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Chapter 2

Temperature Effects in Semiconductors

The changes in temperature described in the previous chapter affect the speed, power, and reliability of our systems. Throughout this book, we will examine all three of these metrics, though the majority of our discussion will be on how temperature affects the speed performance. In this chapter, we discuss the problem of temperature variation at the device and circuit level. In Sect. 2.1, we provide a background on the material dependences on temperature. In Sect. 2.2, the normal and reverse temperature dependence regimes are described. In Sect. 2.3, we explore how these dependences change with technology scaling and the introduction of new processing materials, such as high-k dielectrics and metal gates.

2.1

Material Temperature Dependences

In this section we provide details about the impact of temperature on the MOSFET energy band gap, carrier density, mobility, carrier diffusion, velocity saturation, current density, threshold voltage, leakage current, interconnect resistance, and electromigration. 2.1.1

Energy Band Gap

Temperature affects the properties of electronic systems in a number of fundamental ways. The most fundamental of properties is the energy band gap, Eg, which is affected by temperature according to the Varshni equation [1]
Eg ðTÞ ¼ Eg ð0Þ À

aE T 2
T þ bE

(2.1)

where Eg(0) is the band gap energy at absolute zero on the Kelvin scale in the given material, and aE and bE are material-speciﬁc constants. Table 2.1 [2] provides these
D. Wolpert and P. Ampadu, Managing Temperature Effects in Nanoscale
# Springer Science+Business Media, LLC 2012

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2 Temperature Effects in Semiconductors

Table 2.1 Varshni equation constants for GaAs, Si, and Ge [2]

Material
GaAs
Si
Ge

Eg(0) (eV)
1.519
1.170
0.7437

aE (eV/K)
5.41*10À4
4.73*10À4
4.77*10À4

bE (K)
204
636
235

Fig. 2.1 Energy band gap temperature dependence of
GaAs, Si, and Ge

constants for three material structures. Table 2.1 and (2.1) are used to generate
Fig. 2.1, which shows how the band gaps of the three materials decrease as temperature increases (the labeled points are the band gap of each material at room temperature).

2.1.2

Carrier Density

Carrier densities affect electrical and thermal conductivity, and are a function of the effective density of states in the appropriate band (conduction for n-type, valence for p-type), the Fermi energy level in the material (which is a function of temperature and dopant concentrations), and the temperature as given by the following equations: n ¼ N C eÀ

EC ÀEF kT (2.2)

2.1 Material Temperature Dependences

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Fig. 2.2 Temperature dependence of n in a doped semiconductor

p ¼ N V eÀ

EF ÀEV kT (2.3)

where n is the electron density, p is the hole density, NC is the density of states in the conduction band, NV is the density of states in the valence band, EC is the conduction band energy level, EV is the valence band energy level, EF is the Fermi energy level, k ¼ 1.38·10À23 J/K is the Boltzmann constant, and T is temperature. The temperature dependence of carrier density is shown in Fig. 2.2 for a doped material. In the ionization region, there is only enough latent energy in the material to push a few of the dopant carriers into the conduction band. In the extrinsic region, which is the desired region of operation, the carrier concentration is ﬂat over a wide range of temperatures; in this region, all of the dopant carriers have been energized into the conduction band (i.e. n % ND) and there is very little thermal generation of additional carriers. As the temperature increases, the extrinsic region turns into the intrinsic region, and the number of thermally generated carriers exceeds the number of donor carriers. The intrinsic carrier concentration in a material ni is