There are mainly three symbols for resistors:

These symbols are equivalent. A resistor is a symmetrical two pins component.

Standard resistors are rectangular with a body and connections through contacts at each end:

In order to fit within acceptable dimensions, very long resistors may be drawn with more complex shapes such as L, U, S or W:

These resistor types will be discussed later on.

Resistors in integrated circuits exist in two major types. Types are named after the material they are made from.:

- Bulk resistors made from doped monocrystaline silicon inside the die itself.
- Poly-silicon resistors made from doped polycrystalline silicon over the die in oxide.

In all the figures below, only the final and simplified resistor structure is shown, no details are given on process.

Bulk resistors can be of many different technological types, using various layers. However, from a topological standpoint there are mainly two types:

- P type resistors in a N well
- N type resistors in the P substrate

Occasionally, N type resistors in a P Well may exist in some process.

Following figure shows a P type bulk resistor in a N well. Obviously, an isolation diode exists between resistor body and N well. Voltage on resistor body should always be lower than well voltage for the resistor to behave properly. This is why the N well is often connected to the most positive available voltage, the positive supply.

Here are equivalent schematics for N type (left) and P type bulk resistors.

N Type is located directly in P substrate. One diode is figured at each end. For a better fitting of the distributed diode at very high frequencies, body should be divided in several elements and diodes added at each tap.

P Type is located in an N well located in P sub. Same comment on diodes.

Diodes parameters can be extracted from resistor (and well) geometry using area and perimeter data.

More than one resistor can be fitted in a single well.

Poly resistors have all the same topology:

Only variant is whether resistor is located over N well, P well or P sub.

The model for a resistor is essentially the Ohm's law. U=R ⋅ I

- This model has just one parameter: R, the resistor value.

Temperature coefficients can be added to take into account the temperature dependence of the resistor value. R(T)=R( Tref ) ⋅ ( 1+TC1 ⋅ ( T-Tref )+TC2 ⋅ ( T-Tref ) 2 )

- This expression shows three temperature dependence parameters: Tref, TC1 and TC2
- Tref is the reference temperature. At reference temperature, resistor has nominal value R(Tref).
- TC1 is the first order temperature coefficient
- TC2 is the second order temperature coefficient

In an integrated circuit, resistors are made from a constant thickness thin layer of a given resistivity. The general expression of a constant section resistance versus physical and geometrical parameters is: R= ρ ⋅ L S

- This expression shows three parameters: ρ , L and S
- ρ is the resistive layer resistivity
- L is the resistor length
- S is the resistor section

- For an integrated resistor, section S depends on resistor width W and resistive layer thickness T : S=W ⋅ T

Then, for an integrated resistor: R= ρ T ⋅ L W

This expression is the product of two terms:

- ρ/T is called sheet resistance. It is expressed in ohms per square as it is the resistance of a square resistor for which L=W
- L/W is called the number of squares. It has no unit. It is equivalent to the number of square resistors connected in series. When ratio is smaller than 1 is can be considered as the inverse of the number of square resistors connected in parallel

If several resistors with same width and various length are implemented in a circuit and measured, expected values from above formula should fit a linear law. However, measured data show a linear law with a slope and an offset:

The offset value can be considered as the resistance of a zero length body resistor.

What can this mean?

It means that some extra resistance exists that does not lie in the resistor body. It is usual to call this extra resistance “Head resistance” as it is located at each end of the resistor. In the example above, the body resistance is given by the slope and the length, the head resistance at each end is half the offset. Here, the body resistance is 100 Ω

per unit length and the head resistance is 50 Ω

at each end for a total 100 Ω

The head resistance results from two effects:

- A geometrical effect:

This figure shows a resistor with two connections on the upper side. Constant voltage surfaces are plotted every 5% of the total voltage.

It was simulated using EZMod3D my home made 3D Field Solver.

Current flows perpendicularly to these surfaces. There is clearly an effect around the connections as they do not use the full body width and they are located on the top forcing the current lines to bend. This effect generates some extra resistance.

- A technological effect
- In order for the connection to create ohmic contacts and no parasitic junction, a silicide is created at each resistor end. The silicide traps some of the doping from the body creating a higher resistivity thin layer at the interface

These two effects combine to create the head resistance. Practically, this head resistance is not completely correlated with the body resistance as it results from different process steps. Head and body resistances usually exhibit different temperature coefficients.

As a consequence, if two resistors have to be created with an accurate ratio, playing with the body length is usually not a robust solution: In this case, the ratio is affected by process tolerances and it depends on temperature.

The only solution for the resistor ratio to be accurate is that the body and head contributions are the same for the two resistors.

The only practical approach is connecting a number of identical elementary resistor in series-parallel combinations for implementing each of the two resistors.

This method suffers from an area penalty but achieves a process and temperature independent accurate resistor ratio.

These resistors are made from P silicon inside an N well or from N material inside a P well. Isolation is granted by biasing the well so that the isolation diode is reverse biased. The depleted region thickness depends on the doping and on the reverse voltage. Capacitance between resistor body and well depends on resistor area, depleted region thickness and silicon dielectric constant.

These resistors are made from P or N doped poly-silicon inside silicon dioxide. Isolation is grated by the huge resistivity of oxide. Capacitance depends on resistor area, oxide thickness and oxide dielectric constant.

Parasitic capacitance is distributed along the resistor body. The impedance versus frequency of a capacitance distributed along a resistance with the other pin grounded is:

Resistors are supposed to be linear as Ohm's law states. However, various phenomenon affect this linear behavior and create some non linearity.

- Temperature coefficient. When voltage changes, dissipated power changes and then temperature changes. So resistor value changes.
- For bulk resistors, depleted region thickness modulation. Voltage across isolation junction changes depleted region thickness. At least part of the depleted region extends inside resistor body. When voltage changes, actual resistor section changes so resistor value changes.
- For poly resistors too, voltage coefficients (first order and second order) exist because of field effect and electrostatic forces.

The absolute value of a resistor depends on its length, width, thickness and resistivity. Each of this four parameters suffers from manufacturing tolerances so that finally integrated resistors are not accurate. Tolerance on absolute can be as bad as ± 20% or even ± 30%

In addition, significant temperature coefficients make absolute value sensitive to temperature.

Two identical resistors close to each other in the same silicon die suffer together the tolerance on absolute value from the process. However, even if their absolute value is inaccurate, their relative values are far more accurate. ± 1%is very standard, ± 0.1% is achievable.

Because width and length are perpendicular to each other, they suffer uncorrelated tolerances. For that reason, no matching can be granted for two resistors that are not parallel.

If some power is dissipated in a resistor, its body temperature rises above the surrounding temperature. Here again, a difference exists between bulk and poly-silicon resistors. Oxide thermal conductivity is about 100 times lower than silicon thermal conductivity. Actual thermal resistance can only be simulated with a 3D thermal simulator. However, as a rule of thumb, we can state that poly-silicon resistors can dissipate 100 times less power than bulk resistors for the same size and the same temperature increase.

Resistors are often rectangular objects with one dimension larger than the other one. Sometimes however, they can be close to a square with two identical dimensions. We will analyze both cases.

If the resistor is more or less square shaped, constant temperature surfaces inside silicon are spheres halves:

The thermal resistance between two sphere halves of radius R1 and R2 ( R2>R1 )

Where γ is the silicon thermal conductivity

Example:A 10 μ m ⋅ 10 μ m resistor thermal resistance is:

If the resistor is long, constant temperature surfaces inside silicon are cylinders halves with one sphere quarter at each end:

The thermal resistance between two cylinder halves of length L and radius R1 and R2 ( R2>R1 )

is :

Rth1=1 γ ⋅ π ⋅ L ⋅ ln R2R1This thermal resistance is in parallel with that of the two sphere quarter:

The thermal resistance between two sphere halves of radius R1 and R2 ( R2>R1 )

is :

Rth2=1 γ ⋅ 2 ⋅ π ⋅ (1R1-1R2)This thermal resistance is in parallel with that of the two sphere quarter:

The overall thermal resistance is equal to the two in parallel:

Rth=11Rth1+1Rth2Rth=1 γ ⋅ π ⋅ L ln R2R1+ γ ⋅ 2 ⋅ π (1R1-1R2)For poly resistors, heat has to flow through oxide to reach the silicon bulk. At least for plastic packages, part of the heat flows through oxide to