## Resistor Temperature Coefficient

In general, the temperature coefficient (temp co) of a resistor is positive, meaning that as the temperature increases, the resistor’s value also increases. This happens because the resistivity of the silicon material used to make the resistor increases with rising temperature.

If we examine the electron concentration in, for example, an n-well as the temperature rises, we’ll notice an exponential increase in the number of thermally generated carriers. More carriers might suggest lower resistivity, but there’s more to it.

With the increase in carrier concentration, there’s a reduction in the mobility of these carriers. Mobility describes how easily carriers can move through a material when an electric field is applied.

High mobility means carriers can move quickly through the material. Mobility is typically measured in units of cm^2/V s.

The resistivity of a material depends on the number of free carriers (electrons/cm^3 for n and holes/cm^3 for p).

An important equation in this context relates mobility (n,p), average velocity of carriers (cm/s), and applied electric field (V/cm).

If mobility decreases faster than the carrier concentration increases with rising temperature, we get a positive temperature coefficient. The resistor’s value increases with temperature because mobility is dropping faster than carrier concentration is increasing.

Conversely, if the carrier concentration increases more rapidly with temperature than mobility decreases, we get a negative temperature coefficient. This means the resistor’s value decreases with increasing temperature.

At room temperature, a positive temperature coefficient is typically observed.

**Voltage Coefficient**

The primary contributor to the voltage coefficient is the depletion layer width between the n-well and the p-substrate.

The depletion layer extends into the n-well, effectively altering the sheet resistance of the material.

As the potential (reverse bias) between the n-well and the substrate increases, the thickness of the n-well available for conducting current decreases.

In summary, the temperature coefficient of a resistor depends on the balance between carrier concentration and mobility changes with temperature. If mobility drops faster than carrier concentration rises, you get a positive temp co (resistor value increases with temperature). Conversely, if carrier concentration increases more quickly than mobility decreases, you get a negative temp co (resistor value decreases with temperature). Additionally, the voltage coefficient is influenced by factors like depletion layer width between semiconductor layers.

### Why does the resistivity of the silicon material used to fabricate a resistor generally increase with increasing temperature?

The resistivity of silicon increases with rising temperature because of an exponential increase in the number of thermally generated carriers. However, this increase in carrier concentration is accompanied by a reduction in carrier mobility due to carrier-to-carrier interactions, leading to higher scattering. This results in a positive temperature coefficient, causing the resistor’s value to increase with temperature.

### What does the mobility of carriers in a material represent?

Carrier mobility is a material parameter that characterizes how easily carriers (electrons or holes) can move through the material under the influence of an applied electric field. It affects resistivity because the resistivity of the material depends on both carrier concentration and mobility. Higher mobility indicates that carriers can move more quickly through the material.

### How does the relationship between carrier mobility and carrier concentration influence the temperature coefficient of a resistor?

The temperature coefficient of a resistor can be either positive or negative. If carrier mobilities decrease faster than carrier concentrations increase with temperature, a positive temperature coefficient occurs, causing the resistor’s value to increase with temperature. Conversely, if carrier concentration increases faster than mobility decreases, a negative temperature coefficient results, causing the resistor’s value to decrease with temperature.

### What is the significance of the voltage coefficient (VCR1) in resistor characteristics, and what contributes to it in the case of an n-well resistor?

The voltage coefficient (VCR1) is an important parameter in resistor characteristics. It is mainly influenced by the depletion layer width between the n-well and the p-substrate. As the potential (reverse bias) between the n-well and the substrate increases, the depletion layer extends into the n-well, effectively changing the sheet resistance. This change in resistance with voltage contributes to the voltage coefficient.

### At room temperature, what type of temperature coefficient is typically observed in resistors, and why?

Answer: At room temperature, resistors typically exhibit a positive temperature coefficient. This occurs because, in most cases, the increase in carrier concentration with temperature is not enough to compensate for the decrease in carrier mobility. As a result, the resistor’s value tends to increase with increasing temperature.

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