1. Theory

Definition: A metal-insulator-metal (MIM) capacitor is a parallel-plate capacitor built in the back end from a dedicated thin insulator, such as SiN, SiO2 or a high-k oxide, between a lower metal and an extra top plate; a metal-oxide-metal (MOM) capacitor uses the ordinary interconnect layers as interdigitated fingers coupled mainly through their sidewalls (Niknejad and Boser; SKY130; IHP).
Density: Standard MIM options give about 1 to 2 $fF/ μ m^{2}$ , equivalent to about 17 to 35 $nm$ of SiO2, and MOM stacks of 3 to 5 layers about 0.9 to 1.4 $fF/ μ m^{2}$ ; high-k dielectrics reach 8 to 13 $fF/ μ m^{2}$ (GF180MCU; IHP; Xiong et al.).
Linearity: SiN or SiO2 MIM capacitors have linear voltage coefficients of only about 6 to 30 $ppm/V$ and temperature coefficients of 10 to 20 $ppm/K$ , while high-k dielectrics have quadratic coefficients of hundreds of ppm per square volt (GF180MCU; Xiong et al.).
Low parasitics: Both plates are metal, far from the substrate, so the bottom-plate parasitic and the series resistance are small and the quality factor is high (Niknejad and Boser; GF180MCU).
No extra mask for MOM: MOM capacitors need only the standard metal stack, at the cost of a lower density per layer and a dependence on line width and spacing control (Niknejad and Boser; IHP).
Matching: Adjacent 1 $pF$ MIM capacitors match within about 1 per cent, and lateral-flux structures match better than parallel plates (GF180MCU; Aparicio and Hajimiri).

2. Working Principle

Vertical flux (MIM): The field runs straight through the thin insulator between the two plates, so the capacitance follows the parallel-plate law with a small fringe term at the perimeter (IHP; UPM).
Lateral flux (MOM): The fields run sideways between neighbouring fingers of opposite polarity, so the capacitance grows with the metal thickness and the number of layers and inversely with the spacing, and shrinking line spacing raises it (Aparicio and Hajimiri; Niknejad and Boser).
Voltage dependence: In SiN and SiO2 the polarization is nearly linear in field, while in high-k oxides field-dependent polarization and charge redistribution give a quadratic term, which a stack with a dielectric of opposite sign can compensate (Xiong et al.; background knowledge for the mechanism).
Losses: The plates carry the displacement current along their length, so the series resistance is that of a distributed RC line, and the dielectric adds a small loss (Niknejad and Boser).
Leakage and breakdown: Thin, high-permittivity insulators conduct more and break down at lower fields, which limits the voltage rating (GF180MCU; Xiong et al.).

3. Applications

Precision analog: Switched-capacitor filters, sample-and-hold circuits and charge-redistribution data converters, where linearity and matching matter (Niknejad and Boser).
RF circuits: Matching networks, coupling and resonator capacitors, where the high Q and the RF MIM models are used (IHP; GF180MCU).
Dense decoupling in the back end: High-k MIM and stacked MIM capacitors for supply decoupling above the active circuitry (SKY130; Xiong et al.).
MOM in digital processes: Interdigitated capacitors where no MIM option exists, for example in advanced logic processes (Niknejad and Boser; IHP).

MIM Capacitors

Name	Formula	Related Formulae	Variables/Constants
Capacitance density and equivalent oxide thickness of a MIM capacitor	$C = \frac{ε _{0} ε _{r}}{t _{d}} A, E O T = \frac{3.9 ε _{0}}{C / A}$	$ε_{r} = \frac{( C / A ) t _{d}}{ε _{0}}$	$C : c a p a c i t an ce$ $ε_{0} : v a c uu m p er mi tt i v i t y (8.854 \times 1 0^{- 3} fF / μ m)$ $ε_{r} : r e l a t i v e p er mi tt i v i t y o f t h e in s u l a t or$ $t_{d} : in s u l a t or t hi c k n ess$ $A : pl a t e a r e a$ $E O T : e q u i v a l e n t S i O 2 t hi c k n ess; GF 180 M C U : 1.0 (0.9 t o 1.1), 1.5 (1.27 t o 1.73) an d 2.0 (1.8 t o 2.2) fF/ μ m^{2} o pt i o n s, o n l y o n e p er p r ocess; I H P : 1.5 fF/ μ m^{2}; S K Y 130 : C A P M o v er M e t a l - 3 an d C A P 2 M o v er M e t a l - 4, i d e n t i c a l co n s t r u c t i o n s t ha t c an b e s t a c k e d, m o d e l s v a l i d t o 5 V an d in c l u d in g co n t a c t r es i s t an ce an d b o tt o m - pl a t e p a r a s i t i cs; N ik n e j a d an d B oser : a v er y t hin in s u l a t or o f ab o u t 25 nm g i v es hi g h d e n s i t y an d r e l a t i v e l y l o w ba c k - pl a t e p a r a s i t i cs; X i o n g e t a l . (A L D d i e l ec t r i cs, 80 x 80 μ m) : A l 2 O 3 o f 12.1, 24.7 an d 49.7 nm g a v e 13.19, 7.89 an d 4.62 fF/ μ m^{2}, H f O 2 o f 21.3 nm an a v er a g e o f 12.29 fF/ μ m^{2}, an d c i t e d w or k s 13 fF/ μ m^{2} (H f O 2), 6.05 fF/ μ m^{2} (13 nm A l 2 O 3) an d 11.9 fF/ μ m^{2} (T i O 2/ S i O 2)$ $(co m p u t e d : E O T = 34.5, 23.0 an d 17.3 nm f or t h e 1.0, 1.5 an d 2.0 fF/ μ m^{2} o pt i o n s; X i o n g e t a l .^{'} s A l 2 O 3 v a l u es im pl y ε_{r} = 18, 22 an d 26, ab o u t tw i ce t h e u s u a l v a l u e o f ab o u t 9 f or A l 2 O 3 (ba c k g r o u n d k n o w l e d g e))$
Area and perimeter model	$C = m [C_{a} W L + 2 C_{p} (W + L)]$	$\frac{C _{p er im}}{C _{a r e a}} = \frac{2 C _{p} ( W + L )}{C _{a} W L} = \frac{4 C _{p}}{C _{a} a} (square of side a)$	$C : c a p a c i t an ce$ $m : m u l t i pl i er$ $C_{a} : a r e a c a p a c i t an ce$ $W an d L : pl a t e w i d t h an d l e n g t h$ $C_{p} : p er im e t er (f r in g e) c a p a c i t an ce p er l e n g t h$ $a : s i d e o f a s q u a r e pl a t e; I H P c mim an d r f c mim : C = m (w l \cdot 1.5 \times 1 0^{- 3} + 2 (w + l) \cdot 40 \times 1 0^{- 12}) w i t h w an d l in m e t r es, t ha t i s C_{a} = 1.5 fF/ μ m^{2} an d C_{p} = 0.04 fF/ μ m, d e f a u l t s i z es o f 7 x 7 an d 10 x 10 μ m; U P M : f r in g e f i e l d s a r e ha r d t o m o d e l, so t h e p er im e t er o f t h e pl a t e i s minimi z e d$ $(co m p u t e d : t h e p er im e t er a dd s 1.6 fF t o 150 fF (1.1 p er ce n t) f or 10 x 10 μ m an d 0.32 fF t o 6 fF (5.3 p er ce n t) f or 2 x 2 μ m)$
Voltage and temperature coefficients	$C (V) = C_{0} (1 + β V + α V^{2})$	$C (T) = C_{0} [1 + T C_{1} (T - T_{0})]$	$C (V) : c a p a c i t an ce a t bia s V$ $C_{0} : z er o - bia s v a l u e$ $β : l in e a r v o l t a g e coe f f i c i e n t (pp m / V)$ $α : q u a d r a t i c coe f f i c i e n t (pp m / V^{2}), C (T) : c a p a c i t an ce a t t e m p er a t u r e T$ $T C_{1} : t e m p er a t u r e coe f f i c i e n t; GF 180 M C U (m e a s u r e d f r o m - 6 t o + 6 V) : V c 1 = 7.6 ppm/V (5.7 t o 9.5) an d T C 1 = 13.3 ppm/K (9.9 t o 16.6) f or 1.5 fF/ μ m^{2}, 6 ppm/V an d 10 ppm/K f or 1.0 fF/ μ m^{2}, an d - 28.8 ppm/V an d 18.8 ppm/K f or 2.0 fF/ μ m^{2}; X i o n g e t a l . : α = 303.6 ppm/ V^{2} f or 25 nm A l 2 O 3 in t h e ab s t r a c t an d 329.1 ppm/ V^{2} a t 10 k H z f a l l in g t o 270.5 ppm/ V^{2} a t 1 M H z in t h e t e x t, β c an b e c an ce l l e d b y d i f f er e n t ia l d es i g n, a f r e q u e n cy d i s p er s i o n o f 0.45 p er ce n t f r o m 1 \times 1 0^{3} t o 1 \times 1 0^{6} Hz, an d a T i O 2/ S i O 2 s t a c k w i t h 90 ppm/ V^{2} in c i t e d w or k; N ik n e j a d an d B oser : m e t a l - m e t a l c a p a c i t or s ab o u t 20 ppm/V an d 30 ppm/K$ $(co m p u t e d : 7.6 ppm/V o v er 6 V i s 0.0046 p er ce n t; 303.6 ppm/ V^{2} a t 5 V i s 0.76 p er ce n t, so a hi g h - k d i e l ec t r i c t r a d es tw o or d er s o f ma g ni t u d e o f l in e a r i t y f or i t s d e n s i t y)$
Leakage and breakdown	$I_{l e ak} = J_{l e ak} A, t_{d r oo p} = \frac{C Δ V}{I _{l e ak}}$	$E_{B D} = \frac{V _{B D}}{t _{d}}$	$I_{l e ak} : l e ak a g e c u r r e n t$ $J_{l e ak} : l e ak a g e p er a r e a$ $A : a r e a$ $t_{d r oo p} : t im e f or a s t or e d v o l t a g e t o f a l l b y Δ V$ $E_{B D} : b r e ak d o w n f i e l d$ $V_{B D} : b r e ak d o w n v o l t a g e$ $t_{d} : d i e l ec t r i c t hi c k n ess; GF 180 M C U : a t m os t 1 pA/ μ m^{2} a t 6 V (1.0 fF/ μ m^{2} t y p e a t 16 V), B V_{o x} o f 30 V t y p i c a l an d a t l e a s t 10 V (1.5 fF/ μ m^{2}), 40 an d a t l e a s t 20 V (1.0 fF/ μ m^{2}) an d 24 an d a t l e a s t 10 V (2.0 fF/ μ m^{2}), an d ma x im u m o p er a t in g v o l t a g es o f 10, 20 an d 6.6 V; X i o n g e t a l . : l e ak a g e o f 5.3 \times 1 0^{- 9} A/c m^{2} a t 4.3 V f or 12.5 nm A l 2 O 3, 1.4 \times 1 0^{- 10} an d 2.4 \times 1 0^{- 11} A/c m^{2} f or 25 an d 50 nm, an d b r e ak d o w n o f ab o u t 12 V (4.8 MV/cm) a t 25 nm an d u p t o 21.6 V a t 50 nm; t im e - d e p e n d e n t b r e ak d o w n i s o n t h e E l ec t r i c a l S t r ess p a g e$ $(co m p u t e d : 1 pF o f t h e 1.5 fF/ μ m^{2} M I M co v er s 667 μ m^{2} an d ma y l e ak u p t o 0.67 nA a t 6 V, w hi c h d i sc ha r g es i t b y 1 p er ce n t o f 6 V in 90 μ s; 12 V o v er 24.7 nm i s 4.9 MV/cm)$
Plate resistance and quality factor	$Q = \frac{1}{ω R _{s} C}, R_{s} \approx \frac{R _{□, t o p} + R _{□, b o t}}{3} (square plate fed from one side)$	$f_{S R F} = \frac{1}{2 π L _{s} C}$	$Q : q u a l i t y f a c t or$ $ω : an g u l a r f r e q u e n cy$ $R_{s} : ser i es r es i s t an ce$ $C : c a p a c i t an ce$ $R_{□, t o p} an d R_{□, b o t} : s h ee t r es i s t an ces o f t h e pl a t es (t h e f a c t or 1/3 f or a d i s t r ib u t e d pl a t e f e d a l o n g o n e e d g e i s ba c k g r o u n d k n o w l e d g e)$ $f_{S R F} : se l f - r eso nan ce f r e q u e n cy w i t h ser i es in d u c t an ce L_{s} (ba c k g r o u n d k n o w l e d g e); GF 180 M C U : b o tt o m pl a t e 80 mΩ/ □ (68 t o 92) an d t o p pl a t e 500 mΩ/ □ (400 t o 600) f or t h e 1.0 an d 1.5 fF/ μ m^{2} t y p es; N ik n e j a d an d B oser : d i s pl a ce m e n t c u r r e n t mak es t h e pl a t e c u r r e n t f a l l a l o n g t h e pl a t e, so a d i s t r ib u t e d R C ana l y s i s i s n ee d e d, an d co n t a c t in g b o t h e n d s l o w er s t h e r es i s t an ce f o u r t im es; A p a r i c i o an d H aj imi r i : l a t er a l - f l ux s t r u c t u r es k ee p co m p a r ab l e Q w i t h hi g h er se l f - r eso nan ce f r e q u e n cy t han p a r a l l e l pl a t es$ $(co m p u t e d : R_{s} = 0.19 Ω f or a s q u a r e GF 180 M C U M I M, so a 1 pF c a p a c i t or ha s Q = 165 a t 5 GHz)$

MOM Capacitors

Name	Formula	Related Formulae	Variables/Constants
Lateral-flux MOM capacitor density	$\frac{C}{A} = n_{l a y er s} \frac{ε _{0} ε _{r} t _{m}}{s p} + (\frac{C}{A})_{v er t i c a l}$	$C_{t o t} = C_{a c t i v e} + C_{f ee d}$	$C / A : c a p a c i t an ce p er a r e a o f in t er d i g i t a t e d f in g er s$ $n_{l a y er s} : n u mb er o f m e t a l l a y er s u se d$ $ε_{0} ε_{r} : p er mi tt i v i t y o f t h e in t er - m e t a l d i e l ec t r i c$ $t_{m} : m e t a l t hi c k n ess$ $s : f in g er s p a c in g$ $p : f in g er p i t c h$ $C_{t o t} : t o t a l c a p a c i t an ce$ $C_{a c t i v e} : f in g er - a r r a y p a r t$ $C_{f ee d} : p a r t o f t h e f ee d l in es (p a r a l l e l - pl a t e es t ima t e o f f in g er s i d e w a l l s, ba c k g r o u n d k n o w l e d g e; t h e v er t i c a l t er m i s t h e co u pl in g b e tw ee n l a y er s); I H P (M u e h l ha u s, d e v e l o p m e n t d a t a) : ab o u t 0.88, 1.17 an d 1.36 fF/ μ m^{2} f or 3, 4 an d 5 l a y er s w i t h an 840 x 890 nm u ni t ce l l, s am e - s i d e f ee d coe f f i c i e n t s o f ab o u t 0.97, 1.28 an d 1.46 fF/ μ m o f f ee d w i d t h an d n e a r l y z er o f or o pp os i t e - s i d e f ee d s; N ik n e j a d an d B oser : M O M c a p a c i t or s u se l a t er a l f l ux a t ab o u t minim u m s p a c in g an d man y m e t a l l a y er s, w i t h f e w er l a y er s f or l o w er ba c k - pl a t e p a r a s i t i cs, an d r e a so nab l y g oo d ma t c hin g; A p a r i c i o an d H aj imi r i : l a t er a l - f l ux an d q u a s i - f r a c t a l s t r u c t u r es g i v e l a r g er c a p a c i t an ce, b e tt er ma t c hin g, t i g h t er t o l er an ce an d hi g h er se l f - r eso nan ce t han h or i z o n t a l p a r a l l e l pl a t es; A l l e n : h or i z o n t a l l a t er a l - f l ux m e t a l c a p a c i t or s c an r e a c h ab o u t t e n t im es t h e d e n s i t y o f v er t i c a l - f l ux o n es$ $(co m p u t e d : e a c h a dd e d l a y er a dd s ab o u t 0.2 t o 0.3 fF/ μ m^{2}, so f i v e l a y er s r e a c h 91 p er ce n t o f t h e d e n s i t y o f t h e 1.5 fF/ μ m^{2} M I M w i t h o u t an e x t r a ma s k; 1 pF t ak es 735 μ m^{2} a g ain s t 667 μ m^{2} f or t ha t M I M)$
Series resistance of a MOM capacitor	$R \approx a + b \frac{L}{W}$	$Q = \frac{1}{ω R C}$	$R : tw o - s i d e d ser i es r es i s t an ce$ $a an d b : f i t co n s t an t s$ $L an d W : l e n g t h an d w i d t h o f t h e f in g er a r r a y$ $Q : q u a l i t y f a c t or; I H P (M u e h l ha u s) : a = 0.10, 0.14 an d 0.15 Ω an d b = 0.31, 0.27 an d 0.26 Ω f or 3, 4 an d 5 l a y er s, s in g l e - s i d e r es i s t an ce f i t s s t i l l u n d er in v es t i g a t i o n, an d t h e d a t a a r e c ha r a c t er i z a t i o n g u i d an ce, n o t d e l i v er e d sy mb o l s$ $(co m p u t e d : f or 5 l a y er s, R = 0.41 Ω a t L / W = 1 an d 2.75 Ω a t L / W = 10, so 1 pF ha s Q = 78 an d 12 a t 5 GHz; w i d e, s h or t a r r a y s g i v e t h e hi g h er Q)$

Matching and Parasitics

Name	Formula	Related Formulae	Variables/Constants
Matching of metal capacitors	$σ (\frac{Δ C}{C}) = \frac{A _{C}}{C}$	$σ (\frac{Δ C}{C}) \propto \frac{1}{A}$	$σ (Δ C / C) : s t an d a r d d e v ia t i o n o f t h e r e l a t i v e mi s ma t c h o f tw o i d e n t i c a l c a p a c i t or s$ $A_{C} : ma t c hin g coe f f i c i e n t o f t h e s t r u c t u r e$ $C : c a p a c i t an ce$ $A : a r e a (a r e a l a w, ba c k g r o u n d k n o w l e d g e); GF 180 M C U : a t m os t 1 p er ce n t mi s ma t c h f or a d j a ce n t 1 pF M I M c a p a c i t or s; N ik n e j a d an d B oser : M O M ma t c hin g i s r e a so nab l y g oo d; A p a r i c i o an d H aj imi r i : t h e i r l a t er a l - f i e l d s t r u c t u r es ma t c h b e tt er an d ha v e t i g h t er t o l er an ces t han p a r a l l e l pl a t es; t h e l a y o u t r u l es o f u ni t c a p a c i t or s an d co mm o n ce n t r o i d a r e o n t h e M O S an d P o l y - I n s u l a t or C a p a c i t or s p a g e$ $(co m p u t e d : i f t h e 1 p er ce n t l imi t i s a 3 - s i g ma v a l u e a t 1 pF (a ss u m e d), A_{C} = 0.33 p er ce n t t im es p F^{1/2}, an d 4 pF u ni t s w o u l d ma t c h t o 0.17 p er ce n t a t 1 s i g ma)$
Bottom-plate and substrate parasitics	$C_{b p} = \frac{ε _{0} ε _{r} A}{t _{b p}}$	$α_{b p} = \frac{C _{b p}}{C} = \frac{t _{d}}{t _{b p}} (same dielectric)$	$C_{b p} : c a p a c i t an ce o f t h e b o tt o m pl a t e t o t h e s u b s t r a t e or l o w er m e t a l$ $t_{b p} : d i s t an ce f r o m t h e b o tt o m pl a t e t o t h e s u b s t r a t e$ $α_{b p} : r a t i o t o t h e main c a p a c i t an ce$ $t_{d} : M I M in s u l a t or t hi c k n ess (es t ima t e a ss u m es t h e s am e p er mi tt i v i t y, ba c k g r o u n d k n o w l e d g e); N ik n e j a d an d B oser : M I M c a p a c i t or s ha v e r e l a t i v e l y l o w ba c k - pl a t e p a r a s i t i cs, w hi l e m e t a l c a p a c i t or s in g e n er a l ha v e l a r g e b o tt o m - pl a t e p a r a s i t i cs t ha t l o a d am pl i f i er s; S K Y 130 : m o d e l s in c l u d e t h e b o tt o m - pl a t e t o s u b s t r a t e p a r a s i t i c; I H P : r f c mim ha s a se p a r a t e b o d y n o d e; U P M : b o tt o m - pl a t e, t o p - pl a t e - t o - s u b s t r a t e an d f r in g e p a r a s i t i cs, w i t h f r in g e co m p o n e n t s ha r d t o m o d e l$ $(co m p u t e d : a 25 nm M I M in s u l a t or 5 μ m ab o v e t h e s u b s t r a t e g i v es α_{b p} = 0.5 p er ce n t (a ss u m e d h e i g h t), f a r b e l o w t h e 12 p er ce n t es t ima t e d f or p o l y - p o l y o n t h e M O S an d P o l y - I n s u l a t or C a p a c i t or s p a g e)$

Important Questions

Ques: Is the quadratic voltage coefficient of the 25 nm Al2O3 MIM capacitor 270, 304 or 329 ppm per square volt?

Xiong et al. give all three. The abstract quotes 303.6 $ppm/ V^{2}$ , while the text gives 329.1 $ppm/ V^{2}$ at 10 kHz and 270.5 $ppm/ V^{2}$ at 1 MHz. The values are consistent if the coefficient falls with frequency, by 1.22 times over two decades (computed), as slow polarization or charge redistribution stops following the signal, and the abstract value is probably a value at an intermediate frequency (my reading). The value that applies is the one at the operating frequency of the circuit, so 329 $ppm/ V^{2}$ is the more relevant one for slow signals and 270 $ppm/ V^{2}$ for RF. Even the lowest value changes the capacitance by about 0.7 per cent at 5 $V$ (computed), against 0.005 per cent for a SiN MIM with 7.6 $ppm/V$ (GF180MCU). Reference: L. Xiong, J. Hu, Z. Yang, X. Li, H. Zhang and G. Zhang, ‘Dielectric Properties Investigation of Metal-Insulator-Metal (MIM) Capacitors’, Molecules 27(12), 3951 (2022) (fetch-tool summary); GlobalFoundries GF180MCU PDK documentation, electrical specifications 6.2 ‘MIM Capacitor’ (fetch-tool summary).

Ques: Are Xiong et al.’s Al2O3 capacitance densities consistent with the permittivity of Al2O3?

Not obviously. Their 12.1, 24.7 and 49.7 $nm$ films gave 13.19, 7.89 and 4.62 $fF/ μ m^{2}$ , and $ε_{r} = (C / A) t_{d} / ε_{0}$ gives relative permittivities of about 18, 22 and 26 (computed), while ALD Al2O3 usually has about 9 (background knowledge). The paper itself also shows internal inconsistencies for its TiO2 and HfO2 samples, with ranges and averages that do not match. Either the electrode area or the thickness is different from the stated value, or the films are not pure Al2O3 (my reading). The trend with thickness, an inverse dependence, is still sound, and the leakage and breakdown values, $1.4 \times 1 0^{- 10}$ $A/c m^{2}$ and about 4.8 $MV/cm$ at 25 $nm$ , are typical. The density values should be treated as uncertain until confirmed. Reference: L. Xiong, J. Hu, Z. Yang, X. Li, H. Zhang and G. Zhang, ‘Dielectric Properties Investigation of Metal-Insulator-Metal (MIM) Capacitors’, Molecules 27(12), 3951 (2022) (fetch-tool summary).

Ques: Why does a MIM capacitor need an extra mask, and what does a MOM capacitor give up?

A MIM capacitor needs a dedicated thin insulator and an extra top-plate layer, which GF180MCU offers in 1.0, 1.5 or 2.0 $fF/ μ m^{2}$ versions, only one per process, and SKY130 as CAPM and CAP2M layers that can be stacked. A MOM capacitor uses only the standard interconnect: in IHP’s development data a 3-layer finger array gives about 0.88 $fF/ μ m^{2}$ and 5 layers about 1.36 $fF/ μ m^{2}$ , about 91 per cent of the 1.5 $fF/ μ m^{2}$ MIM (computed). The MOM gives up some density per layer, depends on line width and spacing control, and has larger fringe uncertainty, but it saves the mask and becomes denser as the line spacing shrinks. Niknejad and Boser recommend using fewer layers when the bottom-plate parasitic matters more than density. Reference: GlobalFoundries GF180MCU PDK documentation, electrical specifications 6.2 ‘MIM Capacitor’ (fetch-tool summary); SkyWater Technology, SKY130 PDK documentation, device details ‘MiM Capacitor’ (fetch-tool summary); IHP SG13G2 open PDK documentation, device libraries 3.3 ‘Capacitors’, including V. Muehlhaus, ‘MOM model development notes v4’ (9 November 2022) (fetch-tool summary); A. M. Niknejad and B. Boser, ‘EECS 240 Lecture 2: CMOS Technology and Passive Devices’, UC Berkeley (2006) (capacitor table partly garbled, fetch-tool summary).

Ques: How large is the perimeter contribution to a small MIM capacitor?

IHP models its MIM capacitors as $C = m [C_{a} W L + 2 C_{p} (W + L)]$ with $C_{a} = 1.5$ $fF/ μ m^{2}$ and $C_{p} = 0.04$ $fF/ μ m$ . For a 10 by 10 $μ m$ plate the perimeter adds 1.6 $fF$ to 150 $fF$ , 1.1 per cent, and for a 2 by 2 $μ m$ plate 0.32 $fF$ to 6 $fF$ , 5.3 per cent (computed), so small capacitors deviate from the area law. The UPM notes add that fringe fields are hard to model and recommend minimizing the perimeter. For ratios, unit capacitors of the same size and shape share the same perimeter fraction, so the perimeter term cancels, while ratios built from plates of different shapes carry it as an error, the same reason square unit capacitors are used for oxide capacitors. Reference: IHP SG13G2 open PDK documentation, device libraries 3.3 ‘Capacitors’, including V. Muehlhaus, ‘MOM model development notes v4’ (9 November 2022) (fetch-tool summary); University of the Philippines Microlab, ‘Integrated Capacitors’ class wiki (equations missing in the copy read, fetch-tool summary); MOS and Poly-Insulator Capacitors page of this vault.

Ques: What quality factor can on-chip MIM and MOM capacitors reach?

High values, because both plates are metal. With GF180MCU sheet resistances of 80 $mΩ/ □$ for the bottom and 500 $mΩ/ □$ for the top plate, a square MIM fed from one side has about 0.19 $Ω$ of series resistance, and a 1 $pF$ capacitor has $Q \approx 165$ at 5 $GHz$ (computed, assuming the one-third distributed factor). IHP’s MOM data give a two-sided resistance of $a + b L / W$ , 0.41 $Ω$ for a square 5-layer array and 2.75 $Ω$ for an array ten times longer than wide, so 1 $pF$ has Q of about 78 and 12 at 5 $GHz$ (computed). Wide, short structures fed from both sides therefore give the best Q, and Niknejad and Boser note that contacting both ends lowers the resistance four times. Reference: GlobalFoundries GF180MCU PDK documentation, electrical specifications 6.2 ‘MIM Capacitor’ (fetch-tool summary); IHP SG13G2 open PDK documentation, device libraries 3.3 ‘Capacitors’, including V. Muehlhaus, ‘MOM model development notes v4’ (9 November 2022) (fetch-tool summary); A. M. Niknejad and B. Boser, ‘EECS 240 Lecture 2: CMOS Technology and Passive Devices’, UC Berkeley (2006) (capacitor table partly garbled, fetch-tool summary).

Ques: How much can a MIM capacitor leak, and how high can it be biased?

GF180MCU specifies at most 1 $pA/ μ m^{2}$ at the rated voltage, breakdown voltages of typically 24 to 40 $V$ with minimums of 10 to 20 $V$ , and maximum operating voltages of 6.6, 10 and 20 $V$ for its 2.0, 1.5 and 1.0 $fF/ μ m^{2}$ options, so the thinner, denser dielectric has the lower rating. A 1 $pF$ capacitor of the 1.5 $fF/ μ m^{2}$ type covers 667 square micrometres and may leak up to 0.67 $nA$ at 6 $V$ , which would discharge a stored 6 $V$ by 1 per cent in 90 $μ s$ (computed), an upper bound well above typical leakage. High-k films leak more: Xiong et al. measured $5.3 \times 1 0^{- 9}$ $A/c m^{2}$ at 4.3 $V$ for 12.5 $nm$ Al2O3 and $1.4 \times 1 0^{- 10}$ $A/c m^{2}$ for 25 $nm$ , with breakdown at about 4.8 $MV/cm$ . Reference: GlobalFoundries GF180MCU PDK documentation, electrical specifications 6.2 ‘MIM Capacitor’ (fetch-tool summary); L. Xiong, J. Hu, Z. Yang, X. Li, H. Zhang and G. Zhang, ‘Dielectric Properties Investigation of Metal-Insulator-Metal (MIM) Capacitors’, Molecules 27(12), 3951 (2022) (fetch-tool summary); Electrical Stress page of this vault.

Quartz 5

Explorer

MIM and MOM Capacitors