Unleashing nanofabrication through thermomechanical nanomolding

Description

• , we get the control equation of growth rate (dL/dt): , by integrating which gives: Here L0 is an integration constant related to the loading process and the entry effect in TMNM. In the case of bulk diffusion, where diffusion coefficient is measured by lattice diffusivity (D = DL) and diffusion happens across the whole volume of the nanowire ( = π 2 4 ), we have: In the case of interface diffusion, where diffusion coefficient is measured by interface diffusivity (D = DI) and diffusion is limited in a thin interface layer of a thickness δ between the nanowire and mold ( = π ), we have: Dislocation slip: Nanowires grow by dislocation sliding down into mold in dislocation slip mechanism. Since the moving speed of nanowires in these nanocavities is super slow, by ignoring all inertial forces we have: , where pe is the pressure at the entrance of the nanocavity and I is the shear strength along the nanowire-mold interface. In the process of TMNM, the total molding pressure is consumed partially in the feedstock material and partially in the nanocavities to achieve a free surface (p = 0) at the tips of nanowires. This pressure distribution results in a decreasing gradient from the feedstock to all the way through the nanowire and gives e ≤ . In a growing nanowire, pe increases with increasing L according to eq. S6 until e = . This sets a maximum length in dislocation-based TMNM by: We consider the growth of nanowires in this dislocation slip case is based on dislocation moving in the feedstock material driven by the pressure drop p -pe. The control equation of the growth rate can be written by: , where C is a material related constant, b is the magnitude of the Burgers vector, u is the average propagating velocity of dislocations, Nd the number of dislocations contributing to nanowire growth. Integrating eq. S8 and considering ( =0) = 0 gives: Substitute Lm with eq. S7, then we have: , where α = Cπ is a constant, and = d / π 2 4 is the dislocation density in the feedstock at steady deformation state.

Supplementary Section 2: Scaling results from the theoretical models
Equation S4, S5 and S9 represent the scaling law of the three mechanisms of bulk diffusion, interface diffusion, and dislocation slip. To simplify the discussion, we assume ( =0) = 0. By using the same molding material and keeping T, t, p to be constant, three mechanisms result in very different scaling of L vs. d, that can be readily revealed by experiments. In the case of bulk diffusion, eq. S4 gives ∝ 0 , while in interface diffusion eq. S5 shows This means a constant length in bulk diffusion and a decreasing length in interface diffusion dominated cases when molding diameter is increasing.
In the case of dislocation slip mechanism, we need more discussion to simplify the format of eq. S9. Consider a format of = m (1 − e −k ) where m = 4 I and k = α I are constants resulted from eq. S9, the derivative of eq. S9 gives ( ( ))′ = m(1 − e −k ) + mk e −k > 0 and ( ( =0) )′ = 0 under practical molding conditions, which gives an increasing L with d. To get an intuitionistic idea of the scaling behavior in eq. S9, we approximate it with a power function = n , which is more comparable to the cases in bulk diffusion ( ∝ 0 ) and interface diffusion ( ∝ −1/2 ), and reveal the range of x that can fit eq. S9 well. Considering this power function is equivalent to a linear format ln = ln + (ln ), the approximation above results in a linear fitting to eq. S9 in format: ln = ln m + (ln ) + ln ( A detailed numerical result of L(d) with varies constants m and k is plotted in Fig. S2A, where by tunning these constants the curve eq. S9 is shifting between first and second order power functions.
In practical molding conditions of loading pressure ~ 500 MPa and I ~ 10 MPa, we have m ~ 10 that agrees with the fitting results of experimental data ( Fig. 1C and Fig. S2B). However, the estimation of k remains difficult as there is an unknown constant in its expression.
When using scaling experiments as a tool to reveal the underlying mechanism, we are aiming to distinguish between dislocation slip, interface diffusion or bulk diffusion-controlled growth. The scaling between L and t is also considered as a criterion of dislocation-based from diffusionbased mechanisms. In the case of diffusion-based mechanisms, eq. S4 and eq. S5 give an increasing L with time with ∝ 1/2 . In the case of dislocation-based mechanism, on the other hand, eq. S9 sets a limitation of L and results in a shotting up of length then a quasi-constant state after a short time. This is revealed both numerically and experimentally in Fig. S3.

Supplementary Section 3: Absolute value analysis between bulk and interface diffusion
Besides the scaling behaviors, we also reveal the huge difference in absolute length values expected by bulk and interface diffusion mechanisms, with results summarized in Fig. 1D.
To compare data measured with varies molding conditions (T, p, and t), we first calculated the normalized molding length (L') by ′ = √ 2 8 /k B . In TMNM, L' will be totally decided by molding diameter and the diffusivity of molding material. Furthermore, in the case of bulk diffusion, with eq. S4 we have: In typical molding conditions, we consider d ~ 100 nm, δ ~ 1 nm, and DI ~ 10 4 DL. One can derive from eq. S10-1 and eq. S10-2: For a given molding material (δDI) and molding diameter, the normalized molding length (L') under both diffusion-based mechanisms can be then estimated with eq. S11 and compared with experimental data as shown in Fig. 1D.

Supplementary Section 4: Uniaxial pressure, hydrostatic pressure, and pressure at the entrance
In the theoretical analysis of TMNM, we need to consider three different pressures, the uniaxial compressive pressure (the loading pressure defined by F/A, where F is the loading force and A is the surface area of the feedstock), the hydrostatic pressure inside the feedstock (p), and the pressure at the entrance of nanocavities (pe). The pressure at the entrance of the nanocavities (pe) does not always equal to the hydrostatic pressure (p) in the feedstock, which can be as low as one third of the uniaxial compressive pressure (F/A). In TMNM, pe is not constant but can change with time and the length of the forming nanowires. For derivation of scaling equations (supplementary section 1), the relation between pe and p is demonstrated in the dislocation dominated mechanism (eq. S6). For diffusion dominated cases, however, we approximate pe to p to simplify the scaling behavior of L(t, T, p, d) (eq. S1 to S5). pe = p is a reasonable approximation, but it deserves more mathematical discussion for the pressure and length evolution in TMNM.
In both cases, diffusion and dislocation, pe is not constant but varies with the length of the nanowire. In TMNM, each nanowire experiences a force balance between the applied pressing force at the entrance (peSA) and the resistance from the material-mold interface ( I ): , where SA is the cross-section area of the nanowire, m is the mass of the nanowire, I is the shear strength along the nanowire-mold interface, and v is the moving speed of the nanowire. This leads to: In the dislocation case, the growth of the nanowire is driven by (p -pe). When pe increases with L and approaches the full value of hydrostatic pressure p in the feedstock, the growth rate of nanowire reduces to zero, setting a maximum length to TMNM of m = /4 I (eq. S7). The additional calculations we did for the variation of pe and L vs. time for the dislocation case are plotted in Figure S10.
In the case of interface diffusion, the growth of the nanowire is driven by the pressure gradient that is defined by pe/L. This process can be further divided into two steps: (i) when This gives the growth rate of the nanowire: This constant growth rate results in a linear relation between L and t for interface diffusioncontrolled growth.
(ii) when t > tm, pe = p, and the growth rate is defined by: In contrast to the case of dislocation slip controlled growth, nanowires will keep growing in the atomic diffusion-controlled region down this pressure gradient (Fig. S11). The integration of the equation above leads to a square root relation between L and t, which is shown in eq. (S5). Thus, the relation between L and t in (ii) can be fully represented by: , where L0 is an integration constant, and S is the cross-section area of the interface layer. For cylindrical shaped nanowires, S = πδd, and S/SA = 4δ/d. The variation of pe, pe /L and L vs. time in interface diffusion case are plotted in Figure S11.
For diffusion based TMNM, the growth of nanowires is driven by fast atomic transport through interface diffusion. As a result, tm is a very short time when compared to the experimental time scale of molding (~10 sec by estimation compared to typical experiment times ~300 sec). It is also important to mention that tm is also much shorter for diffusion-controlled growth than for dislocation-controlled growth. Our experimentally determined Lm ~ 1µm when using typical molding diameters (Fig. S3 and S6).
Strictly speaking, eq. S1 to S5 are only valid for t > tm. However, since tm is very short compared to the experimental time scale, the potential difference in describing the L(t, T, p, d) through eq. S1 to S5 can be expected to be small.

Supplementary Section 5: Size effect on the dominating diffusion mechanism
The activation energy for interface diffusion is much lower than that for bulk diffusion, leading to a decreasing difference between DI and DB with increasing temperature. In bulk materials DI and DB cross over at a temperature below the liquidus temperature which leads to a transition of deformation mechanism from grain boundary diffusion to lattice diffusion at high temperatures, such as in diffusional creep. For nanomaterials, as the case in TMNM, size effect also needs to be considered due to a drastic increase of volume portion of interface layers/grain boundary layers.
In TMNM, the ratio between the contribution of interface diffusion and bulk diffusion is: (S14) This equation reveals that Tc is a function of the molding diameter (or grain size) d. For bulk materials, it has been studied for diffusional creep that lattice diffusion can dominate over grain boundary diffusion at high temperature of Tc ~ 0.8 Tm, but for nanocrystalline materials, Tc can increase beyond Tm with decreasing d, so that no transition from interface to bulk diffusion takes place. To show the size dependence of this transition, we calculated the function of Tc/Tm vs. d for some typical materials used in this study, Au, Ag, and Cu. The results are plotted in Figure S12.
This plot reveals that for typical length scales (e.g., d < 1 µm) and temperature range of thermomechanical nanomolding (T < Tm), the diffusion-controlled regime is always dominated by interface diffusion. For large molding dimensions beyond nanomolding (e.g., d > 10 µm), similar to bulk materials, bulk diffusion will dominate at high temperatures.

Supplementary Section 6: Moldability mapping of TMNM
In Fig 2C we plot the moldability of representative materials by predicting their molding aspect ratio (L/d) as a function of molding temperature. Based on the analysis of molding mechanisms in TMNM (eq. 2 and eq. S5), the moldability of materials is defined by their interface diffusivity, eq. S15 predicts an expected aspect ratio in TMNM by assuming certain molding conditions (specifically we use d = 40 nm, a loading pressure of 1 GPa, t = 600 s, and assume δ = 1 nm and = 4 3 3 , where r is the atomic radius). However, with reported data of only a few materials, to find the interface diffusivity is generally challenging. Thus, instead of using interface diffusivity for all materials, we pick bulk diffusivity and enlarge them to reasonable magnitude when interface diffusivity is not available. Specifically, our data treatment follows the principles of: (1) When interface diffusivity (or grain boundary diffusivity) data is available from literature, we use the interface diffusivity for calculating the aspect ratio of nanowires. These materials include: Au, Cu, and Ni.
(2) If there is no interface diffusivity data available from literatures, we use diffusivity data that are measured from fine-grained samples (lateral samples and thin films). With the existence of diffusion short-circuits (40) along their high volume ratio grain boundaries, fine-grained samples are of similar magnitude of diffusivity with interface diffusivity. These materials used include Sn and AuIn2.
(3) If both interface diffusivity and diffusivity data from fine-grained sample are not available, we have to use self-diffusivity data from bulk samples instead. Generally, it is reported that for typical materials the interface diffusivity (or grain boundary diffusivity) is 4 to 6 orders of magnitudes higher than their bulk self-diffusivity at intermediate temperature where TMNM usually applies (28)(29)(30)(41)(42)(43). So, in this case we apply a factor of 10 4 to DB 0 by estimating DI ~ 10 4 DB. Despite the possible difference of the activation energy Q (41), such an estimation presents reasonable length prediction in TMNM (20). These materials include In, Al, Fe, Pt, W (diffusivity fitted from reported experimental results), Sb, Ge, Si, InSb, SnTe, SiO2, and Al2O3.
All data treated can be found in Table S1 with original references.

Supplementary Section 7: Details of cases in TMNM as a toolbox
There are 11 cases listed in Fig. 4 showing the controlled versatile fabrication in TMNM. Below we provide more details of each case. We use A and B to represent two components/layers in the feedstock, and assume A is the faster diffuser without further notice. Case (iv): When using AB solid solutions as feedstock and processing in a diffusion dominated region, one can fabricate pure A nanowires that stand on AB solid solution substrates if DA > DB. In this case, the composition of nanowires are controlled by the kinetic process with competition between diffusion of A atoms and B atoms. Always under the same temperature and pressure gradient, the atomic flux of A and B is solely defined by their diffusivities. And the faster diffuser (A) has higher flux and can eventually change the composition of grown nanowires to pure A. One example of this case is TMNM of Au50Cu50 at 450°C and 40 nm, fabricated nanowires are pure Cu standing on Au-Cu substrate.

Case (v):
When using A-B layered feedstock (A at the top), one can fabricate pure A nanowires standing on pure B substrate. There is no specific requirement on mechanisms in this case. The top layer (A) will always grow first and this case can be achieved upon controlling of molding time. One example of this case is TMNM of Ag-Cu layered feedstock (700 nm Ag at the top) at 300°C and 40 nm, fabricated nanowires are pure Ag standing on pure Cu.

Case (vi):
When using B-A layered feedstock (B at the top) and processing at temperature T > Tc, A  It is also worthy to note that in all the cases above, the structure of heterostructures can be precisely controlled. This ability of precise controlling is another important aspect for further applications. Besides the chemical sequence discussed above, for A-B/B-A heterostructure nanowires, the structure is thoroughly defined by the length of each layer. Consider TMNM with A-B layered feedstock (A at the top) for example, the final length of A part in nanowires (LA) can be precisely calculated with the deposition thickness of A layer in feedstock (tA) and the porosity (a) of the mold upon volume conservation: A = A / Considering the length of B part (LB) is LB = L -LA, LB as well as the location of A-B interface can also be precisely controlled by controlling the total length of nanowires. In a diffusion dominated mechanism, L can be controlled by controlling molding time and pressure (eq. S5). And in a dislocation slip mechanism, one can tune the loading pressure (sometimes also diameter) to tune the maximum length Lm (eq. S9) and apply a reasonable molding time to achieve L = Lm. For example, when using Ag-Cu layered feedstock (200 nm Ag at the top) to fabricate Ag-Cu heterostructure nanowires (case (viii)), typical 40 nm or 80 nm molds with porosity a ~ 15% result in ~1.3 µm Ag parts in fabricated nanowires. Under a molding condition of 80 nm, 600 s, and 1.6 GPa, the total length ~ 6 µm, and Ag-Cu interface is ~ 4.5 µm away from substrate. By reducing the loading pressure to 0.8 GPa, or changing the molding diameter to 40 nm, we can reduce the total length by half (~ 3 µm), and the Ag-Cu interface is tuned to ~ 1.5 µm away from the substrate.  pe (left), pe /L (middle) and L (right) vs. t in interface diffusion dominated mechanism calculated from eq. S12 and S13. Here tm marks the time when pe approaches the full value of the hydrostatic pressure (p). The dashed lines in the L vs. t plot are extrapolated from linear and square root equation (eq. S13).

Fig S12.
Normalized mechanism transition temperature, Tc, from interface to bulk diffusion, as a function of molding diameter. For conditions above the Tc(d) curve bulk diffusion is dominated and for conditions below, interface diffusion dominates. Data of diffusivities are collected from references (28-30, 42, 44, 45).