You can view this as follows: adding heat to a system causes the molecules and atoms to speed up. It may be possible though tricky to reverse the process in a closed system without drawing any energy from or releasing energy somewhere else to reach the initial state.
You can never get the entire system "less energetic" than when it started. The energy doesn't have any place to go. For irreversible processes, the combined entropy of the system and its environment always increases. This view of the second law of thermodynamics is very popular, and it has been misused. Some argue that the second law of thermodynamics means that a system can never become more orderly.
This is untrue. It just means that to become more orderly for entropy to decrease , you must transfer energy from somewhere outside the system, such as when a pregnant woman draws energy from food to cause the fertilized egg to form into a baby.
This is completely in line with the second law's provisions. Entropy is also known as disorder, chaos, and randomness, though all three synonyms are imprecise. Absolute entropy is defined according to the third law of thermodynamics. Here a constant is applied that makes it so that the entropy at absolute zero is defined to be zero. Actively scan device characteristics for identification. Use precise geolocation data.
Select personalised content. Create a personalised content profile. Measure ad performance. As the temperature of a sample decreases, its kinetic energy decreases and, correspondingly, the number of microstates possible decreases.
The third law of thermodynamics states: at absolute zero 0 K , the entropy of a pure, perfect crystal is zero. Using this as a reference point, the entropy of a substance can be obtained by measuring the heat required to raise the temperature a given amount, using a reversible process. Reversible heating requires very slow and very small increases in heat.
Assume the change is reversible and the temperature remains constant. These values have been tabulated, and selected substances are listed in Table However, thermopower is, quite generally, a transport coefficient and its relation to entropy has been shown to be questionable in systems with strongly anisotropic transport for instance In the opposite high-temperature limit, where temperature is the largest energy scale in the system, general relations between the thermopower and derivatives of the entropy can be derived, embodied in the Heikes 8 , 12 , 13 and Kelvin 11 , 14 formulas.
The method we propose here is based on a general observation, which is also an important result of our work: if one weakly couples this system to leads, the conductance of such an interacting system can be put in the form of a non-interacting conductance formula, provided one takes into account a temperature-dependent shift of the chemical potential gate voltage. The thermal response TR , in turn, can be written in a similar manner, where the temperature-dependent shift in the chemical potential produces an extra contribution.
We show that this extra term, which can be determined by comparing the actual thermal response of the system to that of the related non-interacting system which can be estimated using a newly introduced high-temperature version of the original Mott formula 15 , can be used to extract the entropy of such a mesoscopic system even in the case of arbitrary spectrum and degeneracies, and then demonstrate the usefulness of the approach by applying it to several model systems. One big advantage of our formulation is that one can apply it to any such mesosopic system where measurements of both electrical conductance and thermopower are available.
In the process we explain the long standing puzzle of the observation of a non-zero thermopower at the apparent electron-hole symmetry point in the Coulomb Blockade CB valley 16 , This temperature-dependent shift in the chemical potential is given by.
In contrast, our expression indicates that in the case of many levels, which has not been discussed before, the temperature-dependent part of the shift does not depend on which level the electron tunnels through, and what its degeneracy is. A step by step description of the fitting process is detailed in Supplementary Note 3. In the following we demonstrate the usefulness of this formalism in model systems, where one can compare the entropy obtained using the above relation to that calculated directly from thermodynamic considerations, and finally we apply our formalism to available experimental data.
Figure 1b illustrates the correspondence between the TR obtained directly, using Eq. The conductance used in evaluating both terms in the RHS of Eq. In order to construct the estimate for the TR in Fig. The figure displays an almost perfect agreement between the direct calculation of the TR and that obtained by our Ansatz. We have repeated the procedure for QDs of arbitrary degeneracy. We see a perfect agreement even up to large degeneracies.
As mentioned above, some aspects of this simple case of a single degenerate level have been addressed before, and it has been suggested that the thermopower through a single-level QD can be used, e. The advantage of our procedure lies in its application to a multi-level mesoscopic system, such as a multi-level QD, or to a multi-dot system, where the entropy is temperature-dependent.
For simplicity, we assume that the transition through one of the levels dominates the transport, so Eq. As we will demonstrate, even though a single transition dominates the transport, the resulting procedure yields the full entropy change in the system. Figure 1 d and e depict, respectively, the calculated conductance and TR, again using Eq.
The resulting estimate for the entropy change is plotted in Fig. Again we observe excellent agreement between the entropy deduced in our procedure and the direct calculation. In Supplementary Note 4 we discuss our procedure for the case when several transitions are relevant to the total transport. Since Eq. The fitting procedure that corresponds to Eq.
One of the main advantages of our approach, compared, e. The sample is shown in the inset to Fig. The tunnel coupling between the QD and the reservoirs H and C can be controlled symmetrically adjusting the gate voltage applied to gate B1. Gate P, the so-called plunger gate, is used to continuously tune the electrochemical potential of the QD, and consequently the number of electrons on the QD. Gate G is not used in these experiments and is kept at ground at all times. The thermovoltage has a non-zero value in the middle of the valleys around the apparent particle-hole symmetry point arrow.
Interestingly, the data show that at points of apparent particle-hole symmetry in the conductance e. This experimental observation see also refs. Experimental measurements of a conductance and b thermovoltage through the same device as in Fig.
Theoretical NRG calculations of c conductance and d thermopower through a QD with two spin-degenerate levels, with linearly varying level spacing, depicted in the inset to d. The numerical plots were shifted horizontally so that the minima inside the valley for all plots coincided for alignment as in the experimental plots.
The results also indicate a non-zero crossing point arrow. In the following we detail our analysis of these CB peaks. In addition, since the x -axis relation between the conductance measurement Fig. The results of fitting the TR to Eq. As can be seen in the figure, there is a good agreement between the fit and the observed TR in the vicinity of each peak, again using only a few fitting parameters to fit the whole curve see Supplementary Note 3 for a detailed step-by-step of the analysis of the experimental data using our procedure , illustrating the experimental validity of our approach.
This value, however, is not easily and accurately determined in an experiment and thus leads to uncertainties in the absolute values of the entropy changes across the peaks. This means that the first peak signals a transition into a four-fold degenerate state, while the second peak may either correspond to a transition from a four-fold degenerate to a two-fold degenerate state, or from a two-fold degenerate state to a non-degenerate state.
This suggests a deviation from the naive picture of consecutive filling of a four-fold degenerate state. While the degeneracy of these two levels seems fortuitous, such a model, in fact, has been claimed to be generic for transport through QDs 25 , 26 , 27 , and has been invoked to explain the repeating phase jumps in the transmission phase through such a dot 28 , In these works this is caused by two overlapping levels with different tunneling widths. In this scenario, after the second conductance peak the narrow level is doubly occupied, and does not play an additional role in transport, while the wide level is shifted up to overlap with another narrow level, and the process repeats itself.
This explained the repeated phase change across consecutive conductance peaks 28 , 29 , and is, in fact, consistent with the observation that the upshift of the TR from zero at the apparent particle-hole symmetric point happens in consecutive pairs of conductance peaks These data, depicted in Fig.
Similar evolution of the degeneracy as a function of chemical potential has already been observed in quantum nano-tubes This interpretation leads to the observed values of entropy change. Interestingly, this model naturally reproduces the non-zero value of the TR at the seemingly particle-hole symmetric point, which is also visible in the experimental data crossing point in Fig. This anomalous increase of the TR around the middle of the valley is attributed to a non-trivial degeneracy, thus providing a natural explanation that this value of gate voltage does not correspond, in fact, to a particle-hole symmetric point.
An alternative explanation, based on non-linear effects, was suggested in recent work In this work, we have derived a theoretical connection between the entropy and transport coefficients in mesoscopic junctions. This connection relates the TR of a Coulomb-blockaded mesoscopic system with arbitrary many-body levels to the conductance and the entropy change between adjacent CB valleys.
This allowed us to apply the method to experimental data in that regime, which yielded non-trivial, and in fact unexpected information about the entropy in each CB valley. The deduced theoretical model, which described the experimental QD, reproduced the measured thermopower and resolved the long-standing puzzle of a finite TR in the apparent particle-hole symmetric point.
The success of this procedure suggests possible venues to extend this analysis especially towards the study of entropy of exotic states. One direction would be to extend the method to low temperatures, thus enabling the determination the degeneracy of the ground state of the full system. This, for example, is particularly relevant to exotic phases, such as the two-channel Kondo system, where the zero temperature entropy is non zero.
If the TR of this system can be utilized to deduce the entropy of the ground state, this can be a smoking gun for the observation of the two-channel Kondo ground state 32 or other such non Fermi liquid ground states. Such an extension has also been suggested in parallel by Sela et al. In relating the non-interacting conductance and TR we use a high-temperature adaptation of the Mott relation Our sample is designed similar to the one used by Scheibner et al.
The QD is situated on one side of the channel, delimited by gates B1 and B2,while the opposite side of the channel is delimited by the two gates Q1 and Q2, forming a quantum point contact QPC , which is positioned exactly opposite to the quantum dot.
It separates the heating channel H from the reservoir REF, which is kept at ground potential. From here the heat gets removed efficiently by the dilution refrigerator. In this manner we establish a locally enhanced electronic temperature in the channel while the rest of the 2DES remains approximately at base temperature. Hence the measured signal can be attributed fully to the QD. The datasets generated and analysed in the study are available upon request from the corresponding authors.
Andrei, N. Solution of the multichannel Kondo problem.
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