Ultra-Small, High-Frequency, and Substrate-Immune Microtube Inductors Transformed from 2D to 3D

Monolithic on-chip inductors are key passive devices in radio frequency integrated circuits (RFICs). Currently, 70–80% of the on-wafer area of most RFIC chips is occupied by the sprawling planar spiral inductors, and its operation frequency is limited to a few GHz. With continuous scaling of the transistor technology, miniaturization and high frequency operation of inductors have become the bottleneck to meet future demands of wireless communication systems. Here we report on-chip self-rolled-up 3D microtube inductors with extremely small footprint, unprecedented high frequency performance and weak dependence on substrate conductivity. The serpentine metal strips are deposited on an oppositely strained silicon nitrides (SiNx) bilayer. After releasing from the sacrificial layer underneath, the metal/SiNx layer is scrolled into a 3D hollow tubular structure by the strain induced unidirectional self-rolled-up technology. Compared to the planar spiral inductors with similar inductances and quality (Q) factors, the footprint of tube inductors is reduced by as much as two orders of magnitude, and the frequency at peak Q factor improves more than 5 times on doped substrates. The self-rolled-up 3D nanotechnology platform employed here, that “processes in 2D but functions in 3D”, is positioned to serve as a global solution for extreme RFIC miniaturization with improved performance.


Unidirectional Rolled up Mechanism and Technology
Unidirectional rolling up is indispensable for multi-turns tube inductors. When the removal of the exposed sacrificial layer initiates, the opposing built in strain of the bilayer generates a net momentum -the LF SiNx prefers to expand and HF SiNx and Ni/Au to shrink-to lift all the layers up from the substrate. As the Ge is etched away continuously, the layers wrap down to substrate surface and scroll into a spiral cylinder tube architecture. Although the biaxial stresses of amorphous SiNx are isotropic, Ge is only removed away from one side by etchant, and other three sides are covered and protected by SiNx bilayer. Therefore, the scrolling process just happens from the deep trench direction, as shown in fig.1 C and D. On another aspect, during the sacrificial layer is etched away, the net momentum also tears apart the SiNx bilayer at the sidewalls of Ge mesas. The saw-tooth patterns on the edges of mesa and tube are the direct evidence, as shown in Fig.1E. This dynamic tearing process ensures unidirectional multi-turns rolling throughout the wet etching. Once the Ge is removed totally, the rolling and tearing processing terminates, immediately, and the tube inductors are hold by SiNx bilayer and stand on the substrate surface. Figure S2: The optical microscopy and SEM images of self-rolled-up tube inductors with different structural configurations. A1-C1 show 4-strip/3-turn inductor; A2-C2 show 4-strip/9-turn inductor, A3-C3 show 2-strip/15-turn inductor, A4-C4 show 6-strip/15-turn inductor. (A1-A4) the top view of 2D patterns before rolled up by optical microscopy. Longer patterns induce more coiled turns and more metal strips need wider patterns. (B1-B4) the top view of 3D tube inductors by SEM after the Ge layer is released thoroughly. (C1-C4) the cross sectional view of rolled up tube inductors. The outer diameter increases with number of coiled turns, along with the constant inner diameter.

De-embed Methodology
The open-through de-embedding procedure is adopted to calibrate out the parasitic capacitance and resistance from the feedlines (S2, S3). The contact fixture is designed as shown in figure S3 (A-C), and the lumped equivalent circuit model is constructed to represent the physics of parasitic effects, as shown in the insets. As the RF measurement goes up to 40 GHz, feedlines are designed as short as possible to minimize the distribution effect. As shown in inset of figure S3 (A), the admittance π-network is used to model the capacitive effects between the contact pads and the surrounding environment including the substrate and RF ground. Series connected impedance network is used to model the resistance and inductance of the feedlines.
Mathematical procedure to do the open-through de-embedding is shown in Figure S3 (D-F). In the first step, the admittance π-network (open pattern) is abstracted from the original data (DUT). Then, the parasitic resistances and inductances (Z1 and Z2) can be calculated by step 2 ( Figure  In the Figure S4

Equivalent Circuit and Parameter Extraction
After de-embedding, the inductor parameters were extracted based on the π equivalent circuit, as shown in Figure S5. Detailed descriptions of the physical equivalent model can be found in Reference S4. Total inductance and resistance of the tube inductor are represented by serial inductance (L) and serial resistance (R), respectively. The sum of all overlap direct coupling between different turns of metal strip lines represents the inter-turn distribution capacitance (Cc). The substrate parasitic capacitance (Cs) represents the capacitive coupling from coil spiral strip line and connecting line of inductor to ground. Therefore, from the Y parameters admittance matrix, the measured two-port inductor parameters are determined as follows. All the extracted parameters are listed in Table S1. Nc=number of metal strips Rp=R(value at 0.01GHz)/Nc L0=L(value at 0.01GHz)/Nc ω=2πf (5) Figure S5: the π lumped equivalent circuit for the lumped parameters extraction, which is based on the physical structure of tube inductor on doped Si substrate.  As expected, the inductance decreases with the width of metal strips, because the wider metal strip generates the lower current density and weaker average magnetic flux density.

Physical Modeling and Comparison with Experimental Results
Employing the our developed physical model (S4), the measured effective inductances versus frequency from 0.01GHz to 40 GHz are plotted and compared with modeled curves. The device structural parameters, such as inner/outer diameter, metal strips width, number of strips and turns, are input into the model, which are as same as that of the real inductors. Tube inductances with the functions of coiled turns and metal strips are compared in the figure S6 A and B, respectively. Good agreement between the measured and modeled data can be found, which validates the high accuracy and reliability of the physical model on predicting inductance of the tube inductors. Therefore the model can calculate inductance values with more turns. it could be plotted to the number of coiled turns for three different Lc, as shown in the figure S8. Therefore, the cancelling mutual inductance between two adjacent strips could be ignored when the coiled turns under 20 and Lc=20µm, because the ratio is small than 1%. To better illustrate the substrate immunity of tube inductors, FEM simulation by HFSS was performed to compare the magnetic field distribution of a planar spiral inductor and a tube inductor on a silicon substrate, as shown in Fig. S9. The planar spiral inductor (Fig.  S9(A)) has a 1µm by 1µm rectangular cross section, 118µm outer diameter, 18µm inner diameter and 9µm line spacing. The tube inductor (Fig. S9(B)) has an inner radius of with 3.16 µm, a width of 20 µm, and 10 coiled turns of a single conducting strip separated by a 40nm thick SiNx membrane between turns.
In the simulation, the metal spirals were modeled as perfect conductors embedded in a SiNx hollow-cylinder with dimensions specified above. The silicon substrate was meshed and the field was solved in the substrate too. The tube inductor structure with substrate underneath was placed in vacuum with radiation boundary condition in HFSS. Both the E field and H field met the boundary condition at the interface between the substrate and vacuum. All material properties including permeability, permittivity, and conductivity used in the model were set to constant values as shown in Table S2. Since the metal spirals were modeled as perfect conductor, the simulated data only contained information of inductance --1.8 nH for the spiral inductor and 0.19 nH for the 10-turn tube inductor at 5 GHz. Note that a smaller inductance value for the tube inductor was used to save computation time; however, to reach a similar inductance to that of the planar inductor, the tube inductor can be a structure with 10 strips connected in series, which gives the same penetration depth as a single strip structure simulated here. The fields were calculated when both devices fed with 1 Watt RF power. The simulated magnetic field value of the tube inductor and the planar inductor were set to the same minimum value ~1 A/m, in order to compare the penetration level of both structures. By plotting the magnetic field distribution @ 5GHz & 0º, we can compare the field penetration below the substrate and confinement above the substrate for both types of inductors, as shown in Fig. S9. The magnetic field H with 1A/m magnitude penetrate into the substrate about 5 µm for the tube inductor and 50 µm for the spiral inductor.
As can be seen clearly in Fig. S9, significant part of the magnetic field generated by the planar spiral inductor penetrates into the substrate, which will introduce serious substrate effects including capacitive effect and eddy current effect. In contrast, the magnetic field generated by the tube inductor is almost separated from the substrate, which means much less substrate parasitic effects are introduced.   Figure S10: Experimental inductances (solid lines) and Q factors (symbol lines) with the same device structural configuration but on the different substrates are plotted to operation frequency. Black, red and blue curves are for p-Si (ρ=1~5Ω▪cm), p-Si (ρ=10~20Ω▪cm) and c-plane sapphire substrate.
On three different substrates, the tube inductors with the same device structural configurations demonstrate identical inductance values at low frequency range, but different values of Qmax and fQmax.