualities of CuNi from different types of thermocouples is shown in Fig. 3.

Figure 3.

(online color at: Comparison of Seebeck values for varying qualities of constantan. Deviations between literature (Bentley) and calculated values from the characteristics of standardised thermocouples 47 are observed. Whereas (Bentley) refers to the constantan from a standardised type T thermocouple, remaining curves base on different qualities/compositions of constantan according to the standardised thermocouple types J and E.

Comparing the RT value for a nominal same composition of constantan from Ref. 44 and a calculated value from the characteristic of a standardised type T thermocouple 47 gives a deviation of approximately 1 µV K−1 or 2.4% of the nominal value reported in Ref. 44. The absolute Seebeck coefficient for Cu (1.71 µV K−1 at RT) was derived from the Landolt-Börnstein database 43 for this calculation. Calculating the S values for constantan according to type T, E and J thermocouples reveals the sensitivity of this material parameter to the chemical composition and manufacturing process. Although their nominal compositions solely differ below 1%, the maximum deviation for the deduced absolute thermopower for different qualities/compositions of constantan is 3.3 µV K−1 at RT. The thermopower of involved materials was likewise taken from literature here (43: type T copper, type J iron; 44: type E chromel). Although both variants of formula (3) are mathematically equal, the thermopower from the upper Eq. (3) incorporates the absolutely lower Seebeck coefficient of copper (Cu vs. CuNi) and by trend a lower error. Additionally commercial standard class 1 thermocouples possess a limited accuracy and are very sensitive to mechanical stress, when their diameters become small. Since the derived Seebeck coefficient depends upon the differential thermopower of the employed thermocouple it is particularly necessary for in-house-built sensors to conduct calibration measurements repeatedly, ideally under conditions close to the installation situation during application.

2.2 Thermal conditions at the probe tip and induced parasitic thermovoltage

Small point contacts between the probe tip and the sample are desirable, when the thermopower shall be measured with high spatial resolution. However, it may be inconvenient to probe the sample directly by a thermocouple in a scanning device since this might be linked to a lower durability limiting the time of operation but also to a possible drift of sensor characteristics, e.g., when a deformation is occurring at the mechanically loaded thermocouple 48. Several probe designs employed hard metal tips or metal coats to protect the sensing parts. Since a point contact inevitably forms a thermal constriction resistance within the heat transmission path, temperature differences within the probe can arise. Thus only a fraction (ΔTTotal − ΔT2 = T1 − T0) of the measured temperature difference ΔTTotal is actually applied to the sample, see Fig. 4.

Figure 4.

(online color at: Schematic of the measurement setup considering a parasitic temperature drop ΔT2 at the probe tip and its Seebeck coefficient ST. The generated thermovoltage within the probe tip falsifies the measurement of thermopower.

In fact, additional discrepancy must be taken into account if a thermal contact resistance is considered between tip and sample. This error is well known from temperature dependent integral S measurements as the thermal potentiometric error or so called cold finger effect 42. Consequently, a thermal anchorage of the sensor leads is mostly applied in integral measuring devices in order to avoid a significant heat flow over the sensor. Furthermore, the determination of the temperature difference in an integral measuring device is tampered in this context solely by the asymmetric fraction of the temperature drop on both thermocouple test points. In contrast to integral measurements, scanning methods suffer even more by trend, since the full heat flow to build up the temperature difference is passing mainly one sensor, which favours a measurement of an overvalued temperature difference due to stronger asymmetric heat flux conditions.

The temperature gradient within the tip generates a parasitic thermovoltage, which is proportional to the thermopower of the tip material ST and the acting temperature difference ΔT2. Based on Eq. (3) the relative error due to a parasitic voltage contribution of the non-adiabatic probe tip material can be expressed as

equation image((5))

For samples with a high thermopower SS compared to the tip material ST this error is mainly determined by the ratio of the temperature differences. Assuming one-dimensional (1D) heat flow an analytical expression can be formulated for this ratio, which does not account for a thermal contact resistance between the tip and the sample,

equation image((6))

Here, κT and κS denote the thermal conductivities of the tip and the sample, whereas the form resistances RT and RS describe the corresponding geometries independently from material properties. FEM simulations were applied to determine the effective ratio of temperature differences 49. According to Eq. (6) a subsequent calculation of effective form factors was carried out for the studied tip geometries and material properties.

Independence of form resistances from material properties, as initially assumed, is given to certain, but satisfying extent. Remaining deviations between the two sample cases and especially for the truncated cone tip arise from non-isothermal conditions at material transitions, which were explicitly excluded for Eq. (6). Since the bracket term in Eq. (5) almost equals unity for typical TE semiconducting samples (case 1), the ratio of temperature differences nearly represents the relative S error, which is below 1% here. The low thermal conductivity of TE semiconductors favours a low temperature drop in a highly conducting tip material. Thus a variation of error impact is mainly determined by different ratios of form resistances for TE semiconductors. In contrast to this, metal samples may reach relative errors above 20% due to their low Seebeck coefficient and high thermal conductivity (case 2).

Although FEM-based error values are only available for particular tip geometries and contact conditions, they allow for estimations and general trends, when the ratio of contact radius to the tip height is varied, see Fig. 5. A desirably high ratio of form resistances RS/RT necessitates small tip heights or larger contact radii, while the latter at least conflicts with spatial resolution objectives. The implied intersections of the ordinate in Fig. 5 represent form resistance ratios for an infinite tip height of the particular tip geometries.

Figure 5.

(online color at: Ratios of form resistances RS/RT for different tip geometries show almost linear slope with changing ratio between contact radius (rt) and tip height h.

For the cylindrical tip shape this ratio equals to zero, since RT becomes infinite too. The ratio of the truncated cone tip does not exceed a certain value with increasing tip heights. Thus, when larger tip heights are considered in relation to the contact area, a further increase of tip height does not significantly affect the ratio of the corresponding form resistance ratio. The comparison of courses reveals a similar slope and consequently constant absolute differences among considered tip shapes. Hence, the relative difference between form resistances becomes smaller with decreasing tip height, tending the impact of a particular shape to become negligible for small tip heights in relation to the contact radius.

The relative error due to a parasitic thermovoltage within the tip is a time dependent quantity since it is determined by transient temperature gradients within the sample and the tip material. Instantaneously after closing the contact the ratio of form resistances almost equals unity, while the temperature differences adjust according to the thermal properties of the involved materials. As the spread of the temperature field progresses through time, the sample heats up in the vicinity of the tip and the tip's apex cools down while the form resistances tend to their stationary values. Simulation results confirmed the tendency of a higher deviation of the deduced S values at the beginning of thermal field expansion into the sample 49. A systematic deviation to the supposed thermopower was observed for the transient evolution of the S value. For samples with Seebeck coefficients, smaller than those of the tip material, a too high S value is measured in the beginning, while approximating the true value with the course of time. Lower S values are measured in the beginning of the transients for samples with a higher Seebeck coefficient compared to the tip material, while likewise showing convergence to the supposed value when stationary conditions establish.

Local distribution of transport properties within the sample or anisotropy may strongly affect the deduced S value since the effective shape of the temperature field may deviate from a spherical form and gradients of TE properties, which are localised within the thermal gradient zone, may yield a compensating influence to the generated thermovoltage.

2.3 Influence of variation of thermoelectric properties within the sample

For a semiconductor with a primitive band structure in a rigid band approach the calculation of the Seebeck coefficient in dependence upon location is accessible from the assumption of a locally varying charge carrier concentration. This is illustrated in the following for the 1D case:

equation image((7))

Here S is the total Seebeck coefficient, Se,h the partial Seebeck coefficient of the electrons and holes, respectively, n/p the carrier concentration and µe,h is the corresponding mobility. The partial Seebeck coefficient as the contribution of electrons and holes (Se, Sh), respectively, can be expressed in the following way when assuming a single parabolic band for each type of carriers with one pure scattering mechanism, which corresponds to a particular energy dependent relaxation time 50:

equation image((8))


equation image((9))

where Fr(η) is the Fermi–Dirac integral of order r 51. The electron and hole densities can be solved numerically from the Poisson equation

equation image((10))
equation image((11))
equation image((12))

Φ is the electrical potential, while Na/Nd denote the acceptor/donor concentration and ε0/εS are the absolute and relative permittivity. As this derivation yields the theoretical S distribution, it can be considered as the measurable limit on homogeneous materials. In fact this cannot be achieved on inhomogeneous materials due to a compensation of adjacent regions within the sample. Here, it has to be stated, that this compensation cannot be understood as a simple averaging of TE properties in the region of the temperature field, as it has been considered in first approximation frequently.

In this context Bian et al. 20 conducted 3D simulations of a SThEM setup and determined temperature distributions for a nanoscopic tip size being used at a p/n-transition of a GaAs sample. Bian treated the solution of temperature profiles in two regions separately. Around the tip-sample contact and at a distance of a phonon mean free path a constant temperature was supposed within the sample, due to an assumed poor phonon scattering on these length scales. Out of this region, the temperature was solved by finite difference method, since it is governed by the continuum equation of heat conduction. Considering dopant profiles and carrier mobilities from experiments, he deduced the local thermovoltage profile at a planar GaAs p/n-junction in two ways. Initially, thermovoltages were calculated by means of the theoretical S distribution according to Eq. (7) and the outcome of the temperature simulation. Secondly, the region of the tip-sample contact was divided into several cubic cells of a resistor and voltage source network. The resistor network was obtained from the carrier density and the local conductivity of the depletion region, while the voltage sources were derived from the 3D temperature simulation and the local Seebeck coefficients from Eq. (7). Linear equations, based on conservation of current and voltage were solved yielding the node potentials, which finally allowed for the calculation of measurable thermovoltage at an STM tip. The deviation between deduced thermovoltages from the exact S calculation and the distributed network scheme was found to be dependent on the phonon mean free path of the sample and the tip-sample contact dimensions.

As Bian changed the phonon mean free path of the sample, the magnitude of the network-based thermovoltage profiles increased with the decrease of the phonon mean free path and converged to the continuum theoretical result. It was found, that for a highly doped (1019 cm−3) GaAs sample with a phonon mean free path as small as 4 nm solely a small fraction of actual thermovoltage is expectable to be measured. Changes on the contact size showed a strong decrease of measurable thermovoltage with increasing contact dimensions. Combining effects of the phonon mean free path and contact dimensions revealed an occasionally massively lowered thermovoltage. It is interesting to note, that even when the tip size was set to 1 nm, approximately almost half of the theoretical thermovoltage can be expected to be measured within the depletion zone of the studied GaAs p–n junction.

The reason for the deviation of the deduced thermovoltage profiles was found to be the magnitude of a single grid resistance, which decreases much faster than the thermovoltage as the tip moves away from the junction. When the temperature distribution spreads across a change of material properties, the contribution of local TE voltage will be modified due to the non-uniform resistor network. In a simple case of two materials this yields a parallel connection of two voltage sources, which are connected in series with their individual internal electrical resistances. The parallel combination of these sources produces a measurable open circuit voltage, which is determined by the voltage drop due to a compensational internal eddy current flow. Thus, the terminal voltage is mainly determined by the typical lower electrical voltage with a smaller resistance in series.

Sharp property gradients are typically found in p–n or Schottky junctions, implying strongly localised depletion zones within their space charge regions, yielding a characteristic overshoot of the spatial Seebeck coefficient. Basically similar characteristics of thermopower whereas on a much wider local scale could be observed recently on sintered bulk materials from nanoscopic base powders 52. Microscanning by means of the Potential-Seebeck Microprobe (PSM 53) mapped an overshooting S at the studied p–n region. Since a simple mixing of p- and n-type particles would not yield an overshoot in the Seebeck coefficient, particles must have been locally molten during the sintering process and recrystallised at neighbouring grains, forming new charge carrier concentration levels compared to those of the base materials. Formation of inhomogeneities on the micro- or macroscale is well known for crystallised but also observed for sintered TE materials, which are obtained from homogeneous base material powders but are subject to pressure or temperature gradients during compaction. Thus, spatially measured thermopower patterns are influenced by distributed TE, thermal and electrical properties in many TE materials, as they form property gradients in the scale of the thermal gradient.

The impact on thermopower can be shown for microscopic measurements on a rotationally symmetric scheme, incorporating a cylindrical inclusion (S1, σ1) within a surrounding material (S2, σ2), see Fig. 6. No difference in thermal conductivity was considered here, while boundary conditions correspond to those of Table 1, except zero thermopower that was assumed for the tip. The percentage labelling of the ordinate in Fig. 6 indicates the level of thermopower deviation for the given scheme. Here, 100% denotes that the thermopower of the surrounding material is measured, whereas 0% represents an exact measurement of thermopower of the inclusion. Since these values express a weighting of measured Seebeck coefficient in percentage relation between two materials, even a small nominal weighting can yield a considerable absolute error in case of strongly differing thermopower.

Figure 6.

FE modelled impact on measured S for distributed electrical conductivity and thermopower in the thermal gradient region. Assuming a cylindrical inhomogeneity (S1, σ1) within a surrounding material (S2, σ2) reveals the strongest dependence of S on the ratio of the electrical properties if the radii of the inclusion (r1) and the tip contact (rt) are equal. The tip is placed in the centre of the cylindrical inclusion.

Table 1. Steady state 2D FEM simulations of the tip-sample contact distinguish effective form factors RS/RT, ratios of temperature differences ΔT2TTotal, and resulting errors for the deduced thermopower ΔSS/SS due to a parasitic thermovoltage in the tip. Results obtained for different tip geometries and two sample cases with following assumptions: Dirichlet boundary conditions (T2 = 310 K/T1 = 300 K), κT = 197 Wm−1 K−1, ST = 2 µV K−1, contact radius = 40 µm, sample radius/thickness = 4 mm/3 mm.
case 1aΔT2TTotalRS/RTSS/SS|
cone tip0.009540.630.0096
spherical tip0.006440.9380.0064
case 2bΔT2TTotalRS/RTSS/SS|
  • a

    SS = −220 µV K−1/κS = 1.2 Wm−1 K−1;

  • b

    SS = 15 µV K−1/κS = 52 Wm−1 K−1.

cone tip0.2910.6420.252
spherical tip0.2190.9390.189

The strongest dependence on the ratio of electrical conductivities is given for equal radii. For smaller and larger inclusions the influence of the electrical conductivity tends to become small. In the case of equal radii the proportional weighting of the inclusion to the signal is approximately 40% in comparison to the S value of the surrounding material, even if its electrical conductivity is twice as high. Larger inhomogeneities, e.g., with double radius compared to the contact are causing a considerable deviation. For the case of equal electrical conductivities a 75% weighting to the signal arises from the thermopower value of the inclusion.

Tampering of S measurement is also expected for layered structures, which represent a common case of a deposited TE semiconductor on a substrate, see Fig. 7. If the electrical transfer resistance becomes high in this scheme eddy current compensation between layers is hampered. The compensation in case of a good electrical contact depends upon effective localisation of temperature gradients. Whereas a small cylindrical inclusion acts as a parallel material connection with respect to heat flow, this scheme exhibits a predominantly serial connection as long as the film is thin. If the thermal conductivity of the substrate is comparable to the film or lower, a significant temperature at the substrate will be the consequence. As thin films are often deposited on Si-based substrates one has to take into account considerably high thermovoltages which influence the measurement of thermopower of the thin film. If the layer possesses a poor thermal coupling or much higher thermal conductivity, lateral heat conduction has to be considered, which results in a considerably reduced spatial resolution. An influence of differing electrical properties on the S measurement is also given in layered schemes, why substrates with a desirably low electrical and high thermal conductivity improve accuracy of measurements in principle.

Figure 7.

Simulated impact on the measured Seebeck coefficient for a layered structure of a thin film (S1) on a substrate (S2). A high deviation to the Seebeck coefficient of the top layer is observed when the film thickness (d) becomes very small compared to the tip radius (rt). Even for thicker films with a threefold thickness compared to the contact radius the actual Seebeck coefficient of the substrate is still weighted with 10%.

Even for equal electrical and thermal properties the deduced S value of a thin film is considerably influenced by the temperature drop at the substrate, see Fig. 7. The percental labelling of the ordinate indicates the level of S deviation from the pure film value for the given scheme. Here, 100% denotes that the S2 value of the substrate material is measured, whereas 0% represents an exact measurement of S1 of the upper layer.

2.4 Remaining errors, impact level and possible ways to enhance accuracy

Basically, one can state that the smaller the tip-sample contact the higher the accuracy and spatial resolution of S mapping can get potentially. If a probe tip is uniformly reduced in all dimensions, the ratio of form resistances does not change, yielding the temperature drop plus parasitic thermovoltage at the tip unchanged. While reducing the tip dimensions under those conditions one gains spatial resolution, since the temperature gradient is concentrated on a smaller sample volume and reflects the value of local thermopower more accurate. Admittedly, a reduction of tip dimensions for a given probe concept can solely be done to a certain extent without influence on form resistance due to technical constraints on probe shape forming or sensor and gradient heater alignment. When the tip-sample contact size approaches the scale of the phonon mean free path, an additional dependence of the measured thermovoltage on the phonon behaviour becomes apparent. By trend this yields lower thermovoltages compared to results from theoretical calculations based on classical theory of continuous media, which typifies the physical limit for measurements on homogeneous materials. 3D electro-thermal modelling shows, that the lower the phonon mean free path of a sample the higher the spatial resolution and accuracy potentially become on property gradients 20. Contact sizes of STM tips can be made as small as 1 nm and below. If the reduction of tip size introduces a relatively (related to the contact diameter) larger distance of the sensing element to the sample, form resistances and temperature distributions develop disadvantageously and yield a higher parasitic temperature drop and thermovoltage. The thermal potentiometric error also scales with the quality of thermal coupling (thermal interface resistance). Limited contact forces of nanoscopic probe tips and the dimension effect yield a stronger weighting of contact properties compared to microscopic device concepts, which results in a higher underestimation of the Seebeck coefficient. As Majumdar 54 compared quoted S values and results from SThEM experiments using Au[BOND]Ni or Au[BOND]Pt thermocouples, he observed relative errors between 16 and 50% or 33 and 50% for the deduced thermopower. But since TE standard reference materials (SRMs) were a long time coming 55, 56 error quantification for SThEM devices was difficult to conduct, since the uncertainty budget of referenced values was not clear or references provided a wrong set of transport properties. Thus discussion of results on nanoscopic SThEM measurements was frequently limited to thermovoltages, while likely undergoing a considerable thermopower error due to missing experimental traceability for characterisation.

Heat transport in micro-resolving instruments is clearly governed by the continuum equation of heat conduction. Thus any dependence of the heat transport on the phonon mean free path can be neglected, which possibly could arise from nanoscopic fluctuations of doping concentrations or structural defects. This makes micro-devices ideal candidates to reveal significance of the induced errors by poor sensor coupling.

A thermopower scanning was performed on the Seebeck coefficient SRM (SRM-3451) from the National Institute of Standards and Technology (NIST). A small remaining amount of inhomogeneity of thermopower according to the statistical distribution was observed. Half value full width (HVFW) between 2.5 and 4.4% of the mean value was found whereas about 1% was found in highly uniform Czochralski bismuth telluride crystals 57. Measurements were conducted at RT on the PSM apparatus employing the principle shown in Fig. 1.

While changing the tip material or sensor alignment the sample thermocouple and mounting was kept the same. We used commercially available thermocouple wires (type-T) of 127 µm (0.005 inch) in diameter, laser welded to a tungsten and copper tip of comparable dimensions. Furthermore, one scan was conducted with a crossed-wire geometry without welding, as suggested by Iwanaga et al. 42, establishing a direct contact between the sample and the thermocouple (Fig. 8).

Before discussing quantitative S results one has to bear in mind that the referenced values of this SRM are solely valid for full compliance to the measurement instruction described in the corresponding certificate. This instruction specifies the use of a particular sample holder design, which is related to a commercial temperature depending integral characterisation device. By that this SRM suits only conditionally as a meaningful reference to scanning methods or to differing integral device concepts.

However, a relative comparison among employed tip designs becomes deducible anyway. Comparing mean values of the statistical thermopower distribution documents a continuous convergence to the referenced value as thermal coupling of the sensor to the sample improves.

Using a tungsten tip with a thermal conductivity of 173 Wm−1 K−1 58, the peak value of the Gaussian distribution is measured at −178 µV K−1, which is 11% lower than for the copper tip with a conductivity of 386 Wm−1 K−1 58. While for these two metal tips comparable form resistances can be assumed, this is not given for the third case of a directly contacting thermocouple. The apex of a cylindrical base body from Al2O3 was designed as a truncated cone guiding the thermocouple wires, which form a crossed alignment at the apex plateau of the body. Unless the form resistance ratio between this tip and the sample cannot be compared to the ratios of metallic tips, the parasitic temperature drop seems drastically lowered compared to the tungsten tip. This yields a further increase of the thermopower expectation of 3.3% compared to the copper tip, finally yielding a 9.5% deviation to the referenced value. These results reflect a variation of 15% for the expectation of thermopower in dependence on tip design and material. The deviation to the thermopower falsification, which was expected from FEM simulations in Section 2.2, must be attributed to the eligible temperature drop on the contact between the tip and the sample.

Due to thermal field expansion both, nano- and microscopic measuring devices have larger effective probe diameter as could be expected from nominal contact radii. Considering inhomogeneity sizes below contact dimensions rather plane heat source characteristics must be assumed. Additional impact arises from non-homogeneous temperature distributions at the tip-sample contact, to which micro-resolving devices are exposed practically more than SThEM setups. 3D simulations of a microprobe tip have shown a strong blurring and geometric distortions expressed by local overshoots on small inhomogeneities due to surface effects on temperature gradients around the contact area, see Fig. 9. The near-surface heat conduction differs from the bulk conditions since the half-space scheme initially yields a non-spherical distribution of isothermals in the near field. 2

Figure 8.

(online color at: Gaussian fits of normalised frequency distributions for micro-resolved thermopower measurements on NIST SRM-3451. Measurements were conducted with metal tips (tungsten, copper) and direct contacting of thermocouple wires (crossed 42). Results for dissimilar tip designs show a varying expectation of thermopower and differences to the reference value due to different sensor coupling to the sample.

Figure 9.

(online color at: 3D-FEM simulated characterisation of a spatial Seebeck pulse (base: 0 µV K−1, 2 µm width; peak: 200 µV K−1, 1 µm width) for differing contact diameter and time of measurement after closing the contact. Material properties are shown in Table 2. Inset: temperature distribution close to contact moment reveals thermal gradients, which overweight the local thermopower on side regions of the tip. Geometrical imaging fails for narrow pulses, while distortion decreases for smaller tips. A faster signal evaluation improves shape imaging and deduced absolute S values.

Table 2. Supposed material properties for transient FEM simulation of a spatial Seebeck pulse. Model information: Dirichlet boundary conditions (T2 = 298 K/T1 = 293 K), tip thermopower equals zero, no contact resistance.
 density (kg m−3)specific heat (J kg−1 K−1)th. conductivity (Wm−1 K−1)el. resistivity (µΩm)
metal tip89203854001.7 × 10−2
TE sample77401541.29.8

This particularly influences very fast measurements and yields a precursory over-evaluation of thermopower gradients as the tip approaches an inhomogeneity, while a lagging overshoot of spatial thermopower is observed when the tip departs from a plateau, see Fig. 9. Delayed measurements are decreasingly governed by near field surface effects and thus overshoots vanish against an advanced spatial temperature field expansion. While similar plateau values are measurable, the spatial slew rate suffers clearly from a later evaluation.

The actual shape and magnitude of a spatial S pulse is basically reproduced as long as the contact is significantly smaller than the inclusion. The transient signal shows a massive decrease of slew rate and absolute accuracy with the course of time. Interesting to note that even for a threefold expansion compared to the contact size a very fast measurement is required, since a 34% lowered value of the thermopower plateau is already prevalent 3 µs after contact occurs. A promoted raise of damping, as the inclusions size falls below the contact length, is attributed to an increased electrical node potential difference within the thermal gradient region. An opposite sign of thermopower for adjacent regions, as shown in Fig. 10, yields a contrary generated thermovoltage and by trend a higher compensation by eddy currents in practice.

Figure 10.

(online color at: Simulated transients of measured thermopower for a Cu tip with a 6 µm contact in the middle of a spatial thermopower pulse function between −100 and +100 µV K−1 of varying width. The decrease of the ratio between pulse width and contact length yields a gradually increasing influence of the surrounding material, whereas a massive increase of damping is observed for the thermopower of the inhomogeneity, when its size falls below the contact length.

As long as the contact length is kept approximately 30% smaller than the lateral size of inhomogeneity, a relatively small deviation for thermopower is observed within the first 500 ns of transient signals. High temporal signal sampling opens the opportunity to enhance the spatial slew rate and the absolute accuracy. This can be done to a certain extend for thermally low conducting TE materials, but becomes challenging for metallic samples due to smaller time constants and low terminal voltages, taking low-noise broadband amplifiers to their limits. Since TE semiconductors possess low thermal diffusivity and high thermopower, technically realisable enhancement can be obtained from fast measurements.

As accuracy and resolution of S scanning methods are exposed to a comprehensive field of impact parameters, a system-theoretical description of the signal transfer characteristic can be taken into account. Since every device concept incorporates its particular apparatus function a deconvolution approach is often considered to obtain refined measurement results by a post processing of raw data. This was shown for instance by Hansen et al. 59, who obtained a submicron resolution for the electrical conductivity, while using a 1.1 µm electrode spacing for the employed four-point scanning method.

Fourier-analysis on 1D courses of thermopower yields a sum of simpler trigonometric functions in the space domain. Thus any spatial Seebeck function can be expressed by its spectral function in the frequency domain, which corresponds here to an imaginary spatial domain. A thermopower pulse function within the space domain is represented by its spectral rect-function. This rect-function is treated as the Fourier-transformation of the spatial input signal to the scanning device. The output signal represents the Fourier-transformed spatial thermopower result, which forms from a convolution of input signal and apparatus function. This allows for a reverse transformation in case of known apparatus function.

As can be seen from Fig. 11, harmonic sine functions are exposed to higher magnitude damping as the frequency of spatial input signals rises. Setting the spectral magnitude of the convoluted signal into relation to the transformed original sine function allows for the calculation of damping over a broad frequency range, yielding the apparatus transfer function as shown by the red marks in the inset of Fig. 11. Secondly the transfer function was calculated for a spatial thermopower pulse function. The original rectangular pulse is convoluted to a blurred pulse of reduced slew rate and magnitude in the space domain. Both spectra have been set into relation yielding a second transfer function, which is shown by the continuous line in the inset of Fig. 11. Since the spatial pulse composes from particular harmonics within the frequency domain, the damping must be equivalent, which was confirmed by these FEM simulations. Thus a deconvolution of thermopower mappings seems to be theoretically feasible.

Figure 11.

(online color at: Simulated damping of harmonic sine functions of thermopower in the space domain. Sine function damping increases with increase of spatial frequency. Damping was simulated for several sine functions yielding a low pass characteristic for the transfer function in the frequency domain. Inset: comparison of transfer functions from damping of harmonics and rectangular input signal shows good agreement within the frequency baseband.

In practise postprocessing by deconvolution requires a high signal sampling and very good signal integrity, which demands low noise levels and sophisticated amplifying technologies. A principle proof for a deconvolution of spatial thermopower was given on PSM measurement data recorded on an eutectic zone of a Bi2Te3 grown crystal 60. Although a relatively good accordance to pulse widths measured by energy-dispersive X-ray spectroscopy (EDX) could be observed after deconvolution of PSM data, more information about the dependency of the transfer function on distribution conditions of transport properties and contact properties is necessitated in order to establish a reliable postprocessing scheme in terms of accuracy and traceability. Additionally, the reconstruction of the original spatial input signal by deconvolution is only possible for compliance to the Nyquist sampling theorem, which at least requires a double sampling frequency compared to the highest emerging frequency of the spatial input signal. Since the sampling is performed by thermal excitation and thus exposed to spatial broadening it yields an aliasing of spectral functions when higher frequencies are found in the spatial thermopower functions. This yields a breakdown of this deconvolution approach as can be seen from the inset of Fig. 11, showing that a good accordance of both transfer functions is actually given for the spectral base band but not for higher frequencies.

While summarising error impact levels, which were obtained from FEM-based simulations on microscopically resolving measuring schemes, one can state that thermopower measurements on homogeneous TE semiconductors are theoretically exposed to relative errors of approximately 3–5%. Simulations for metals showed an accumulated relative error of up to 75% for the deduced absolute value of thermopower. Even for TE semiconductors measurements can yield a 75% tampered thermopower result on transition regions of thermopower. Comparing micro- and nano-resolving probes gives nominally better preconditions for higher absolute accuracy and spatial resolution to SThEM concepts. But since the elusive temperature drop between sensor and sample represents a major error source, a higher deviation for every scale region is most likely. Comprehensive comparison among thermopower measurement results from integral and spatially micro-resolving instruments estimates the relative error to be in a range of 10–15% for TE semiconductors, whereas a significantly higher level must be assumed for regions of property gradients or metals as it is derived from FEM results. Final comparison of absolute accuracy for micro- and nano-resolving devices is difficult to conduct, since little documentation is found on SThEM device details and discussions on absolute accuracy of thermopower determination. Literature values point out, that SThEM concepts are exposed to relative errors of up to 50%, but unfortunately not giving any information on the type of studied material 54.

3 Thermopower probing in practise

Detection of different phase impurities by X-ray diffraction (XRD) typically distinguishes between concentrations of at least a few percent, while chemical variations of less than 1% represent the resolution limit from scanning electron microscopy (SEM). Since TE properties can massively change in the interval of ∼0.1 at.% EDX often fails to give insights to distribution conditions and stability of TE multiphase systems. The strong dependency of the Seebeck coefficient on materials structure and carrier concentration features a high sensitivity of thermopower scanning devices, which is completed by a high grade of reproducibility in practise.

On the other hand, main difficulties on absolute accuracy arise from tip-sample interaction and from the level of sample inhomogeneity. Even the characterisation of homogeneous materials, which is not additionally tampered by compensation effects within the sample, is not only exposed to an offset error due to an elusive temperature drop between the tip and the sample. As thermal coupling depends on sample roughness, surface covering, form resistances and contact pressure typically a Gaussian distribution is observed on thermopower scans. From this, a basic problem arises particularly for inhomogeneous materials as a distribution broadening could also be caused by spatial sub-phases below the resolution limit.

3.1 The Zn4Sb3 challenge

The investigation of degradation mechanisms of Zn4Sb3 is an excellent application example as it demands on accuracy and resolution, while likewise revealing the necessity and deficiencies of thermopower probing.

This material usually forms multiple sub phases within a narrow compositional window, from which the β-Zn4Sb3 phase provides the most desirable set of transport properties and thermal stability. Additionally a 1:1-phase (ZnSb) can form with a lower electrical conductivity and a Klevtsova phase (stoichiometrically similar to the β phase) 61, which has an undesirably higher thermal conductivity compared to the β-phase. Besides this, phase mixtures are possible, which can incorporate precipitations of elemental Zn of varying concentrations, see Fig. 12.

Figure 12.

(online color at: Schematic of thermopower frequency distribution in Zn4Sb3 identifies undesired phases with poor transport properties and degradation indicating phases by means of individual median thermopower levels. Phase distribution over the entire range cannot be investigated by a single independent method, but mainly distinguished by Seebeck probes. Correlation of spatial thermopower to synchrotron measurements and data from EBSD and XRD point out to a complex degradation mechanism indicated by the content of Zn-depleted β sub-phases with elemental precipitations of zinc.

The occurrence depends on the synthesis conditions, which yields a different ratio of formed phases. Samples from direct current-assisted short time sintering processes show additionally a gradation of thermopower due to electromigration of Zn in the DC field 62. While this rich distribution of phases appears for small changes in the Zn/Sb ratio, it results in a wide variation in the Seebeck coefficient typically in the range from 40 to 160 µV K−1.

Thus, the absolute values and the frequency distribution from the S-scan in Zn4Sb3 provide valuable information on presence of phases, homogeneity and microstructure, which is expressed by the peak positions and profiles (asymmetry, broadening). Sharp distributions are occasionally found for the β- and Klevtsova-phase if their formation is not dispersed below the length scale of effective probe diameter but agglomerates on a larger scale. Since both phases have similar mean S values and narrow distributions an initial indication of different phases arose by observed humps on the frequency distributions at particular S values. This indication was proved by electron backscatter diffraction (EBSD) 63 but could neither be resolved by regular powder XRD nor EDX. In Fig. 12, the closeby yet distinct S values for these phases are indicated in the scale. The Zn-depleted β sub-phases range in the interval of 70–120 µV K−1 is characteristic of the amount of interstitial zinc. The frequency distribution in such specimens shows a characteristic undulation with multiple shallow maxima each indicative of distinct S values. Such quantised nature of thermopower values confirms the presence of distinct sub-phases. The observation of these different phases is further helpful in understanding the degradation mechanism since the degradation state is characterised by a continuous Zn loss from the β-Zn4Sb3 phase. Overall, S-scans in Zn4Sb3 can provide a quick check on the nature of existing phases and thus its thermal stability. Such information cannot be obtained from other routine experiments due to the co-existence of multiple phases within a narrow compositional window.

The efficiency of the S-scan is however limited due to possible errors in estimation of the absolute values discussed in the previous sections and also the resolution in case of phase mixtures below the resolving limit of the PSM (can result in intermediate S values).

Although a distinct growth of a particular phase can be recognised by integral S measurements, basically no information is provided on the progress of degradation, since necessary information on the relative content ratio of phases is missing. Likewise no single independent method (EDX, EBSD, XRD) can distinguish between the variety of phases. While these methods are characterised to be complex, time consuming and expensive, todays S scanning on the micrometer range basically already provides sufficient resolution to show contrast between all phases and to give insights on the current state of degradation. On the other hand, S scanning is likely exposed to a tampering of absolute thermopower values. Since the desired β-Zn4Sb3 phase is bordered by undesired phases this particularly becomes problematic when the phase alignment becomes very close, which occasionally can arise from different synthesis or operation conditions. In order to predict the degradation state and phase distributions more accurate, further improvement on spatial resolution and absolute accuracy is desired, which should yield the necessitated separation of adjacent distribution peaks of thermopower on several material classes.

3.2 Development of SRMs for thermopower

As previously mentioned, a thermopower SRM was released in the beginning of 2012 by the NIST, enabling members of the TE materials community to reliably calibrate integral Seebeck coefficient measurements for the first time. In compliance with a particular measurement scheme this Bi2Te3-based SRM provides certified S values for the low temperature interval from 10 to 390 K. Increased focus has been placed on high temperature TE applications in the recent past by companies and research institutions. Thus, significant effort is undertaken to develop suitable SRMs in order to support TE researchers independently from employed integral measuring principles by provision of stable SRMs for temperatures up to 1000 K 56.

However, the development of thermopower SRMs is ultimately relying on the proof of functional homogeneity on single specimens and batches from reproductive syntheses. This can solely be achieved by S scanning methods.

Frequently high purity materials are considered to suit as SRMs for thermopower. Irrespective that undesirably low S values or high thermal conductivities are typically provided hereby, smallest amounts of impurities or uncertainties during the synthesis process (e.g., by non-uniform temperature conditions) potentially cause a massive change in thermopower. This was observed on a high purity Ge sample, which shows a distinct modulation of thermopower along isothermals of the cooling process, see Fig. 13. The corresponding frequency distribution of this sample is shown in the lower diagram. As can be seen from comparison to the frequency distribution of an Fe0.95Co0.05Si2 sample, which is currently under development as a potential high temperature SRM at DLR 64, a pronounced broadening excludes the high purity Ge from being used as an SRM. Although spatial resolution is usually not required on an exceptionally high level for SRM development, a precocious detection of grain growth, phase transformation or formation of precipitations can be recognised as mandatory for the assessment of functional stability. These material changes usually cannot be detected directly by integral methods but can even tamper their thermopower results due to the cold finger effect 42, particularly when the expansion of inhomogeneities reaches the length scale of contact dimensions of employed sensors. Thermopower probing allows for the detection of unstable material behaviour, which makes it a valuable tool not only for TE materials research.

Figure 13.

(online color at: Upper diagram: PSM homogeneity analysis of Ge (99.99%) as a potential SRM for thermopower. Lower diagram: Corresponding frequency distribution of S for the Ge sample in comparison to the distribution of an Fe0.95Co0.05Si2 sample.

3.3 Aspects from TE material and device technology

Local variations of thermopower are well known from several TE materials and synthesis routes. Partly this is favoured by a self-doping material nature (Bi2Te3), which yields massive changes on thermopower due to small compositional differences. Furthermore, thermodynamic processes can benefit a formation of multiple and dispersed sub-phases, such as given for instance in Zn4Sb3 or AgPbmSbTem + 2n. Melt syntheses or crystal growth processes are comparably exposed to a strong formation of inhomogeneities due to reaction kinetics. Even when rapid cooling is considered to preserve particular phase compositions, inhomogeneous thermopower distributions can frequently not be avoided due to non-uniform temperature conditions in ampoules during quenching. Usually this necessitates a homogenisation by milling and subsequent sintering processes for compaction. Since sintering processes similarly suffer from non-uniform synthesis conditions, mainly given by temperature and pressure gradients within pressing dies, the synthesis of homogeneous specimens with respect to thermopower is usually difficult. XRD and SEM analyses frequently give no indication of inhomogeneity on TE materials due to their fine microstructure or chemical variations below the resolution limit 24, 65. But basic questions related to TE materials performance, functional stability and degradation mechanisms are ultimately relying on information about local distributions of transport properties. Thus, spatially resolving characterisation methods are essential for the development of new and efficient TE materials and for improvement of known material classes in the context of fundamental research and technology.

Development of TE sensors and generators is likewise supported by probing methods in a wide range. Functionally graded materials (FGM) are considered for enhancement of conversion efficiency by expansion of application temperatures in TEGs or linearisation of sensor characteristics and enhancement of sensitivities 66–70. FGM approaches are ultimately relying on local property information since a proof of effective property gradation and feedback to numerical system simulation has to be given in order to improve device performance in the course of development 71, 72. Scanning devices frequently offer additional capabilities, such as measurements of contact resistances and the electrical conductivity 73. Thus, they represent valuable tools for the development of new joining technologies. Comfortable and fast assessment of effectiveness and stability of diffusion barriers becomes accessible not only for TEG, but likewise supporting developments on fuel cells, electro ceramics or photovoltaic systems for instance.

4 Conclusions

The thermopower is a distinct indicator for TE materials efficiency and stability. It gives insights to many materials due to its strong dependency on the charge carrier concentration and microstructure. The spatial resolution and absolute accuracy of thermopower probing systems is determined by a comprehensive set of methodical factors and has to be discussed in relation to absolute transport properties of samples and their grade of functional inhomogeneity.

By trend several systematic errors are less severe as typical TE materials possess a low thermal conductivity and high S values. Since metals generate lower thermovoltages, a higher tampering of thermopower can be deduced from inaccuracy of electronic measuring devices. The corresponding error adds to a comparatively small level below 1% for TE semiconductors, while metals are exposed to an error between 2 and 40%.

Considerable tampering of deduced thermopower is additionally caused by parasitic thermovoltages within the probe tip. If thermal contact resistances are neglected, FEM simulations report a relative error below 1% for TE semiconductors, whereas an error of up to 25% is predicted for metals. Summing up errors from FEM simulations on homogeneous materials yields a total falsification of 3–5% for TE semiconductors and occasionally up to 75% for metals, when additionally uncertainties in characteristics of employed sensors are taken into account.

As material property gradients lie within the catchment of excitatory temperature gradients, they yield a compensational influence on thermovoltages. This compensation causes an overestimation of inhomogeneity regions, which have a higher electrical conductivity and typically lower thermopower. The associated impact on accuracy of thermopower determination exceeds expectable errors for homogeneous materials occasionally by far.

Additionally, thermal contact resistances yield an elusive temperature drop, which tends thermopower to be systematically underestimated. AFM- or STM-based nano-resolving concepts show errors up to 50% 54. The impact for micro-resolving device principles was derived from measurements on a SRM and comprehensive comparisons to results of integral measuring devices on manifold materials. The expectable total error of absolute thermopower in microscale probing systems lies in the range of 10–15% in dependence of thermal properties of the samples.

However, thermopower probing already provides a valuable contribution to TE materials and systems research. Although spatially resolved thermopower characterisation cannot provide the absolute accuracy of integral measurement principles to date, it clearly offers advantage over competing structural and compositional analytic methods. The easy sample preparation and low costs of operation and purchase benefit thermopower probing methods in comparison to EDX, XRD or SEM systems. Nevertheless significant effort must be undertaken to enhance accuracy and resolution. Future efforts should address the development of suitable reference materials, adapted to requirements of spatial thermopower measurements. This would yield an expansion of application fields, rise of acceptance and open the way to establish TE probing as a traceable and standardised characterisation technique.


Financial support from the German Federal Ministry of Education and Research is gratefully acknowledged for funding of the “TESt”-project (03X3550B). Likewise the authors would like to thank the Fraunhofer Institute for Physical Measurement Techniques for preparation and provision of the Germanium sample.

Biographical Information

Pawel Ziolkowski is a PhD student in the group of Prof. Eckhard Müller at the German Aerospace Center (DLR) where he works on spatially resolved measurement techniques for transport properties and on thermoelectric (TE) standardisation involving aspects of characterisation techniques for TE generators and the development of high temperature reference materials for the thermopower. In 2008 Pawel received the innovation award from the DLR for the product development of the PSM. He received his diploma in electrical engineering from the RWTH Aachen University, where he worked under the supervision of Prof. W. Mokwa. There, his work focused on impedance spectroscopy based measurement concepts for the determination of the long-time reliability of protection coatings for bio-sensors.

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Biographical Information

Eckhard Müller, leading the thermoelectrics (TE) division at the German Aerospace Center (DLR), has been active in the field of thermoelectrics for already 25 years. He is Board member of the European Thermoelectric Society and Advisory Board member of the German Thermoelectric Society. He received his PhD in physics from the Martin Luther University Halle-Wittenberg (MLU) in Halle/Saale, where he worked on the band structure and charge carrier scattering in (Bi1 − xSbx)2Te3 based solid solutions. After working as a scientist at Karlsruhe University in a NASA headed project developing an IR sensor for the Cassini mission to Saturn, he joined DLR in 1996. Silicides for thermal sensors, simultaneous measuring of thermoelectric properties at elevated temperature, continuum theoretical description of TE materials and materials research on skutterudites and chalcogenides have been his research topics at the DLR Institute of Materials Research. Since September 2012 he is an appointed Professor for Thermoelectric Materials at the Justus Liebig University (JLU) in Gießen (Germany) in addition to his function as the head of the Thermoelectrics Department at DLR.

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