1. Theory

Definition: A diffused or implanted resistor is a doped region of the silicon, such as an n+ or p+ source/drain diffusion, an ion-implanted layer or a CMOS well, isolated from its surroundings by a reverse-biased pn junction and contacted at both ends (Kraft; Allen).
Sheet resistance: A layer of depth $x_{j}$ has a resistance per square $R_{s} = 1/ (q μ N x_{j})$ , so the value is set by the doping and the depth and the layout sets only the number of squares $L / W$ (Kraft; Camenzind).
Range of values: Heavily doped source/drain diffusions give about 10 to 200 $Ω/ □$ , implanted layers 500 to 2000 $Ω/ □$ and wells about 1000 to 3000 $Ω/ □$ ; silicide must be blocked, since salicided diffusion falls to about 5 to 7 $Ω/ □$ (Allen; SKY130; GF180MCU; Niknejad and Boser).
Temperature coefficient: The resistance rises with temperature as the mobility falls, by about +400 to +2000 $ppm/K$ for doped diffusions and up to 4000 to 8000 $ppm/K$ for lightly doped wells (Allen; Kraft).
Voltage coefficient: The isolating junction depletes part of the resistor body, so the resistance depends on the voltage to the isolation, weakly for heavy doping (about 100 to 200 $ppm/V$ ) and strongly for wells (about 8000 to 20 000 $ppm/V$ ) (Allen; Kraft; TI).
Accuracy: Absolute values vary by about 15 to 50 per cent from run to run, but ratios of identical, nearby resistors match to about 0.1 per cent with careful layout (Allen; Kraft; Niknejad and Boser).
Pinched resistor: Squeezing a layer between two junctions gives a very high sheet resistance in a small area, at the cost of a large spread, low breakdown and strong nonlinearity (Camenzind).

2. Working Principle

Conduction: Majority carriers drift along the doped layer between the two contacts; the current is ohmic at low fields, and the junction to the substrate or well carries only leakage as long as it stays reverse biased (Kraft).
Isolation: The resistor body must stay reverse biased to its surroundings, for example an n+ resistor in a p-substrate tied to the lowest potential, so its terminals must never forward-bias that junction (Kraft; background knowledge).
Head and contact effects: Current crowds where it enters the contacts, so each head adds about 0.2 to 0.4 squares and the contact resistance (Camenzind; Kraft).
Depletion modulation: Raising the reverse bias between the resistor and its isolation widens the depletion region into the resistor and narrows the conducting layer, which raises the resistance; lateral depletion from neighbouring wells dominates in well resistors (TI; Kraft).
Distributed parasitics: The junction capacitance is spread along the body, so at high frequency the resistor behaves as an RC line whose effective resistance falls (Niknejad and Boser).
Thermal and mechanical effects: Self-heating, contact Seebeck voltages of 0.2 to 1.4 $mV/K$ and piezoresistance from package stress shift precision values (Camenzind; Allen).

3. Applications

High-value bias resistors in small area: Well and pinched resistors give tens to hundreds of kilo-ohms where accuracy and linearity are not needed (Allen; Camenzind).
ESD and ballast resistors: Unsalicided diffusions in series with I/O and ballast fingers of output transistors, where the resistor shares the junction physics of the protected devices (background knowledge).
Piezoresistive sensors: Diffused resistors in thin silicon membranes are the sensing elements of piezoresistive pressure sensors and strain gauges, using the stress sensitivity that is a drawback in precision circuits (Allen; background knowledge).
Current status: For precision analog work, polysilicon and thin-film metal resistors have largely replaced diffused ones, because of their lower voltage coefficient and parasitic capacitance (Allen; Niknejad and Boser).

Sheet Resistance and Layout

Name	Formula	Related Formulae	Variables/Constants
Sheet resistance of a diffused or implanted layer	$R_{s} = \frac{ρ}{x _{j}} = \frac{1}{q μ N x _{j}}$	$R_{s} = [q \int_{0}^{x_{j}} μ (N (x)) N (x) d x]^{- 1} (non-uniform profile)$	$R_{s} : s h ee t r es i s t an ce in Ω/ □$ $ρ : r es i s t i v i t y$ $x_{j} : j u n c t i o n d e pt h (t hi c k n ess o f t h e co n d u c t in g l a y er)$ $q : e l e m e n t a r y c ha r g e$ $μ : maj or i t y - c a r r i er m o bi l i t y a t t h e d o p in g N, N (x) : n e t d o p in g p r o f i l e (t h e in t e g r a l f or m i s ba c k g r o u n d k n o w l e d g e); K r a f t : R_{s} = 1/ (q μ_{n} N_{D} x_{j}), d r ain an d so u r ce d i f f u s i o n s 10 t o 100 Ω/ □, n + 20, p + 100 an d n - 8000 Ω/ □ a s e x am pl es; C am e n z in d : bi p o l a r ba se ab o u t 200 an d e mi tt er a s l o w a s 5 Ω/ □, C M O S p + ab o u t 150 Ω/ □; S K Y 130 : n + 120 (108 t o 132), p + 197 (166 t o 228), n - w e l l 950 (550 t o 1350), p - w e l l 3050 an d d ee p n - w e l l 2200 Ω/ □; GF 180 M C U (u n s a l i c i d e d, 5 μ m w i d e a t 100 μ A) : n + 60 (45 t o 75), p + 185 (145 t o 225) an d n - w e l l 1000 (800 t o 1200) Ω/ □, an d s a l i c i d e d n + an d p + o n l y 6.3 an d 7 Ω/ □; N ik n e j a d an d B oser : s a l i c i d e d d i f f u s i o n ab o u t 5 Ω/ □, n - w e l l ab o u t 1000 Ω/ □$ $(co m p u t e d : n + a t 1 \times 1 0^{20} c m^{- 3}, μ = 90 c m^{2} / (V s) an d x_{j} = 0.15 μ m (a ss u m e d) g i v es 46 Ω/ □; 1000 Ω/ □ n ee d s a 0.84 μ m l a y er a t 1 \times 1 0^{17} c m^{- 3} (μ = 740) or 2.8 μ m a t 2 \times 1 0^{16} c m^{- 3} (μ = 1100), a ss u m e d m o bi l i t i es)$
Resistance from the layout	$R = R_{s} (\frac{L}{W _{e f f}} + 2 n_{e n d}) + 2 \frac{R _{c}}{n _{c}}$	$W_{e f f} = W_{d r a w n} + Δ W$	$R : t o t a l r es i s t an ce$ $L : l e n g t h b e tw ee n co n t a c t s$ $W_{e f f} : e l ec t r i c a l w i d t h$ $n_{e n d} : e n d cor r ec t i o n o f e a c h h e a d in s q u a r es$ $R_{c} : r es i s t an ce o f o n e co n t a c t$ $n_{c} : co n t a c t s p er h e a d$ $W_{d r a w n} : d r a w n w i d t h$ $Δ W : w i d t h bia s f r o m l a t er a l d i f f u s i o n an d l i t h o g r a p h y; K r a f t : R = R_{s} L / W, so L = 7 W g i v es 7 R_{s}, an d co n t a c t r es i s t an ce a dd s t o t h e t o t a l; C am e n z in d : ab o u t 0.4 s q u a r es p er e n d f or a minim u m - w i d t h r es i s t or an d ab o u t 0.2 f or w i d e co n t a c t s, an d a s q u a r e m e a s u r es R_{s} w ha t e v er i t s s i z e; S K Y 130 : e l ec t r i c a l n + w i d t h 0.157 μ m an d p + 0.144 μ m f or a d r a w n 0.14 μ m$ $(co m p u t e d : f or a 10 - s q u a r e minim u m - w i d t h r es i s t or, tw o 0.4 - s q u a r e e n d s a dd 8 p er ce n t; t h e S K Y 130 n + w i d t h bia s o f + 0.017 μ m i s 12 p er ce n t o f a 0.14 μ m l in e; 100 kΩ t ak es 105 s q u a r es o f S K Y 130 n - w e l l b u t 833 o f n +)$

Temperature and Voltage Dependence

Name	Formula	Related Formulae	Variables/Constants
Temperature coefficient of resistance	$R (T) = R_{0} [1 + T C_{1} (T - T_{0}) + T C_{2} (T - T_{0})^{2}]$	$T C_{1} = \frac{1}{R} \frac{d R}{d T}$	$R (T) : r es i s t an ce a t t e m p er a t u r e T$ $R_{0} : v a l u e a t t h e r e f er e n ce T_{0}$ $T C_{1} : f i r s t - or d er coe f f i c i e n t$ $T C_{2} : seco n d - or d er coe f f i c i e n t; A l l e n : d i f f u se d (so u r ce / d r ain) + 1500 ppm/K, i o n im pl an t e d + 400 ppm/K, n - w e l l 4000 ppm/K; K r a f t : d i f f u se d ab o u t 2000 ppm/K in hi s t ab l e, n + 1500 an d n - 8000 ppm/K, f a l l in g a t hi g h er d o p in g; N ik n e j a d an d B oser : N + an d P + p o l y r es i s t or s in ser i es c an c an ce l t h e i r T C; t h e r i se w i t h t e m p er a t u r e co m es f r o m t h e f a l l in g maj or i t y - c a r r i er m o bi l i t y a t f u l l i o ni z a t i o n (ba c k g r o u n d k n o w l e d g e)$ $(co m p u t e d : o v er 100 K, + 15 p er ce n t a t 1500 ppm/K, + 40 p er ce n t a t 4000 an d + 80 p er ce n t a t 8000 ppm/K in t h e l in e a r a pp r o x ima t i o n)$
Voltage coefficient and depletion modulation	$R (V) = R_{0} (1 + V C_{1} V + V C_{2} V^{2})$	$R (V_{R}) \approx \frac{R _{s, 0}}{1 - x _{d} ( V _{R} ) / t} \frac{L}{W}, x_{d} = \frac{2 ε _{s} ( V _{bi} + V _{R} )}{q} \frac{N _{A}}{N _{D} ( N _{A} + N _{D} )}$	$R (V) : r es i s t an ce a t a v o l t a g e V a cr oss or f r o m t h e r es i s t or t o i t s i so l a t i o n$ $V C_{1} an d V C_{2} : f i r s t - an d seco n d - or d er v o l t a g e coe f f i c i e n t s (V C an d B C in N ik n e j a d an d B oser)$ $R_{s, 0} : u n d e pl e t e d s h ee t r es i s t an ce$ $x_{d} : d e pl e t i o n d e pt h in t o t h e r es i s t or l a y er o f t hi c k n ess t$ $ε_{s} : p er mi tt i v i t y o f S i$ $V_{bi} : b u i l t - in v o l t a g e$ $V_{R} : r e v er se bia s t o t h e i so l a t i o n$ $N_{A} an d N_{D} : d o p in g o f t h e i so l a t i o n an d o f t h e r es i s t or l a y er (o n e - d im e n s i o na l es t ima t e, ba c k g r o u n d k n o w l e d g e); A l l e n : d i f f u se d ab o u t 200, im pl an t e d ab o u t 800 an d n - w e l l ab o u t 8000 ppm/V (10000 in hi s s u mma r y); K r a f t : 100 t o 500 ppm/V f or d i f f u se d, n + 100, p + 1100 an d n - 20000 ppm/V, an d w e l l r es i s t an ce r i ses w i t h r e v er se bia s; C am e n z in d : ab o u t 1 p er ce n t p er 5 V in a 200 Ω/ □ ba se l a y er, s t r o n g es t in l i g h tl y d o p e d l a y er s; T I : l a t er a l d e pl e t i o n f r o m t h e w e l l j u n c t i o n c a u ses t h e hi g h V C R o f n - w e l l r es i s t or s, an d s p a c in g t h e w e l l s r ai ses t h e b r e ak d o w n f r o m ab o u t 12 t o ab o u t 35 V$ $(co m p u t e d : a 1 μ m n - w e l l a t 1 \times 1 0^{17} c m^{- 3} o v er a 1 \times 1 0^{15} c m^{- 3} s u b s t r a t e (a ss u m e d) i s d e pl e t e d b y 0.010 μ m a t 0 V an d 0.027 μ m a t 5 V, ab o u t 3600 ppm/V f r o m t h e b o tt o m j u n c t i o n a l o n e; 5 V c han g es R b y 0.05 p er ce n t a t 100 ppm/V, 4 p er ce n t a t 8000 an d 10 p er ce n t a t 20000 ppm/V)$
Pinched resistor	$R_{p in c h} = R_{s, p in c h} \frac{L}{W}, R_{s, p in c h} = \frac{1}{q μ N ( t _{c h} - x _{d} )}$	$V_{B R} \approx 6 V (base pinch, emitter-base junction)$	$R_{p in c h} : r es i s t an ce o f a l a y er s q u eez e d b e tw ee n tw o j u n c t i o n s$ $R_{s, p in c h} : i t s s h ee t r es i s t an ce$ $t_{c h} : t hi c k n ess o f t h e u n d e pl e t e d c hann e l w i t h o u t bia s$ $x_{d} : t o t a l d e pl e t i o n in t r u d in g f r o m b o t h j u n c t i o n s$ $V_{B R} : b r e ak d o w n v o l t a g e (f or m i s ba c k g r o u n d k n o w l e d g e); C am e n z in d : t h e ba se p in c h r es i s t or ha s a l o w b r e ak d o w n o f ab o u t 6 V, ab o u t 10 : 1 v a r ia t i o n an d r e a c h es 100 kΩ in ab o u t a t r an s i s t or a r e a, an d t h e e p i p in c h r es i s t or p in c h es o f f ab o v e ab o u t 5 V, t h e n a c t s a s a c u r r e n t so u r ce o f a f e w μ A w i t h ab o u t 8 : 1 v a r ia t i o n$ $(co m p u t e d : n o n e; t h e l a r g e s p r e a d f o l l o w s f r o m R_{s, p in c h} d e p e n d in g o n t h e s ma l l d i f f er e n ce t_{c h} - x_{d} (m y r e a d in g))$

Parasitics, Matching and Thermoelectric Effects

Name	Formula	Related Formulae	Variables/Constants
Junction capacitance and distributed RC	$C_{p a r} = C_{j} A + C_{j s w} P, C_{j} = \frac{C _{j 0}}{( 1 + V _{R} / V _{bi} ) ^{m}}$	$τ_{E} \approx \frac{R C _{p a r}}{2} (Elmore delay of a uniform RC line)$	$C_{p a r} : p a r a s i t i c c a p a c i t an ce o f t h e r es i s t or b o d y t o i t s i so l a t i o n$ $C_{j} : a r e a j u n c t i o n c a p a c i t an ce p er a r e a a t r e v er se bia s V_{R}$ $C_{j 0} : i t s z er o - bia s v a l u e$ $m : g r a d in g coe f f i c i e n t$ $C_{j s w} : s i d e w a l l c a p a c i t an ce p er l e n g t h$ $A an d P : a r e a an d p er im e t er$ $τ_{E} : E l m or e d e l a y (ba c k g r o u n d k n o w l e d g e; j u n c t i o n c a p a c i t an ce o n t h e P N J u n c t i o n p a g e); K r a f t : t h e p a r a s i t i c p n j u n c t i o n t o t h e s u b s t r a t e g i v es l e ak a g e an d a v o l t a g e - d e p e n d e n t d e pl e t i o n c a p a c i t an ce; A l l e n : t h e p a r a s i t i c c a p a c i t an ce o f d i f f u se d r es i s t or s i s v o l t a g e d e p e n d e n t, an d n - w e l l r es i s t or s ha v e l a r g e p a r a s i t i cs; N ik n e j a d an d B oser : r es i s t or s a r e d i s t r ib u t e d R C l in es, a 10 kΩ r es i s t or o f 5 μ m w i d t h f a l l s t o ab o u t 5 kΩ a t 1 GHz (R_{s} = 100 Ω/ □, C = 3.45 \times 1 0^{- 5} F / m^{2}), an d in d u c t an ce i s n e g l i g ib l e u p t o ab o u t 10 GHz f or s ma l l s t r u c t u r es$ $(co m p u t e d : 100 kΩ o f n - w e l l w i t h an a ss u m e d 40 fF t o t h e s u b s t r a t e g i v es τ_{E} = 2 ns; C = 3.45 \times 1 0^{- 5} F / m^{2} i s 0.0345 fF/ μ m^{2})$
Matching of resistor pairs	$σ (\frac{Δ R}{R}) = \frac{A _{R}}{W L}$	$σ (\frac{Δ R}{R}) \propto \frac{1}{W} (fixed number of squares)$	$σ (Δ R / R) : s t an d a r d d e v ia t i o n o f t h e r e l a t i v e d i f f er e n ce b e tw ee n tw o i d e n t i c a l r es i s t or s$ $A_{R} : ma t c hin g coe f f i c i e n t o f t h e l a y er$ $W an d L : w i d t h an d l e n g t h (a r e a l a w, ba c k g r o u n d k n o w l e d g e); A l l e n : r e l a t i v e a cc u r a cy o f 2 p er ce n t a t 5 μ m an d 0.2 p er ce n t a t 50 μ m w i d t h f or d i f f u se d r es i s t or s, 0.15 p er ce n t a t 50 μ m f or im pl an t e d o n es an d ab o u t 5 p er ce n t f or n - w e l l, a g ain s t ab so l u t e a cc u r a c i es o f \pm 35, \pm 15 an d \pm 40 p er ce n t; K r a f t : ab so l u t e t o l er an ce 10 t o 50 p er ce n t b u t ma t c hin g 0.05 t o 0.1 p er ce n t, b e tt er t han 0.1 p er ce n t w i t h g oo d l a y o u t; C am e n z in d : r es i s t or s ma y b e u p t o 25 p er ce n t o f f t a r g e t w hi l e ma t c h e d o n es s hi f t t o g e t h er, an d 0.5 p er ce n t or b e tt er n ee d s ab o u t t e n t im es minim u m w i d t h; N ik n e j a d an d B oser : 0.1 t o 1 p er ce n t w i t h c a r e f u l l a y o u t, d u mm y e n d r es i s t or s an d co mm o n - ce n t r o i d pl a ce m e n t$ $(co m p u t e d : A l l e n^{'} s t e n f o l d im p r o v e m e n t f or a t e n f o l d w i d t h a t t h e s am e n u mb er o f s q u a r es i s t h e 1/ W sc a l in g o f t h e a r e a l a w)$
Thermoelectric offset at the contacts	$V_{t h} = S_{c} Δ T_{c}$	$V_{t h} \to 0 for equal head temperatures$	$V_{t h} : t h er m oe l ec t r i c (S ee b ec k) v o l t a g e o f a r es i s t or w h ose tw o h e a d s d i f f er in t e m p er a t u r e b y Δ T_{c}$ $S_{c} : S ee b ec k coe f f i c i e n t o f t h e se mi co n d u c t or - m e t a l co n t a c t p ai r; C am e n z in d : 0.2 t o 1.4 mV/K d e p e n d in g o n d o p in g an d m e t a l; N ik n e j a d an d B oser : l a y o u t s t ha t c an ce l t h er m oe l ec t r i c o f f se t s a r e p a r t o f p r ec i s i o n p r a c t i ce$ $(co m p u t e d : a 1 K d i f f er e n ce b e tw ee n t h e h e a d s g i v es 0.2 t o 1.4 mV, co m p a r ab l e t o t h e o f f se t o f a p r ec i s i o n am pl i f i er)$

Important Questions

Ques: What sets the sheet resistance of a diffused resistor?

The doping, the mobility and the depth of the layer, through $R_{s} = 1/ (q μ N x_{j})$ for a uniform layer (Kraft), or the integral of $q μ N$ over the depth for a real profile. For an n+ diffusion with $1 0^{20}$ $c m^{- 3}$ , a mobility of 90 $c m^{2} / (V s)$ and a depth of 0.15 $μ m$ this gives about 46 $Ω/ □$ (computed, assumed values), in the range of 10 to 100 $Ω/ □$ that Kraft and Allen give for source/drain diffusions. The SKY130 process specifies 120 $Ω/ □$ for n+ and 197 for p+, and GF180MCU 60 and 185 for unsalicided diffusion, so p+ is typically higher than n+ because hole mobility is lower. A silicide layer shorts the diffusion to about 5 to 7 $Ω/ □$ , so resistors need a silicide block. Reference: M. Kraft, ‘ELEN0037 Microelectronic IC Design, Lecture 2: Technological Aspects’, University of Liege (year not stated) (fetch-tool summary); P. E. Allen, ‘ECE 6440 Frequency Synthesizers, Lecture 040: Integrated Circuit Technology - II’, Georgia Institute of Technology (2003) (fetch-tool summary); SkyWater Technology, SKY130 PDK documentation, device details ‘res_generic’ parameter table (fetch-tool summary); GlobalFoundries GF180MCU PDK documentation, electrical specifications 5.1 ‘Sheet Resistance’ (fetch-tool summary).

Ques: Is the voltage coefficient of an n-well resistor 8000, 10 000 or 20 000 ppm per volt?

The sources give all three. Allen lists about 8000 $ppm/V$ in his resistor table and 10 000 in his summary table, and Kraft gives 20 000 $ppm/V$ for an n- layer. The values differ because the coefficient depends on the well doping and depth, on the substrate doping and on how the voltage is applied, so none is universal. A one-dimensional estimate for a 1 $μ m$ well at $1 0^{17}$ $c m^{- 3}$ over a $1 0^{15}$ $c m^{- 3}$ substrate gives about 3600 $ppm/V$ from the bottom junction alone (computed, assumed doping), and TI notes that lateral depletion from neighbouring wells adds more. About $1 0^{4}$ $ppm/V$ is the more likely order for a typical n-well (my reading), and 5 $V$ then changes the resistance by about 5 per cent. Reference: P. E. Allen, ‘ECE 6440 Frequency Synthesizers, Lecture 040: Integrated Circuit Technology - II’, Georgia Institute of Technology (2003) (fetch-tool summary); M. Kraft, ‘ELEN0037 Microelectronic IC Design, Lecture 2: Technological Aspects’, University of Liege (year not stated) (fetch-tool summary); Texas Instruments, ‘Diffusion resistor with reduced voltage coefficient of resistance and increased breakdown voltage using CMOS wells’, US Patent 10,128,145 B2 (2018) (fetch-tool summary).

Ques: Why does the resistance of a diffused resistor rise with temperature?

Because the carrier density is fixed by the doping while the mobility falls with temperature through lattice scattering, so the resistivity rises (background knowledge). The effect is stronger in lightly doped layers, where lattice scattering dominates, and weaker in heavily doped ones, where ionized-impurity scattering is less temperature sensitive. This is what the sources show: Allen gives +1500 $ppm/K$ for source/drain diffusions, +400 $ppm/K$ for implanted layers and 4000 $ppm/K$ for n-wells, and Kraft gives 1500 $ppm/K$ for n+ and 8000 $ppm/K$ for an n- layer and states that the coefficient falls at higher doping. Over 100 $K$ a well resistor changes by 40 to 80 per cent (computed), so wells are unsuited to references and to filters that must be stable over temperature. Reference: P. E. Allen, ‘ECE 6440 Frequency Synthesizers, Lecture 040: Integrated Circuit Technology - II’, Georgia Institute of Technology (2003) (fetch-tool summary); M. Kraft, ‘ELEN0037 Microelectronic IC Design, Lecture 2: Technological Aspects’, University of Liege (year not stated) (fetch-tool summary).

Ques: Why do resistor ratios match much better than absolute values?

Because most variations are common to neighbouring resistors. The doping, the depth and the width bias change from run to run by tens of per cent, Allen gives $\pm 35$ per cent for diffused resistors and Camenzind up to 25 per cent off target, but two identical resistors side by side shift almost together, so their ratio is preserved. What remains is random variation, which falls with area: Allen’s relative accuracy improves from 2 per cent at 5 $μ m$ width to 0.2 per cent at 50 $μ m$ , the tenfold gain expected from $1/ W L$ at a fixed number of squares (computed). Kraft gives 0.05 to 0.1 per cent with good layout, and Niknejad and Boser 0.1 to 1 per cent, using dummy end resistors and common-centroid layouts against gradients. Reference: P. E. Allen, ‘ECE 6440 Frequency Synthesizers, Lecture 040: Integrated Circuit Technology - II’, Georgia Institute of Technology (2003) (fetch-tool summary); H. Camenzind, ‘Designing Analog Chips’, chapter ‘Resistors in IC Processes’ (All About Circuits online edition) (fetch-tool summary); M. Kraft, ‘ELEN0037 Microelectronic IC Design, Lecture 2: Technological Aspects’, University of Liege (year not stated) (fetch-tool summary); A. M. Niknejad and B. Boser, ‘EECS 240 Lecture 2: CMOS Technology and Passive Devices’, UC Berkeley (2006) (some tables scrambled in the copy read, fetch-tool summary).

Ques: Why must the end corrections and contacts be included in a short resistor?

Because the current spreads into and crowds at the contacts, so the heads do not behave like the uniform body. Camenzind gives an end correction of about 0.4 squares per head for a minimum-width resistor and about 0.2 squares for wide contacts, and Kraft notes that the contact resistance adds to the total. For a resistor of 10 squares at minimum width the two heads add about 8 per cent (computed), more than many matching budgets. The electrical width also differs from the drawn width: SKY130 specifies 0.157 $μ m$ for a drawn 0.14 $μ m$ n+ line, 12 per cent wider (computed). Long, wide resistors therefore approach the ideal $R_{s} L / W$ , while short or narrow ones must be modelled with these corrections, which the PDK models include. Reference: H. Camenzind, ‘Designing Analog Chips’, chapter ‘Resistors in IC Processes’ (All About Circuits online edition) (fetch-tool summary); M. Kraft, ‘ELEN0037 Microelectronic IC Design, Lecture 2: Technological Aspects’, University of Liege (year not stated) (fetch-tool summary); SkyWater Technology, SKY130 PDK documentation, device details ‘res_generic’ parameter table (fetch-tool summary).

Ques: Why does an integrated resistor lose resistance at high frequency?

Because its parasitic junction capacitance is spread along its length, so it is a distributed RC line rather than a lumped resistor. Niknejad and Boser show that a 10 $kΩ$ resistor of 5 $μ m$ width with 100 $Ω/ □$ and a capacitance of $3.45 \times 1 0^{- 5}$ $F/ m^{2}$ falls to about 5 $kΩ$ at 1 $GHz$ , and that the inductance is negligible up to about 10 $GHz$ for small structures. Kraft and Allen add that the capacitance of a diffused resistor depends on the voltage, since it is a depletion capacitance. A high-value well resistor is the worst case: 100 $kΩ$ with an assumed 40 $fF$ to the substrate has an Elmore delay of about 2 $ns$ (computed), so it cannot be used as a precise resistor above tens of megahertz. Reference: A. M. Niknejad and B. Boser, ‘EECS 240 Lecture 2: CMOS Technology and Passive Devices’, UC Berkeley (2006) (some tables scrambled in the copy read, fetch-tool summary); M. Kraft, ‘ELEN0037 Microelectronic IC Design, Lecture 2: Technological Aspects’, University of Liege (year not stated) (fetch-tool summary); P. E. Allen, ‘ECE 6440 Frequency Synthesizers, Lecture 040: Integrated Circuit Technology - II’, Georgia Institute of Technology (2003) (fetch-tool summary).

Ques: What is a pinched resistor, and why is its value so poorly controlled?

A pinched resistor is a layer squeezed between two junctions, such as the base of a bipolar process under an emitter diffusion, or the epitaxial layer under a base diffusion. Its conducting thickness is the small difference between the layer thickness and the depletion of the two junctions, so its sheet resistance is very high but depends strongly on both the process and the voltage (my reading of the form). Camenzind gives the base pinch resistor a breakdown of only about 6 $V$ , set by the emitter-base junction, a variation of about 10:1, and up to 100 $kΩ$ in about a transistor area. The epitaxial pinch resistor pinches off above about 5 $V$ and then behaves as a current source of a few microamperes, with about 8:1 variation. Reference: H. Camenzind, ‘Designing Analog Chips’, chapter ‘Resistors in IC Processes’ (All About Circuits online edition) (fetch-tool summary).

Ques: Why can a diffused resistor create an offset of a millivolt?

Because each metal-to-semiconductor contact is a thermocouple. If the two heads of a resistor are at different temperatures, the Seebeck voltages of the two contacts no longer cancel, and Camenzind gives 0.2 to 1.4 $mV/K$ depending on the doping and the metal. A difference of only 1 $K$ between the heads, easily produced by a nearby power transistor, then gives 0.2 to 1.4 $mV$ (computed), comparable to the input offset of a precision amplifier. Niknejad and Boser list layouts that mitigate thermoelectric offsets among precision practices: the heads of matched resistors are placed on the same isotherm, and resistors are split so that the thermoelectric voltages of opposite contacts cancel. Stress from the package adds piezoresistive shifts, which Allen notes for diffused resistors. Reference: H. Camenzind, ‘Designing Analog Chips’, chapter ‘Resistors in IC Processes’ (All About Circuits online edition) (fetch-tool summary); A. M. Niknejad and B. Boser, ‘EECS 240 Lecture 2: CMOS Technology and Passive Devices’, UC Berkeley (2006) (some tables scrambled in the copy read, fetch-tool summary); P. E. Allen, ‘ECE 6440 Frequency Synthesizers, Lecture 040: Integrated Circuit Technology - II’, Georgia Institute of Technology (2003) (fetch-tool summary).

Quartz 5

Explorer

Diffused and Implanted Resistors