As temperature increases, the resistivity of metal increases, giving it a positive temperature coefficient of resistance.
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Semiconductors have a negative temperature coefficient of resistance.
The resistivity of extrinsic semiconductors is greater than intrinsic semiconductors.
The temperature dependence of the resistivity of semiconductors plays a significant role in their application in electronics
Electrical conductivity describes the ease at which electric current can pass through a material, and is an important parameter of a material. Conductors are the materials that allow current to pass through them. Those which block the current flow are called insulators. There are materials that fall between conductors and insulators when the current that flows through them is taken as a relative parameter. Such materials are known as semiconductors.
In all these materials, the current flow can be directly related to the conductivity of the material, which is the reciprocal of resistivity. Resistivity is a material property, and it is temperature-dependent. The temperature dependence of the resistivity of semiconductors plays a significant role in their application in electronics. In this article, we will explore why this is.
Resistivity is an intrinsic property of a material. It is a constant for material under a given temperature. The resistivity of a material can be defined as the resistance of the material of unit cross-sectional area and unit length. The resistivity of a material is independent of its length and area.
The relationship between the resistance of a material and resistivity is:
Note that R is the resistance, is the resistivity of the material, l is the length, and A is the cross-sectional area of the material. The unit of resistivity is ohm meter.
The conductivity is the reciprocal of resistivity. When the resistivity of a material is high, then its conductivity is very low, and vice versa. Considering this relationship, it can be said that the resistivity of metals or conductors is very low. Arranging the materials in the ascending order of resistivity, the order is conductors, semiconductors, and insulators.
Next, we will explore how temperature affects resistivity.
The resistivity of a material is temperature-dependent. The temperature dependence of the resistivity is different for conductors, semiconductors, and insulators. Let&#;s discuss how resistivity varies in conductors and insulators before discussing semiconductors.
In conductors, as temperature increases, the atoms start vibrating heavily, leading to the collision of free electrons and other electrons. This collision causes a loss of energy from free electrons, which are responsible for the current flow. The reduction in the movement, or drift velocity, of the electrons due to energy drain increases the resistivity of the conductors, especially metals. As the temperature increases, the resistivity of the metal increases as well, giving it a positive temperature coefficient of resistance. At high temperatures, the conductor resistivity increases and conductivity decreases.
Insulators shift to the conduction zone with an increase in temperature. The resistivity of an insulator decreases with temperature, resulting in an increase in conductivity. Insulators exhibit a negative temperature coefficient of resistance.
Next, we will explore the temperature dependence of the resistivity of semiconductors.
In semiconductors, the energy gap between the conduction band and valence band decreases with an increase in temperature. The valence electrons in the semiconductor material gain energy to break the covalent bond and jump to the conduction band at high temperatures. This creates more charge carriers in the semiconductor at high temperatures. The higher concentration of charge carriers decreases the resistivity of the semiconductor. As the resistivity of the semiconductor decreases with an increase in temperature, it becomes more conductive. A semiconductor exhibits excellent conductivity at high temperatures.
The graph below shows the relationship between resistivity and temperature in a semiconductor. Semiconductors have a negative temperature coefficient of resistance. This property is utilized for the application of semiconductors in electronics. When an external voltage is applied, the temperature of the semiconductor crystal increases, which, in turn, increases the density of the thermally generated carriers in it. More electron-hole pairs are generated, allowing an easy flow of current through the semiconductor.
Resistivity vs. temperature in semiconductors
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A semiconductor&#;s performance is enhanced by doping it with donor or acceptor impurities. Such semiconductors are called extrinsic semiconductors. The resistivity of extrinsic semiconductors is greater than intrinsic (undoped or pure) semiconductors.
The temperature dependence of the resistivity of semiconductors is greatly beneficial; the semiconductor electronics we use today are possible only due to the negative temperature coefficient of resistance. Cadence offers a suite of design and analysis tools to help build solid-state electronic circuits.
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The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in Figure 1 is easy to analyze, and, by so doing, we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder&#;s electric resistance [latex]\boldsymbol{R}[/latex] is directly proportional to its length [latex]\boldsymbol{L}[/latex], similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again similar to the flow of fluid through a pipe). In fact, [latex]\boldsymbol{R}[/latex] is inversely proportional to the cylinder&#;s cross-sectional area [latex]\boldsymbol{A}[/latex].
For a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the resistivity [latex]\boldsymbol{\rho}[/latex] of a substance so that the resistance [latex]\boldsymbol{R}[/latex] of an object is directly proportional to [latex]\boldsymbol{\rho}[/latex]. Resistivity [latex]\boldsymbol{\rho}[/latex] is an intrinsic property of a material, independent of its shape or size. The resistance [latex]\boldsymbol{R}[/latex] of a uniform cylinder of length [latex]\boldsymbol{L}[/latex], of cross-sectional area [latex]\boldsymbol{A}[/latex], and made of a material with resistivity [latex]\boldsymbol{\rho}[/latex], is
[latex]\boldsymbol{R =}[/latex] [latex]\boldsymbol{\frac{\rho L}{A}}[/latex].
Table 1 gives representative values of [latex]\boldsymbol{\rho}[/latex]. The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. Conductors have varying but large free charge densities, whereas most charges in insulators are bound to atoms and are not free to move. Semiconductors are intermediate, having far fewer free charges than conductors, but having properties that make the number of free charges depend strongly on the type and amount of impurities in the semiconductor. These unique properties of semiconductors are put to use in modern electronics, as will be explored in later chapters.
Material Resistivity [latex]\boldsymbol{\rho (\Omega \cdot \;\textbf{m})}[/latex] Conductors Silver [latex]\boldsymbol{1.59 \times 10^{-8}}[/latex] Copper [latex]\boldsymbol{1.72 \times 10^{-8}}[/latex] Gold [latex]\boldsymbol{2.44 \times 10^{-8}}[/latex] Aluminum [latex]\boldsymbol{2.65 \times 10^{-8}}[/latex] Tungsten [latex]\boldsymbol{5.6 \times 10^{-8}}[/latex] Iron [latex]\boldsymbol{9.71 \times 10^{-8}}[/latex] Platinum [latex]\boldsymbol{10.6 \times 10^{-8}}[/latex] Steel [latex]\boldsymbol{20 \times 10^{-8}}[/latex] Lead [latex]\boldsymbol{22 \times 10^{-8}}[/latex] Manganin (Cu, Mn, Ni alloy) [latex]\boldsymbol{44 \times 10^{-8}}[/latex] Constantan (Cu, Ni alloy) [latex]\boldsymbol{49 \times 10^{-8}}[/latex] Mercury [latex]\boldsymbol{96 \times 10^{-8}}[/latex] Nichrome (Ni, Fe, Cr alloy) [latex]\boldsymbol{100 \times 10^{-8}}[/latex] Semiconductors Carbon (pure) [latex]\boldsymbol{3.5 \times 10^5}[/latex] Carbon [latex]\boldsymbol{(3.5 - 60) \times 10^5}[/latex] Germanium (pure) [latex]\boldsymbol{600 \times 10^{-3}}[/latex] Germanium [latex]\boldsymbol{(1-600) \times 10^{-3}}[/latex] Silicon (pure) [latex]\boldsymbol{}[/latex] Silicon [latex]\boldsymbol{0.1-}[/latex] Insulators Amber [latex]\boldsymbol{5 \times 10^{14}}[/latex] Glass [latex]\boldsymbol{10^9 - 10^{14}}[/latex] Lucite [latex]\boldsymbol{>10^{13}}[/latex] Mica [latex]\boldsymbol{10^{11} - 10^{15}}[/latex] Quartz (fused) [latex]\boldsymbol{75 \times 10^{16}}[/latex] Rubber (hard) [latex]\boldsymbol{10^{13} - 10^{16}}[/latex] Sulfur [latex]\boldsymbol{10^{15}}[/latex] Teflon [latex]\boldsymbol{>10^{13}}[/latex] Wood [latex]\boldsymbol{10^8 - 10^{11}}[/latex] Table 1. Resistivities [latex]\boldsymbol{\rho}[/latex] of Various materials at 20ºCA car headlight filament is made of tungsten and has a cold resistance of [latex]\boldsymbol{0.350 \;\Omega}[/latex]. If the filament is a cylinder 4.00 cm long (it may be coiled to save space), what is its diameter?
Strategy
We can rearrange the equation [latex]\boldsymbol{R = \frac{\rho L}{A}}[/latex] to find the cross-sectional area [latex]\boldsymbol{A}[/latex] of the filament from the given information. Then its diameter can be found by assuming it has a circular cross-section.
Solution
The cross-sectional area, found by rearranging the expression for the resistance of a cylinder given in [latex]\boldsymbol{R = \frac{\rho L}{A}}[/latex], is
[latex]\boldsymbol{A =}[/latex] [latex]\boldsymbol{\frac{\rho L}{R}}[/latex]
Substituting the given values, and taking [latex]\boldsymbol{\rho}[/latex] from Table 1, yields
[latex]\begin{array}{r @{{}={}} l} \boldsymbol{A} & \boldsymbol{\frac{(5.6 \times 10^{-8} \;\Omega \cdot \textbf{m})(4.00 \times 10^{-2} \;\textbf{m})}{0.350 \;\Omega}} \\[1em] & \boldsymbol{6.40 \times 10^{-9} \;\textbf{m}^2} \end{array} .[/latex]
The area of a circle is related to its diameter [latex]\boldsymbol{D}[/latex] by
[latex]\boldsymbol{A =}[/latex] [latex]\boldsymbol{\frac{\pi D^2}{4}}[/latex]
Solving for the diameter [latex]\boldsymbol{D}[/latex], and substituting the value found for [latex]\boldsymbol{A}[/latex], gives
[latex]\begin{array}{r @{{}={}} l} \boldsymbol{D} & \boldsymbol{2 (\frac{A}{p})^{\frac{1}{2}} = 2(\frac{6.40 \times 10^{-9} \;\textbf{m}^2}{3.14})^{\frac{1}{2}}} \\[1em] & \boldsymbol{9.0 \times 10^{-5} \;\textbf{m}} \end{array}.[/latex]
Discussion
The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because [latex]\boldsymbol{\rho}[/latex] is known to only two digits.
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