DIODE VS TEMPERATURE

WHAT YOU ARE LOOKING AT

A diode biased at a constant current has a voltage across it that falls almost perfectly linearly with temperature. That is the basis of every temperature sensor and every bandgap reference you will build, so it is worth knowing where the slope comes from and how linear “almost perfectly” really is.

The chain is short. The intrinsic carrier concentration n_i(T) comes from the density of states and the bandgap, and it is a ferocious function of temperature — look at the log axis. The saturation current goes as n_i². And then

VD = (kT/q)·ln(IC/IS)

puts the two temperature dependences against each other: kT/q rising, ln(1/I_S) falling much faster. The result is the falling, nearly straight line in the third panel. The fourth panel is what is left after subtracting the best straight line through it — a couple of millivolts of curvature, and that curvature is exactly the error a first-order bandgap reference cannot correct.

The slope is not a constant of nature; it depends on where the diode is biased, through dVD/dT = (VD − Eg/q − 3kT/q)/T. The familiar −2 mV/K belongs to a diode sitting near 0.7 V. With the script's numbers — 1 µm² of junction and 10¹⁹ doping on both sides — I_S comes out extremely small, V_D lands near 0.91 V, and the slope is only −0.95 mV/K. The last readout evaluates that formula so you can watch it track the fitted slope as you move the bias. And notice where the fitted line extrapolates to at 0 K: about E_g/q + 3kT/q, which is where a bandgap reference gets both its name and its output voltage.

THINGS WORTH TRYING

• Compare the three n_i models. The rule of thumb — doubles every 11 K — is a straight line on a log axis by construction; the density-of-states calculation is not, because of the T^3/2 prefactor. All three now agree to within about 30 % at room temperature, and the density-of-states one gives 9.0×10⁹ cm^-3 at 27 °C against a measured 9.65×10⁹.
• That agreement is recent. Until this was written the script's calculation returned 3.6×10⁹, a factor of √6 low, because it used (m_lm_t²)^1/3 for the electron density-of-states mass and left out the 6^2/3 that accounts for silicon's six equivalent conduction band minima. The density of states counts every valley. That one missing factor put m_n^* at 0.33 m₀ instead of 1.08 m₀, which is a factor of six on I_S and about 46 mV on V_D. A good reminder that a model agreeing with itself is not the same as a model agreeing with silicon.
• Move I_C over its whole range. V_D moves by 60 mV per decade, as it must, and the slope dV_D/dT gets shallower as the bias goes up: a hotter-biased diode is a less sensitive thermometer. Two diodes at different currents give you the ΔV_BE that a bandgap adds back in.
• Watch the max non-linearity readout as you change I_C or the area. A few millivolts over the industrial range is typical, and the residual is curved rather than random, which is why curvature correction is a standard trick and why it works.
• Change the bandgap and watch the 0 K intercept follow it. The script holds E_g constant with temperature; a real bandgap narrows by about 50 mV between −40 and 125 °C, which adds to the residual curvature you are looking at rather than cancelling it.

THE PYTHON

def calc_ni(T):
    mn = (0.98*0.19*0.19)**(1/3)*m0
    mp = 0.81*m0                       # heavy hole
    Nc = 2*np.sqrt(np.power((2*pi*k*T*mn)/(h*h),3))
    Nv = 2*np.sqrt(np.power((2*pi*k*T*mp)/(h*h),3))
    ni = np.sqrt(Nc*Nv)*np.exp(-Eg/(2*k*T))
    return ni*cm3

I_s = q*A*n_i_adv**2*(1/NA*np.sqrt(Dn/tau_n) + 1/ND*np.sqrt(Dp/tau_n))
Vd  = k*T/q*np.log(I_c/I_s)

line = np.polynomial.polynomial.polyfit(T,Vd,1)
vd_lin_err = Vd - (T*line[1] + line[0])

Source: ex/vd.py, based on Streetman. The diffusion constants are fixed at their 300 K values (D_n = 36, D_p = 12 cm²/s), as in the script; the antenna leakage example adds their temperature dependence.