In this section we present the main results about the transport properties, the spin polarization and the thermoelectric effects in complex silicene structures. In particular, we show that these properties display a self-similar behavior for both spin orientation: spin up (SU) and down (SD). Moreover, we will characterize mathematically the self-similar patterns by deriving scaling expressions numerically in the case of the conductance and spin polarization and analytically for the Seebeck coefficient. We will also show, as in the case of graphene complex structures^{8}, that structural parameters such as the generation of the Cantor-like structure *N*, the height of the barriers (Delta ) and the length of the system *w* are directly involved in the scaling expressions (self-similar patterns). It is important to remark that all our numerical results are computed for the *K* valley ((eta =+1)), since the results for the (K’) valley ((eta =-1)) can be obtained straightforwardly by simply reversing the spin orientation, see Fig. 1b,c. So, from now on, we will suppress the valley index (eta ) in the scaling expressions.

### Transport properties

In first place, we address the self-similar patterns that arise in the linear-regime conductance once silicene is nanostructured in Cantor-like fashion. As we can see in Fig. 2 the conductance curves of different generations of the Cantor-like structure have similar envelopes. The similarity is presented for both spin components, SU and SD, Fig. 2a,b, respectively. The main difference between the conductance curves is an offset in the vertical axis, which tells us that we can go from one curve to the other with an appropriate scale factor. Determining the scale factors of the self-similar conductance patterns can be tricky. However, we can obtain them by following the protocol reported for graphene complex structures^{8}. In particular, we can use an auxiliary conductance-related quantity to unravel the precise scale factors. The guidelines of the protocol and specific results of the auxiliary quantity for complex silicene structures can be found in the Supplementary information. So, in the rest of this subsection we will focus on the conductance scaling rules without forgetting that there is a protocol to obtain them. In the case of the self-similar patterns between generations the scaling expression is given as

$$begin{aligned} {mathbb {G}}^{sigma }_{N}(E_{F})approx frac{left[ {mathbb {G}}^{sigma }_{N+1}(E_{F})right] ^{alpha _{sigma }}}{(2)^{alpha _{sigma }-1}}, end{aligned}$$

(13)

where the exponent (alpha _{sigma }) depends on the structural parameters and the spin component and *N* represents the generation number. The conductance in this expression and the coming ones is given in terms of the fundamental conductance factor, ({mathbb {G}}^{sigma }(E_F)=G^{sigma }(E_F)/G_{0}). In the specific cases shown in Fig. 2 the exponents take values of (alpha _{+1}=1.83) and (alpha _{-1}=1.89). The results of the scaling can be appreciated in Fig. 2c,d. As we can see the scaled curves *N*8 (dashed-blue lines) match quite well with the reference ones *N*7 (solid-black lines). Here, we would like emphasize that the exponents of the spin components are not equivalent because the corresponding conductance curves are dissimilar. This is related to the spin-dependent silicene band structure, which results in fundamental differences between the complex barrier structures for SU and SD.

Now, it is turn to explore the scaling associated to the height of the barriers. When we consider complex silicene structures with different heights in the barriers the conductance curves look quite similar. See for instance the results (Fig. 3) for complex structures with barriers of (Delta =2) and (Delta =1). In these cases the generation and the length of the system are *N*7 and (w=30). As we can notice the conductance curves are practically the same except for the vertical offset between them, see Fig. 3a,b. By following our protocol and using the auxiliary conductance-related quantity we can derive a general expression that connects the conductance patterns of complex structures with different barrier heights, namely:

$$begin{aligned} {mathbb {G}}^{sigma }_{Delta }(E_{F})approx frac{left[ {mathbb {G}}^{sigma }_{frac{Delta }{2}}(E_{F})right] ^{beta _{sigma }}}{(2)^{beta _{sigma }-1}}, end{aligned}$$

(14)

where (beta _{sigma }) is a non-constant exponent that depends on the structural parameters as well as the spin component. For the cases presented in Fig. 3 the exponents take values (beta _{+1}=2.87) and (beta _{-1}=3.02). As we can see in Fig. 3c,d the scaling is fairly good for both SU and SD. Here, the scaled curve corresponds to (Delta =1), while the reference one to (Delta =2). It is important to mention that the fundamental differences between the scaled and reference curves take place in the low-energy side, being more important for SD.

Regarding the length of the system, we can also obtain self-similar conductance patterns for complex structures with different lengths. In Fig. 4 we show the conductance versus de Fermi energy for systems with lengths (w=20) and (w=10). The generation and height of the barriers are *N*7 and (Delta =2). Figure 4a,b illustrate the corresponding conductance patterns for SU and SD, respectively. Our results, at first glance, indicate that these patterns cannot be directly connected. In fact, the curves envelopes are not at all similar. However, by contracting the energy axis and choosing an appropriate exponent it is possible to connect the conductance patterns, see Fig. 4c,d. The general expression for this scaling is given as

$$begin{aligned} {mathbb {G}}^{sigma }_{w}(E_{F})approx frac{left[ {mathbb {G}}^{sigma }_{frac{w}{2}}(frac{E_{F}}{2})right] ^{gamma _{sigma }}}{(2)^{gamma _{sigma }-1}}, end{aligned}$$

(15)

with (gamma _{sigma }) the scaling exponent. For the cases of Fig. 4 the exponents are (gamma _{+1}=gamma _{-1}=3.2). As we can see in Fig. 4c a good matching for SU is obtained, except for (E_F<0.4E_0). For SD we obtain similar results. However, the coincidence between the scaled and reference curve is substantially better for (E_F>1.2E_0), see the dashed-blue curve in Fig. 4d. Here, it is important to remark that the exponents for both spin components are the same. However, in general, they depend on the structural parameters and the spin component.

We consider that the present results are remarkable because in principle conductance-related quantities such as the spin polarization and the Seebeck coefficient could display self-similar characteristics as well. Moreover, even though we know the scaling rules for the conductance patterns there is no guarantee that the scaling for the spin polarization and the Seebeck coefficient can be obtained following the conductance protocol and that the scale factors be directly related to the conductance ones.

### Spin polarization

In second place, we analyze the results of the so-called conductance spin polarization (P_{C}). By using Eq. (11) it is possible to calculate (P_{C}) as a function of the Fermi energy for different *N*, (Delta ) and *w*. The corresponding spin polarization curves also present self-similar characteristics. However, in the present case, there is no an auxiliary quantity that helps us to unveil the scaling rules. So, we proceed numerically by testing different scale factors in order to connect the self-similar patterns. To illustrate our scaling results we use the same generations, heights of the barriers and lengths of the system as in the conductance case.

In Fig. 5a,b we show the conductance spin polarization patterns for generations *N*7 and *N*8, the other structural parameters are the same as in Fig. 2. As we can notice the spin polarization spectra look quite similar, to be specific, we can see similar curve envelopes for energies greater than (E_{0}). A better perspective of the self-similar characteristics is presented in the subview. Here, it is important to remark that the spin polarization can take both positive and negative values, as a consequence, a dilatation transformation is implemented instead of an exponent as in the conductance rules. With these considerations, the general expression that connects the self-similar spin polarization patterns between two generations is given as

$$begin{aligned} P_{C,N}(E_{F})approx 2left[ P_{C,N+1}(E_{F})right] , end{aligned}$$

(16)

resulting in a simpler relation than in the conductance case. A comparison between the reference and scaled pattern (dashed-blue lines) is illustrated in the subview of Fig. 5b. As we can notice a good matching is obtained.

The spin polarization curves for different heights in the barriers also display self-similar characteristics. In Fig. 6a we show the spin polarization results for (Delta =2) and (Delta =1). We have considered the same structural parameters as in Fig. 3. As we can see the curves are not at all similar at low energies. However, as the energy increases the resemblance between the spin polarization patterns improves, see the subview in Fig. 6a. By testing different scale factors, it is possible to go from one curve to the other with the following expression

$$begin{aligned} P_{C,Delta }(E_{F})approx 4left[ P_{C,frac{1}{2} Delta }(E_{F})right] . end{aligned}$$

(17)

In this case, the scale factor is twice the one found for the scaling between generations. In the subview of Fig. 6b we show the concrete results of applying Eq. (17). In some energy intervals the scaling works reasonable well, while in others, mainly at low energies, the coincidence between the scaled and reference curve is far from good.

Systems with different lengths also manifest self-similar spin polarization patterns. The results for (w=20) and (w=10) are shown in Fig. 7a. The generation and the height of the barriers are the same as in Fig. 4. As in the conductance case, at first instance, there are no self-similar characteristics. So, in order to connect the spin polarization curves it is necessary to scale the Fermi energy as well as to choose appropriately the proportional factor between the curves. In specific, the scaling relation is given as

$$begin{aligned} P_{C,w}(E_{F})approx 8left[ P_{C,frac{1}{2} w} left( frac{E_{F}}{2} right) right] . end{aligned}$$

(18)

Here, the scale factor is twice the one for the scaling between heights. The results of the scaling according to Eq. (18) are shown in Fig. 7b. In the subview we can see more details about the matching between the scaled and reference curves. As in the other cases, the scaling improves as the Fermi energy increases.

These results are quite interesting because despite (P_{C}) is an intricate average of the conductance spin components the self-similar characteristics prevail for all the structural parameters. Furthermore, the three scaling rules proposed for the spin polarization work reasonably well as the Fermi energy increases. It is worth mentioning that we obtain similar results for the so-called tunneling spin polarization (P_T), which is equivalent to (P_{C}) but defined in terms of the transmittance. The details can be found in the Supplementary information.

### Thermoelectric effects

Finally, we study the thermoelectricity for the electron spin components due to the hot and cold contacts that generate a temperature gradient along the complex silicene structures. In particular, we have focused our attention in the well-known Seebeck coefficient. To compute it, we first have to redefine it as ({mathbb {S}}^{sigma }(E_F)=S^{sigma }(E_F)/S_{0}), with (S_{0}=pi ^{2} k_{beta }^{2} T/3e) the fundamental thermopower unit. In order to be consistent, we used the same structural parameters than the ones for the conductance and the spin polarization.

Regarding generations, Fig. 8 shows the Seebeck coefficient results for *N*7 and *N*8. Fig. 8a,b correspond to SU and SD, respectively. As we can notice the Seebeck coefficient curves are remarkably similar, see the subviews. Even, the resemblance is superior to the one found for the spin polarization curves. So, once we are sure that this quantity reflects evidence of self-similarity, the matter now is limited to deal with the scalability. As we have corroborated with the spin polarization there is no a route or procedure to derive the scaling relations. In the present case, we take advantage of the direct relation between the Seebeck coefficient and the conductance in order to obtain the scaling expressions. In particular, we use the definition of the Seebeck coefficient for an arbitrary generation *N*

$$begin{aligned} left. {mathbb {S}}^{sigma }_{N}(E) = S_{0}dfrac{partial ln left[ {mathbb {G}}^{{sigma }}_{N}(E)right] }{partial E} right| _{E=E_F}, end{aligned}$$

(19)

by substituting the scaling relation for ({mathbb {G}}^{{sigma }}_{N}(E)) (Eq. 13)

$$begin{aligned} left. {mathbb {S}}^{sigma }_{N}(E) approx S_{0}dfrac{partial ln left[ frac{[{mathbb {G}}^{sigma }_{N+1}(E)]^{alpha _{sigma }}}{(2)^{alpha _{sigma }-1}}right] }{partial E} right| _{E=E_F}, end{aligned}$$

(20)

and using the properties of the logarithmic function we can obtain the expression that connects the Seebeck coefficient between generations

$$begin{aligned} {mathbb {S}}^{sigma }_{N}(E_F)approx alpha _{sigma }left[ {mathbb {S}}^{sigma }_{N+1}(E_F)right] . end{aligned}$$

(21)

This expression is surprisingly simple, mathematically equivalent to the scaling relations found for the spin polarization. However, the scaling is much better for the Seebeck coefficient. In Fig. 8c,d we show how Eq. (21) works for SU and SD. As compare with the spin polarization scaling (see Fig. 5), the matching between the scaled and reference Seebeck curves is fairly good for both spin components, being superior for SU. It is also interesting to note that the scale factors are the same for both the conductance and the Seebeck coefficient, however for the latter the scale factors come in multiplicative fashion.

The Seebeck coefficient of complex structures with different barrier heights also display self-similar characteristics. In Fig. 9, we show the Seebeck coefficient results for (Delta =2) (solid-black lines) and (Delta =1) (solid-blue lines). Figure 9a,b correspond to SU and SD, respectively. To obtain the scaling relations we follow the previous analytic procedure. In specific, the scaling relation of the Seebeck coefficient for complex structures with different barrier heights is given as

$$begin{aligned} {mathbb {S}}^{sigma }_{Delta }(E_F)approx beta _{sigma }left[ {mathbb {S}}^{sigma }_{frac{1}{2} Delta }(E_F)right] , end{aligned}$$

(22)

where (beta _{sigma }) is the same exponent that connects ({mathbb {G}}^{sigma }_{Delta }) and ({mathbb {G}}^{sigma }_{frac{1}{2} Delta }). The results of the scaling for SU and SD are shown in Fig. 9c,d, respectively. As in the conductance case (Fig. 3) the factors adopt the values (beta _{+1}=2.87) and (beta _{-1}=3.02). As we can see in the subviews of Fig. 9c,d, a good coincidence between the scaled and reference curves is obtained.

In the last place, we explore the Seebeck coefficient results of complex structures with different lengths. In Fig. 10 we show the corresponding results for (w=20) (solid-black lines) and (w=10) (solid-blue lines). Figure 10a,b correspond to SU and SD, respectively. The generation and the height of the barriers are the same as in Fig. 4. As in the previous cases, the self-similar Seebeck coefficient patterns can be connected with an appropriate scaling relation. By using the definition of the Seebeck coefficient (Eq. 12) and the scaling rule of the conductance between lengths (Eq. 15) we can obtain the corresponding scaling relation for the Seebeck coefficient

$$begin{aligned} {mathbb {S}}^{sigma }_{w}(E_F)approx gamma _{sigma }left[ {mathbb {S}}^{sigma }_{frac{1}{2}w}left( frac{E_{F}}{2}right) right] , end{aligned}$$

(23)

with (gamma _{sigma }) the same exponent that connects ({mathbb {G}}^{sigma }_{w}) and ({mathbb {G}}^{sigma }_{frac{1}{2}w}). The results of the scaling for SU and SD are shown in Fig. 10c,d, respectively. The scale factors take values of (gamma _{+}=gamma _{-}=3.2), which are the same as in the conductance case, see Fig. 4. As we can appreciate in the subviews the scaling works quite well, being better for SU. In this case, initially, the curves are far from similar, but once the energy axis is contracted and the Seebeck coefficient multiply by the correct factor, we recover in high degree the reference curve.

We consider that the present results are quite interesting because as far as we know it is the first time that a conductance related quantity like the Seebeck coefficient presents self-similar characteristics. Moreover, that we can obtain the scaling rules straightforwardly through the analytic relation between the conductance and Seebeck coefficient.