NANO LETTERS
Thermopower Enhancement in Nanowires via Junction Effects
2009 Vol. 9, No. 2 617-622
Nicolas B. Duarte,†,‡ Gerald D. Mahan,§ and Srinivas Tadigadapa*,†,‡,| Department of Electrical Engineering, Department of Physics, and The Materials Research Institute, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16802 Received September 22, 2008; Revised Manuscript Received December 22, 2008
ABSTRACT We present thermopower measurements on free-standing, straight and “junctioned” gold nanowires using a micromachined thermoelectric workbench. Measurements on straight 70 nm diameter gold nanowires show a thermopower similar to that of bulk gold; however for “junctioned” gold nanowires we observed a hitherto unreported peak in the thermopower near room temperature. The observed enhancement can be explained by approximating the “junctioned” nanowires as tunnel junctions in combination with Coulombic effect of the electrons crossing the junction. The electron transfer across the barrier can be expected to be stochastic in nature. Under thermal equilibrium conditions and in the absence of temperature gradient across the tunnel junction, the time averaged random fluctuation of the electrons across the tunnel junction results in a net zero voltage. However, in the presence of a temperature gradient across the junction, the time averaged fluctuation of the electrons across the junction is now offset by the tunnel junction thermoelectric effect and is measured by the lock-in amplifier. A hundredfold enhancement in the ZT of “junctioned” nanowires has been observed for the gold nanowires measured over several samples.
Nearly five decades after the last significant developments in thermoelectric materials, in the form of doped bismuth telluride semiconductors led by Abraham Ioffe, the field has been once again energized by recent developments in nanoscale materials and structures.1-6 For example, PbTe/ PbSeTe quantum dot superlattices and Bi2Te3/Sb2Te3 superlattices have produced thermoelectric figures of merit (ZT) of 4.5 and 1.9 times their bulk values at room temperature.7,8 More recently, silicon nanowires (20 nm diameter) and roughened silicon nanowires have been shown to exhibit 100fold enhancement in room temperature ZT as compared to bulk silicon ZT of 0.01,9,10 while the use of nanoparticles in the formation of a bulk BiSbTe has improved its ZT to 1.4.11 The most important aspect of these latter improvements is that they exploit the dependence of electrical and thermal transport properties upon sample geometry at the nanoscale which could be independently combined with other methods to increase ZT to realize thermoelectric generators and coolers that could potentially rival the efficiencies of their mechanical counterparts. In this paper we present a previously unreported observation of a near room temperature 10-fold enhancement * Corresponding author: Srinivas Tadigadapa, Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802. Tel: (814) 865 2730. Fax: (814) 865 7065. E-mail:
[email protected]. † Authors Nicolas Duarte and Srinivas Tadigadapa have contributed equally to this work. ‡ Department of Electrical Engineering. § Department of Physics. | The Materials Research Institute. 10.1021/nl802882h CCC: $40.75 Published on Web 01/23/2009
2009 American Chemical Society
in the thermoelectric power of “junctioned” nanowires of polycrystalline gold. Thermoelectric materials are characterized by the dimensionless thermoelectric figure of merit, ZT ) (S2σ/κ)T, where S is the thermoelectric power, σ is the electrical conductivity, κ is the thermal conductivity, and T is the absolute temperature. In order to achieve a high ZT, one requires a high thermoelectric power, a high electrical conductivity, and a low thermal conductivity. Unfortunately the three transport properties are not independent of each other. Increasing the thermoelectric power for most materials also leads to a simultaneous decrease in the electrical conductivity. Similarly an increase in the electrical conductivity leads to a comparable increase in the electronic contribution to the thermal conductivity as per Wiedemann-Franz law.12 Consequently, with known conventional solids, a limit in ZT is rapidly approached when an improvement of any of the three parameters, namely, electrical conductivity, thermoelectric power or thermal conductivity is attempted.3 However, nanostructures and nanomaterials present unique situations that can reduce and even remove some of these dependencies. The nature of nanowires promises improvements in the ZT values via multiple methods. Most recent one-dimensional (1-D) enhancement studies have focused on improving ZT by decreasing thermal conductivity through size reduction9 or surface roughening.10 Dresselhaus et al. have proposed that the thermopower itself can be enhanced as a result of discrete peaks in the density of states near the Fermi energy that leads to large values for the energy derivative of the
Figure 1. An individual unit of our thermoelectric workbench. Sets of units are placed across from each other to increase the number of test sites per unit area. The heater bridge (seen here going from bottom left to top right) is released from the Si substrate by use of XeF2. Three “sensing fingers” are also visible approaching the bridge. Inset shows the workbench with a straight gold nanowire bridging the gap between the thermally clamped finger and the heater bridge.
electrical conductivity.5,13 Other systems of considerable interest are point contacts and tunnel junctions.14-17 Unlike bulk contacts where a secondary material is necessary to measure thermoelectric voltage, the thermopower of point contacts has been observed by making a point contact of a material with itself.15 This would indicate that the point contact itself has a thermopower different than, but dependent on, the bulk material from which the point contact was made. In tunnel junctions a thermoelectric voltage arises due to the higher average energy of the electrons on the hot side in comparison to the cold side resulting in preferential tunneling of electrons toward the cold side of the junction. The thermopower in tunnel junctions is found to be insensitive to the barrier width and to image charges.17 In this work we will present the thermopower measurements on free-standing, junctioned and straight gold nanowires using a micromachined thermoelectric workbench. Measurements on junctioned gold nanowires exhibit a peak in the observed thermopower near room temperature which has not been reported thus far. We hypothesize that the junctioned nanowire configuration may be approximated as a tunnel junction. For temperatures near and above the Debye temperature of gold, the tunneling thermopower of gold increases linearly with temperature. However above a transition temperature, determined by the barrier height and width, the tunneling conductance of the devices increases rapidly and the thermopower begins to decrease as T-1.18,19 As we will show, the temperature dependence of the thermopower near room temperature requires the consideration of the tunnel junction effect in combination with Coulomb blockade effect of the electrons across the junction for a complete understanding of the behavior. The micromachined workbench developed to measure the thermopower in nanowires is shown in Figure 1. Each test unit consists of three substrate-clamped Ni “sensing fingers” and a 40 µm long freestanding Ni heater bridge separated by a 1 µm gap allowing for a controlled temperature gradient 618
to be applied to any nanowire crossing the gap such as shown in Figure 1 inset. These features are each 2000 Å thick and 5 µm wide. The device structure allows for three distinct advantages. First, it allows up to three unique testable nanowire structures in each 17 µm span that runs parallel to the heated bridge and a total of 144 test sites on a 5 mm × 5 mm die. Since the drop dispense method of nanowire deposition is inherently a random process, this high density of test sites significantly improves the yield of testable nanowires without further manipulation. Second, the thermal isolation of the bridge greatly improves the magnitude of the thermal gradient generated across the nanowire resulting in a larger measured thermoelectric signal. Third, by suspending the nanowire under test, we are able to prevent substrate-coupling effects.20,21 The temperature of the bridge is monitored via its ac resistance. The fabrication process and calibration of the workbench are described in detail in the Supporting Information and in previously reported work.22 For the experiments described here gold nanowires with an average diameter of 70 nm and a length of 1-3 µm synthesized by electrodeposition in the pores of “tracketched” polycarbonate membranes were obtained as a suspension in isopropyl alcohol.23 The nanowire suspension was diluted and was thoroughly dispersed using a 30 s ultrasonication. Deposition of a 1 µL drop via syringe was followed by room temperature/pressure evaporation of the solvent, resulting in a surface density of about one nanowire per 5 µm2. Presence of nanowires spanning the gap was verified first by optical microscope inspection followed by performing an I-V characterization using a Keithley 4200 analyzer. On test sites with gold nanowires, a linear I-V characteristic is obtained indicative of an Ohmic contact. It should be noted that as-deposited nanowires required a 5 V potential sweep prior to obtaining the linear I-V characteristics which is most likely caused by a native nickel oxide. In order to prevent nanowire destruction at the moment of oxide breakdown, the current compliance was set to 10 µA. No further clamping was performed since thermopower measurements were made using a high impedance lock-in amplifier.24 The measurement of the thermopower was made using the 2ω method. Heating the bridge at a frequency of 106.2 Hz (well below the 1 ms thermal time constant of the heater) the open circuit thermoelectric voltage was monitored at the second harmonic of the heating frequency using a SR830 DSP lock-in amplifier with time constant setting of 3 s. Several tests were performed to verify that the obtained 2ω signal was thermoelectric in origin and not due to unanticipated electrical coupling.22 Details of the measurement settings and additional characterization tests are described in the Supporting Information. Thermoelectric measurements were made with substrate temperature ranging from 10 K to room temperature (295 K). To obtain the thermopower at each temperature point, the amplitude of the 106.2 Hz heater voltage was swept from 0.3 to 1 Vrms with the 2ω thermoelectric voltage measured at each 0.1 Vrms point. Thermoelectric measurements were performed by a custom LabView Nano Lett., Vol. 9, No. 2, 2009
Figure 2. Directly connected and “junctioned” (or series connected) configuration of nanowires. (a) A bundle of directly connected nanowires. While the path of electrical connection is unclear, it is certain that there exists at least one nanowire that has a direct uninterrupted connection between the heating bridge (bottom) and sensing finger (top). (b) Another example of a straight nanowire lying across the heater bridge-finger gap. (c) No single nanowire is visible connecting the finger (top left) and the heater (highlighted in red). The most likely electrical path is in the encircled region with wires connected in series. (d) Two wires meet in the middle of the gap. From the orphan wire segment on the left it appears as though an electrostatic discharge destroyed part of a wire that was directly crossing the gap leaving behind the “knot” structure in the center. This device resulted in the weakest observed enhancement.
program at 50 equally spaced temperature points between 10 and 200 K and 10 points between 200 and 295 K. The use of fewer temperature points at higher temperatures was chosen because of reduced temperature control above 200 K in the closed cycle refrigerator used in this work. The drop dispensed nanowires could be obtained in two distinct configurations. Panels a and b of Figure 2 show scanning electron microscopy (SEM) photographs of a single nanowire sample spanning across the gap created by the freestanding heater and the thermally clamped nickel finger while panels c and d of Figure 2 show the case where multiple nanowires join together in series to span the gap. The samples shown were obtained as is with no attempt at manipulating the nanowires, and thus in any given device the number of junctions formed or the orientation of any specific nanowire was random. Ignoring any radiation or heat losses through the surrounding gas, we use the 1-D heat conduction equation to obtain the temperature profile along the heater wire. The thermal conductivity of the nickel wire was experimentally measured using the 3ω-method25 as a function of temperature in the 10-295 K range and was used in the model. Additionally, the temperature profile of the heater wire was also mapped using a scanning thermal microscopy (SThM). Assuming the nickel fingers to be clamped at ambient temperature, the slope of the linear fit of thermoelectric voltage versus the temperature difference as a function of ambient temperature was obtained. Figure 3 shows the temperature dependence of the measured thermopower for a Ni-Au nanowire thermocouple for the two distinct configurations of nanowires. The blue line represents data from 70 nm Au nanowires that have at least Nano Lett., Vol. 9, No. 2, 2009
Figure 3. Thermopower for junctioned and straight gold nanowires spanning across a heated nickel bridge and a Ni finger at various ambient temperatures. The data represent the average of several different measurements made on nanowires made in the two configurations. The error bars show the maximum spread in the acquired data. Thermoelectric contributions from nickel have not been removed in this data and are considered to be constant from one device to the next. Inset shows the typical I-V characteristic of a “junctioned” nanowiresfluctuations near the origin arise due to discrete charge transfer effects over the measurement period. The device resistance of 1.368 kΩ is mainly dominated by the resistance of the long connection paths from the nickel fingers to the bonding pads on the edges of each chip.
one nanowire making direct contact between the Ni heater bridge and the clamped Ni finger. This direct contact nanowire data agrees with literature values for bulk Ni-Au thermocouples, which is expected since the diameter of the nanowires is too large to exhibit any low-dimensional transport effects. As additional verification we tested a lithographically defined, evaporated gold thin film square (5 µm wide, 200 nm thick) and obtained the similar thermopower temperature profile and is described in detail in Supporting Information. When the contact between the heater and sensing finger consists of a series of nanowire segments with no direct path, the black line profile in Figure 3 is obtained. It should be noted that all samples were tested for electrical continuity prior to thermoelectric measurements since the apparent electrical conduction path could not always be clearly identified from the SEM photographs. At low temperatures, “junctioned” samples show the characteristic increase in thermopower due to phonon drag. However, compared to bulk gold phonon drag thermopower values, an enhanced effect is observed which may be attributed to diffusion thermopower under strong electron boundary scattering in the junction region.26 Given the nature of the junction in the presented system and the likely mechanical instabilities involved, it is conceivable that in this temperature region the system acts as a constricted (limited conduction channels) system with enhanced boundary scattering resulting in the enhanced phonon drag thermoelectric effect. Additionally the fall off from the phonon drag peak is not as steep resulting in some increase of S compared to the bulk values all the way up to the Debye temperature. Above the Debye temperature, until about 225 K, the thermopower of the series connected nanowires matches the thermopower of a direct 619
nanowire to within measurement error. Near room temperature a new peak is observed which has not been reported thus far. These experiments have been performed during both substrate heating and cooling. Upon subtracting the thermopower of Ni (∼-15 µV/K) from the measured peak value, we find a nearly 10 times increase in the thermopower of junctioned gold nanowires as compared to the direct gold nanowires. Furthermore, we found that in general the magnitude of the enhancement in the thermopower near room temperature in the series junctioned nanowires increased with the number of junctions. Since an exact electrical pathway was difficult to identify for each device, no quantitative correlation between the number of junctions and the observed increase has been attempted at this time. It is our hypothesis that the drop-dispense method of deposition of nanowires on the thermal workbench naturally creates tunnel junctions with possible mechanical instabilities. In many of the physical configurations obtained, individual gold nanowires end up lying upon or crossing each other as the solvent from which they are dispensed evaporates. Since the wires being used are gold, we do not expect a native oxide to have formed at the junction between wires. Additionally individual wires tested for surface contaminants via energy dispersive X-ray spectroscopy (EDX) showed no trace of contamination. Because we are dealing with nanosized cylindrical objects, it is our theory that the source of this barrier comes from the physical shape of the contact and any trapped molecules between the nanowires interface. While the source of this barrier remains an interesting question, we will explain how such a tunnel junction might explain the enhancement in the observed thermopower. When forming the “junctioned” configuration, gold wires are expected to lie upon each other in close proximity; however it is conceivable that due to the random topographical and mechanical anchoring conditions of the individual wires, small gaps between the two overlying gold wires could be easily created. The extremely small gap between the gold wires is expected to result in the observed thermopower dependence on temperature. Leavens and Aers derived the analytical expressions for vacuum tunneling thermopower.16 Since one side of the junction is connected to the heater wire (i.e., hot) and the other is clamped at ambient temperature, this temperature difference across the tunnel junction results in the development of a thermoelectric voltage. Vacuum tunneling thermopower can be defined as V(∆T) ∆Tf0 ∆T
S ≡ Limit
(1)
where V(∆T) is the voltage induced due to a vanishing temperature difference, ∆T, across the tunnel junction. If T0 is the temperature on one side of the junction and T0 + ∆T is the temperature on the other, the tunneling current density for small ∆T and V(∆T) can be approximated as 620
J(T0, ∆T, V, d) )
∂J ∂(∆T)
|
|
Vf0
Vf0
∂J V(∆T) ∂V ∆Tf0
∆T + ∆Tf0
(2)
where d is the tunnel barrier width. Using the fact that for an open-circuit voltage measurement with no externally applied bias voltage (i.e., Vapplied ) 0), the current across the tunnel junction J ) 0, we can define the tunneling thermopower S as S)
V(∆T) ∆T
|
) ∆Tf0
-∂J ∂(∆T)
|
Vf0 ∆Tf0
⁄
|
∂J Vf0 ∂V ∆Tf0
(3)
The tunnel current can be given by the expression J ) J1f2 - J2f1 )
4πmeffe
∞
Emax
∫ T (E ) dE ∫ (f (E + ⊥
h3
⊥
Emin
1
|
0
eV(∆T)) - f2(E|)) dE|
(4)
where J represents the net difference in the current flowing between side 1 and side 2 and vice versa of the tunnel junction. u (E⊥) is the transmission coefficient of the junction and only depends upon the energy of the electrons perpendicular to the interface (E⊥). Using the Wentzel-KramerBrillouin approximation, the tunneling transmission coefficient can be given by the expression
(
x2
∫ √2m
2 px
T (E⊥) ) exp -
eff(W(x) - E⊥(x))
(
1
exp -
)
dx =
2∆x√2meff √(EF1 + φav - E⊥ p
)
(5)
In this work since gold nanowires are considered, EF1 represents the Fermi energy of gold which is ∼4 eV. φav represents the average barrier height in the tunnel junction, and ∆x is the barrier width. An electron between two closely spaced parallel electrodes polarizes both of them. The image charges created within the electrodes effectively reduce the barrier height by rounding off the sharp corners.27 For electrodes made from the same material, i.e., identical Fermi energies and with no potential applied across them, the maximum of the rectangular barrier height is lowered by27 (φi)max ) φ0 -
10 ( K∆x )
(6)
where K is the dielectric constant of the barrier material (K ) 1 for vacuum). For example, the peak height of the barrier can be lowered by as much as 0.5 eV for a 20 Å wide barrier and the barrier shape is expected to become pronouncedly parabolic from the original square shape. The supply function can be integrated analytically for arbitrary voltage and temperature difference and can be written as ∞
∫ (f (E + eV(∆T)) - f (E )) dE ) k 1
0
2
|
[{
ln
(
1 + exp
|
EF1 - E⊥ - eV kBT1
B
×
)} ⁄ { T1
(
1 + exp
EF2 - E⊥ kBT2
)} ] T2
(7)
Substituting eq 5 and 7 in eq 4 and taking the partial derivatives and substituting in eq 3, it is possible to estimate the tunneling thermopower arising due to a temperature difference across the junction, and Vapplied ) 0, for the tunnel junction. For a gold-vacuum-gold tunnel junction, the Nano Lett., Vol. 9, No. 2, 2009
Figure 4. Calculated tunneling thermopower arising in the limit of a vanishing temperature difference between the two sides of a tunnel junction. An average barrier height of 3 eV, barrier width of 20 Å, and a dielectric constant K ) 1 have been assumed in this calculation.
energy barrier is expected to be the work function of the material which is φgold ∼ 4 eV. If we assume that barrier width is ∼20 Å, then (φi)max is 3.5 eV and φav ∼ 3 eV. Figure 4 shows the calculated S as function of temperature in the 200-300 K. A linearly increasing tunneling thermopower as a function of temperature is observed, the magnitude of which matches well with the measured thermopower.28 The calculated tunneling thermopower is much larger than the value of diffusive thermopower of bulk gold in the same temperature region and therefore dominates the observed behavior of the device. This appears to be the trend until the tunneling electron flux across the junction begins to dominate at high temperatures. In this high temperature region, the thermopower is expected to behave as T-1 since the tunnel conductance increases as T2 in this regime.18 The transition temperature at which the behavior changes between the two regimes is a material-dependent temperature parameter and depends upon the barrier height and barrier width. This can be clearly seen in the experimental results as a flattening and eventual decrease in the thermopower as the temperature increases in the 300 K range. What is very interesting is the fact that if we assume the two gold nanowires to form a parallel plate capacitor with ε0 ) 8.85 × 10-12 F/m, and an overlapping area of 20 nm × 20 nm, then the capacitance of the tunnel junction for the given barrier width can be calculated to be ∼1.8 aF. This implies that for every electron that jumps across the tunnel barrier, energy of e2/2C eV is required which can be calculated to be 45 meV. This value is larger than the thermal energy of 25.9 meV for electrons at 300 K. Thus electron transfer across the barrier can be expected to be stochastic in nature and can be seen in the I-V characteristics of the junctioned devices near zero bias (Figure 3 inset).29 Furthermore, the transfer of a single electron across the 1.8 aF capacitor electrodes will result in a potential difference of 90 mV, whereas the experimentally measured thermoelectric voltage is of the order of 100 µV. Since subelectronic charge transfer cannot be expected to occur, this result can be meaningfully interpreted in terms of the average potential Nano Lett., Vol. 9, No. 2, 2009
difference arising due to fluctuation of electrons across the barrier over the period over which the potential is measured using the lock-in amplifier. Under equilibrium conditions and in the absence of temperature gradient across the tunnel junction, the time-averaged random fluctuation of the electron across the tunnel junction results in a net zero voltage. However, in the presence of a temperature gradient across the tunnel junction, the time-averaged fluctuation of the electron across the junction is now offset by the tunnel junction thermoelectric effect and can be measured by the instrument. Since control of the temperature in the 200-300 K range with the current cryostat was difficult, we measured only 10 data points in this range. We were clearly able to observe the rise in the thermopower near room temperature in all cases of junctioned nanowires. The expressly particular nature of this phenomenon easily explains why it has not been previously reported. In the earlier “point contact” thermopower studies the size of the point contact was changed by modulating the force applied to the contact.15 While this will certainly change the size of the point contact, it would be impossible to create a contact with a tunnel junction similar to what we have observed. From our theory this enhancement should not be limited to cylindrical contacts and may present itself in nanowire-tothin film contacts as well; however all previous studies have attempted to improve thermal and electrical contact between the nanowire and measurement device by “clamping” the nanowire with the device itself or with focused ion beam deposited contacts thereby eliminating the possibility of the existence of a tunnel junction. Two-probe electrical resistance measurement of a straight single nanowire sample over the desired temperature range showed a resistance-temperature profile which matches well with that reported in literature for bulk gold to within measurement error.30 From our two-probe I-V measurements of the finger-nanowire-bridge, we did not see any difference between the “junctioned” and the straight nanowire samples to within 100 Ω (this includes the resistance of the nickel interconnect wires which dominate the resistance of the device). Three-probe electrical I-V measurements were performed at room temperature on three independent multiwire samples, consisting of both series and parallel configurations, yielding two segment resistances per wire. Assuming the contact resistances were the same for each contact point, the gold nanowire resistivity averaged 0.06 µΩ m with results ranging from 0.02 to 0.08 µΩ m. This is only slightly above the literature value for bulk gold resistivity of 0.022 µΩ m.31 Temperature-dependent measurements of thermal conductivity of the nanowires have not yet been completed for series junctioned and direct nanowires; however, some important observations can already be made. First off, inclusion of junctions is most likely to result in a reduction in the thermal conductivity which can only increase the value of ZT. Since no appreciable difference in the overall resistivity of the junctioned or straight nanowires was observed, we can estimate an overall increase in ZT of ∼100 times. More accurate four-probe resistivity measurements and thermal conductivity measurements on the gold 621
nanowires are required to verify this enhancement in junctioned gold nanowires. To the best of our knowledge this is the first reporting of a temperature-dependent increase in thermopower at near room temperature in gold nanowires. The presence of the room temperature peak in “junctioned” nanowires has been exceedingly consistent and has been verified on multiple independent samples. Furthermore, the enhancement is observed only in junctioned gold nanowire samples as opposed to straight wire samples, and since all the measurements have been made on identical nickel thermoelectric workbenches and using identical measurement setup, the variations can only be attributed to the difference in the two configurations of the samples. To summarize, the “junctioned” configuration is best interpreted as a narrow tunnel junction with a temperature-dependent complex dielectric constant. In the region from and above the Debye temperature of gold, the thermopower increases linearly with temperature. The tunneling electron flux is small, i.e., the junction is predominantly capacitive and is best represented by the imaginary part of the junction dielectric constant. Near and above a transition temperature, the thermopower begins to level off and eventually decrease. In this temperature range the conductance (real part of the dielectric constant) begins to dominate and the thermopower is expected to decrease as T-1. This transition temperature is a material property of the tunnel junction depending upon the barrier height and the junction width. Further investigation into the phenomenon is required to develop a complete quantitative theory for the observed enhancement. It also remains to be seen if this phenomenon only applies to metals or if it can be applied to high thermopower semiconductors. Acknowledgment. This work was made possible by support from NSF-NIRT Grants No. DMR 0304178 and ECCS 0609243. Some of the work presented here was performed at the National Nanotechnology Infrastructure Network at Penn State. We would also like to thank Dr. Peter Eklund for use of the cryostat. Supporting Information Available: The fabrication and detailed performance testing of the thermoelectric workbench.This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Hicks, L. D.; Dresselhaus, M. S. Phys. ReV. B Condens. Matter 1993, 47 (24), 16631. (2) Mahan, G. D.; Sofo, J. O. Proc. Natl. Acad. Sci. U.S.A. 1996, 93 (15), 7436–7439.
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NL802882H
Nano Lett., Vol. 9, No. 2, 2009
