**Appendix A**

#### *Appendix A.1. Note from Section 2.1*

A very good approximation of the actual *T* profile and hence the SpAv of ρ and κ in accordance with a real material can be calculated in a straightforward way from the *T*-dependent properties without using an iterative solution [18] for *<sup>T</sup>*(*x*). This may considerably simplify the estimation of appropriate SpAvs as CPM property values. Δ *<sup>T</sup>*Joule(*x*) can be obtained analytically from the CPM case, Δ *<sup>T</sup>*κ(*T*) from a integration of the Fourier equation and Δ *T*Thomson and from a 1TD α(*T*) model by a single integration.

#### *Appendix A.2. Material Data and Boundary Conditions*

**Figure A1.** Temperature-dependent thermoelectric material properties of representative material classes: (**a**) thermal conductivity, (**b**) Seebeck coefficient, (**c**) electrical resistivity and (**d**) figure of merit. Since SnSe has much higher resistivity, the scale for it is given on the right y-axis. All the raw experimental data taken from the literature [20–24] were fitted with appropriate polynomials (usually 3rd or 4th order). For SnSe, a 9th order polynomial fit was used owing to the complex T dependence and hence shows an unusually high *zT*max. However, this does not affect the physics discussed and hence these fitted data were used throughout the manuscript.


**Table A1.** Temperature range of analysis for all materials of Figure A1.

#### *Appendix A.3. Additional Information*

Appendix A.3.1. Finding Individual Contributions to the Total *T* Profile

The partial *T* profiles are each found by equating κ(*T*) ∂2*T* ∂*x*<sup>2</sup> in Equation (1) to each of the other corresponding terms, assuming isothermal boundary conditions and fixing all coefficients in the equation according to the total *T* profile *<sup>T</sup>*(*x*). Thus, solving for the respective partial *T* profile reduces to a double integration, where the first step provides the total amount of each partial heat contribution to the thermal balance.

As the partial *T* profiles can have opposite signs in amplitude and partially compensate for many of the common TE materials (however, not always), the *T* profiles of a real material and the CPM may be quite close, as in the example of *n*-type Mg2*X*, Figure 2b.

.

#### Appendix A.3.2. Contributions to Qin

As both Joule and Thomson heat, after appearing inside the leg, will flow out, physically, as Fourier heat, we have to consider in this discussion the pure Fourier heat *Q*F,h = *K* Δ *T* (with *K* = κ<sup>−</sup><sup>1</sup> −1*A*/*L*), which is merely related to the thermal resistance of the leg and is constant along the leg, separately from the Joule- and Thomson-related contributions. Accordingly, *Q*in is composed of

$$
\dot{Q}\_{\rm in} = \dot{Q}\_{\rm F,h} + \dot{Q}\_{\rm \pi,h} - \dot{Q}\_{\rm \tau,h} - \dot{Q}\_{\rm J,h} \tag{A1}
$$

.

.

.

.

.

.

The real Joule- and Thomson-related contributions, − *Q*<sup>τ</sup>,<sup>h</sup> and −*Q*J,h, to the inflowing hot side heat are calculated by splitting the overall temperature profile *<sup>T</sup>*(*x*) into additive partial *T* profiles, each related to one of the individual physical contributions. Partial Thomson *T* profiles of example materials are plotted in Figure A2b. Evaluating −κh· *ddx* Δ *T*Thomson h and −κh· *ddx* Δ *<sup>T</sup>*Joule h from the partial *T* profiles gives . *Q*<sup>τ</sup>,<sup>h</sup> and . *Q*J,h, respectively.

Figure A2a shows the relative contribution of each heat to . *Q*in: *Q*F,h . *Q*in , *Q*<sup>π</sup>,<sup>h</sup> . *Q*in ,<sup>−</sup> *Q*J,h . *Q*in ,<sup>−</sup> *Q*<sup>τ</sup>,<sup>h</sup> . *Q*in . This comparison reveals that Joule and Thomson heat contribute about 1–5% to . *Q*in, usually flowing out, with their contributions being roughly of the same order. Figure A2a also shows the fraction of Thomson heat and Joule heat distributed to the hot side ( . *Q*<sup>τ</sup>,<sup>h</sup> . *Q*τand . *Q*J,h . *Q*J).

 In order to illustrate example situations of the distribution of Peltier and Thomson heat along the leg, α(*T*) graphs for p-Mg2*X* and Bi**2**Te3 are given in Figure A2c,d, respectively. Due to the bowed shape of the α(*T*) graph and relatively close values of αh to αc for p-Mg2*X*, the di fference between . *Q*<sup>π</sup>,<sup>h</sup> and . *Q* CPM <sup>π</sup>,h is almost negligible, but . *Q*<sup>τ</sup>,<sup>h</sup> amounts to more than twice the amount of . *Q*<sup>π</sup>,<sup>h</sup> − . *Q* CPM <sup>π</sup>,h . Nevertheless, this did not a ffect the e fficiency deviation δηmax too much, as . *Q*<sup>τ</sup>,<sup>h</sup> is quite small in absolute terms. In the case of Bi2Te3, . *Q* CPM <sup>π</sup>,h is even higher than . *Q*<sup>π</sup>,<sup>h</sup> again due to the curved shape of <sup>α</sup>(*T*), a ffecting the position of αTAv. However, . *Q*<sup>τ</sup>,<sup>h</sup> almost completely compensates for this Peltier heat di fference, keeping the influence on the e fficiency deviation negligible.

#### *Appendix A.4. Thomson Heat Distribution and Entropy*

With the TEG leg, we discuss the entropy flow in a reversible system of Peltier heat transport and Thomson heat exchange which is running on a non-equilibrium temperature background mainly fixed by the continuous flow of Fourier heat. As released Thomson heat will be transported as Fourier heat but is small in relation to the Fourier heat background (see Figure A2a), which is driven by the temperature di fference and the thermal resistance of the TE leg, we will treat the variation of the temperature profile by the conducted Thomson heat as insignificant for the following consideration.

In the steady state, the entropy of the system remains constant; there is a continuous entropy production by the dissipative heat transport from hot to cold and the balancing continuous entropy export by transmitted Fourier heat (plus a negligible fraction arising from outflowing Joule heat). Assuming ideal outer current leads with α = 0, there is no other entropy exchange at the hot and cold sides.

**Figure A2.** (**a**) Ratio of individual heat contributions to *Q*in (Equation (A1)) calculated from the corresponding partial temperature profiles (for comparison, all quantities are counted as positive when flowing into the element) (left *y*-axis), and distribution factors (right *y*-axis) for Thomson and Joule heat. (**b**) Thomson *T* profiles for all example materials (**c**) . *S*(*T*) diagram for p-Mg2X showing the area between *<sup>I</sup>*α(*<sup>T</sup>*h) and *I*αTAv (corresponding to the Peltier heat difference between the CPM and real case), which is very small due to the shape of <sup>α</sup>(*T*). The position of the first peak in the Thomson partial *T* profile is marked as a brown vertical line. (**d**) . *S*(*T*) diagram for Bi2Te3,where αTAv > <sup>α</sup>(*<sup>T</sup>*h). Hence, . *Q* CPM <sup>π</sup>,h is higher than . *Q*<sup>π</sup>,h.

.

In the CPM, we have a constant convective entropy flow α*I* throughout the element, equal to the absorbed and released entropy rate α*I* by absorption and release of Peltier heat at the terminals. In a real material, the absorbed entropy rate <sup>α</sup>h*I* equals the convective entropy flow at the hot side, and, likewise, the amount of <sup>α</sup>c*I* at the cold side. The variation of α along the leg drives local Thomson heat production *d* . *Q*τ = *T d*α*dT IdT* = *TId*<sup>α</sup>, contributing an entropy flow increment *d* .*S* = *Id*<sup>α</sup>. Thomson heat flows to the hot and cold sides and the related total entropy exchange is (<sup>α</sup>h − <sup>α</sup>c)*<sup>I</sup>* = Δα*I*. It distributes by the fraction *x*h to the hot and cold sides:

$$
\Delta \dot{S}\_{\mathsf{T}, \mathsf{h}} = \mathsf{x}\_{\mathsf{h}} (\alpha\_{\mathsf{h}} - \alpha\_{\mathsf{c}}) I \text{ and } \Delta \dot{S}\_{\mathsf{T}, \mathsf{c}} = (1 - \mathsf{x}\_{\mathsf{h}}) (\alpha\_{\mathsf{h}} - \alpha\_{\mathsf{c}}) I. \tag{A2}
$$

Driven by the gradient of the partial Thomson temperature profile, all Thomson heat released at one side of a maximum (or minimum) of this profile will be exchanged to this side of the leg. With the temperature *T*<sup>τ</sup>,ex of this position and its Seebeck coefficient <sup>α</sup>τ,ex = <sup>α</sup>(*<sup>T</sup>*<sup>τ</sup>,ex), the shares of the entropy exchange which are bound to each of the sides are

$$
\Delta \dot{S}\_{\mathbf{r}, \mathbf{h}} = (a\_{\mathbf{h}} - a\_{\mathbf{r}, \mathbf{ex}})I \text{ and } \Delta \dot{S}\_{\mathbf{r}, \mathbf{c}} = (a\_{\mathbf{r}, \mathbf{ex}} - a\_{\mathbf{c}})I. \tag{A3}
$$

Multiplying both by the respective temperature of the side yields total Thomson heat:

$$\dot{Q}\_{\mathsf{T}} = T\_{\mathrm{h}} (a\_{\mathrm{h}} - a\_{\mathsf{T},\mathsf{ex}})I + T\_{\mathrm{c}} (a\_{\mathsf{T},\mathsf{ex}} - a\_{\mathsf{c}})I = \{T\_{\mathrm{h}}a\_{\mathrm{h}} - T\_{\mathrm{c}}a\_{\mathsf{c}} - a\_{\mathsf{T},\mathsf{ex}}(T\_{\mathrm{h}} - T\_{\mathrm{c}})\} I = \Delta \dot{Q}\_{\mathsf{T}} - \Delta I\_{\mathrm{c}} \tag{A4}$$
 
$$I a\_{\mathsf{T},\mathsf{ex}} \Delta T.$$

Comparing Equation (A4) with the energy balance of reversible heat . *Q*τ = Δ . *Q*π − *IV*0, we can conclude that

$$
\alpha\_{\pi,\text{ex}}\Delta T = V\_0 = \overline{\alpha}\Delta T,\text{ thus }\alpha\_{\pi,\text{ex}} = \overline{\alpha} \tag{A5}
$$

This gives us a rule for the temperature intervals over which the Thomson heat is flowing to either side of the leg. Consequently, Thomson heat has to be integrated from the crossing point of the curve of the Seebeck coe fficient α(*T*) with its temperature average α. As a reversible approximation, this result is approximate and not strict as we have neglected here that dissipative processes are involved when Thomson heat is conducted to the leg sides. Below we will analyze these changes and find that these are small, and thus the rule stated here on the position of <sup>α</sup>τ,ex, although not strict, is a good guide for estimates of the distribution of Thomson heat. Indeed, as observed by comparison to exact numerical calculations, this rule is almost perfectly fulfilled for all the example materials.

Within this reversible approximation, the Thomson heat flowing to the hot side is obtained as . *Q*<sup>τ</sup>,<sup>h</sup> = *<sup>T</sup>*h(<sup>α</sup>h − <sup>α</sup>)*<sup>I</sup>*. This would be equivalent to a complete compensation of the Peltier heat di fference between reality and the CPM, i.e., the vanishing axial redistribution of reversible heat which is consistent with the simplifying assumption that the Thomson heat flowing to the outside is transmitted free of dissipation, i.e., equivalent to reversible heat. Here, the (additional) *T* gradient related to the flow of Thomson heat is neglected, whereas an underlying *T* profile related to an independent heat flow (here, the background of Fourier heat transfer) does, in e ffect, not contribute to its dissipation. We will see below that this happens as Thomson heat flowing to di fferent sides will contribute almost compensating shares to the entropy balance. What is neglected here is that the Thomson heat itself when flowing to the ends of the leg will dissipate, according to the slight shift of the inner *T* profile it is causing. Above, this *T* offset was separated and called the partial *T* profile due to Thomson heat, Δ *<sup>T</sup>*Thomson(*x*). Additionally, this omission will contribute to a weak deviation from the position rule <sup>α</sup>τ,ex = α.

The dissipative part of the entropy transport to the sides of the leg is related to the *T* drop or step-up between the location where an increment of Thomson heat *d* . *Q*τ is released and the side temperature, *T*h or *T*c. The entropy increment is released over a segmen<sup>t</sup> of the leg with the *T* increment *dT* is *d* . *S* = *Id*α = *d* . *Q*τ *T* . With the transfer to the cold side, for example, the transmitted increment of Thomson heat *d* . *Q*τ increases its entropy up to *d* . *S*c = *d* . *Q*τ *T*c , and the according entropy gain is

$$d\Delta \dot{S}\_{\rm c} = d\dot{S}\_{\rm c} - d\dot{S} = \frac{d\dot{Q}\_{\rm \tau}}{T\_{\rm c}} - \frac{d\dot{Q}\_{\rm \tau}}{T} = \frac{d\dot{Q}\_{\rm \tau}}{T T\_{\rm c}}(T - T\_{\rm c}) = d\dot{S} \frac{T - T\_{\rm c}}{T\_{\rm c}} \tag{A6}$$

Summing over all Thomson heat flowing to that side, we have

$$
\Delta \dot{S}\_{\text{c}} = \frac{1}{T\_{\text{c}}} \int\_{la\_{\text{c}}}^{\overline{da}} (T - T\_{\text{c}}) d\dot{S} = \frac{I}{T\_{\text{c}}} \int\_{a\_{\text{c}}}^{\overline{a}} (T - T\_{\text{c}}) da = \frac{I}{T\_{\text{c}}} \int\_{T\_{\text{c}}}^{T\_{\overline{a}}} T \frac{da}{dT} dT - I(\overline{a} - a\_{\text{c}}) \tag{A7}
$$

Multiplying with the cold side temperature, Δ . *Q*<sup>τ</sup>,<sup>c</sup> = *T*cΔ . *S*c = *T*α *T*c *T d*α*dT dT* − *<sup>I</sup>*(α − <sup>α</sup>c)*<sup>T</sup>*c gives us the amount of Thomson heat that is just the di fference from the Peltier–Thomson heat balance of the CPM, Δ . *Q*<sup>τ</sup>,<sup>c</sup> = . *Q*<sup>τ</sup>,<sup>c</sup> − . *Q* CPM π,c − . *Q*<sup>π</sup>,<sup>c</sup> , i.e., the part that we have identified as uncompensated Peltier–Thomson heat in a real material. Note that . *Q* CPM π,c contains merely completely reversible exchange of Peltier heat. Thus, the incomplete compensation of the Peltier–Thomson heat balance can be understood as an e ffect of the partly dissipative character of the exchange of the Thomson heat in a

.

real system when conducted to the side. Accordingly, with the same consideration for the hot side, with *d* . *S*h = *d* . *Q*τ *T*h, we obtain

$$
\Delta \dot{S}\_{\text{h}} = d \dot{S}\_{\text{h}} - d \dot{S} = \frac{d \dot{Q}\_{\text{\pi}}}{T\_{\text{h}}} - \frac{d \dot{Q}\_{\text{\pi}}}{T} = \frac{I}{T\_{\text{h}}} \int\_{\overline{\pi}}^{\alpha\_{\text{h}}} (T - T\_{\text{h}}) da = \frac{I}{T\_{\text{h}}} \int\_{T\_{\text{\pi}}}^{T\_{\text{h}}} T \frac{d \alpha}{dT} dT - I(\alpha\_{\text{h}} - \overline{\alpha}) \tag{A8}
$$

i.e., Δ *S*h gives a negative contribution to the entropy balance. This sounds contradictory to the second law of thermodynamics but it is not, as the Thomson heat is not really flowing from a lower to a higher temperature but, when released, reduces the *T* gradient of the underlying background of flowing Fourier heat, thus reducing the Fourier heat flow by the amount of "upstreaming" Thomson heat.

The hot and cold side entropy changes together give

$$
\Delta \dot{S} = \Delta \dot{S}\_{\text{h}} + \Delta \dot{S}\_{\text{c}} = \frac{l}{T\_{\text{h}}} \int\_{\overline{\alpha}}^{\alpha\_{\text{h}}} (T - T\_{\text{h}}) d\alpha + \frac{l}{T\_{\text{c}}} \int\_{\alpha\_{\text{c}}}^{\overline{\alpha}} (T - T\_{\text{c}}) d\alpha = \frac{l}{T\_{\text{c}}} \int\_{T\_{\text{c}}}^{T\_{\text{H}}} T \frac{d\mu}{d\Gamma} dT + \tag{A9}
$$

$$
\frac{l}{T\_{\text{h}}} \int\_{T\_{\text{H}}}^{T\_{\text{h}}} T \frac{d\alpha}{d\Gamma} dT - I(\alpha\_{\text{h}} - \alpha\_{\text{c}}).
$$

With 1*T*c ααc *Td*α>α − αc and 1*T*h αh α *Td*α<αh − α we ge<sup>t</sup> *IT*c *T*α *T*c *T d*α*dT dT* + *IT*h *Th T*α *T d*α*dT dT* <sup>≈</sup>*<sup>I</sup>*(<sup>α</sup>h − <sup>α</sup>c) and thus Δ . *S* ≈ 0. Hence, assuming <sup>α</sup>τ,ex = α, the entropy balance of the inner Thomson heat transfer as an offset of a much larger background Fourier heat flow is almost zero. This indeed confirms our approach to deduce a rule for the local distribution of Thomson heat based on a reversible approximation,i.e.,assuming Δ . *S* ≈ 0,butalsoshowsthattheruleisnotcompletelystrict.
