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Band Gap Energy in Silicon AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 7, NO. 1 Band Gap Energy in Silicon Jeremy J. Low, Michael L. Kreider, Drew P. Pulsifer Andrew S. Jones and Tariq H. Gilani Department of Physics Millersville University P. O. Box 1002 Millersville, Pennsylvania 17551 USA Received: December 26, 2007 Accepted: April 16, 2008 ABSTRACT The band gap energy Eg in silicon was found by exploiting the linear relationship between the temperature and voltage for the constant current in the temperature range of 275 K to 333 K. Within the precision of our experiment, the results obtained are in good agreement with the known value energy gap in silicon. The temperature dependence of Eg for silicon has also been studied. I. INTRODUCTION e − Eg where a = and b = . Hence, The study of the band gap structure of kC kC a semiconductor is important since it is ⎛ b ⎞ directly related to its electronic properties. Eg −= ⎜ ⎟e . (3) As a consequence, it has attracted a ⎝ a ⎠ considerable interest in undergraduate Therefore, if we find constants a and b by laboratories [1-3]. Several different methods experimentally measuring the temperature have been discussed to determine the band and voltage across the diode, we can find gap energy of semiconductors [1-4]. A diode the band gap energy of a semiconductor, is also used as a temperature sensor [5], in which the linear relationship between the provided that E g does not depend strongly temperature, T and the forward voltage on temperature. For silicon, the variation of is exploited. drop across the p-n junction, V E g with temperature is weak [6] in the It has been shown that with certain temperature range of our interest, 273 K to simplifications, the −VT relationship for a 335 K. diode can be written as [4] II. EXPERIMENTAL METHODS ⎛ e ⎞ ⎛ Eg ⎞ T = ⎜ ⎟V − ⎜ ⎟ (1) ⎝ kC ⎠ ⎝ kC ⎠ Figure 1(a) shows the schematic diagram of the circuit used for −VT data where e is the charge of electron, k is measurements for the base-collector Boltzmann constant, E is band gap junction of an npn transistor (MPS2222AG) g while Figure 1(b) depicts the use of a silicon energy and C is a function of I , the current diode (1N914). The current, I was kept through the diode. Therefore, if I is kept constant using the circuit [7] shown in the constant, then dotted box in Figure 1(a). The variable resistor was adjusted to obtain 10 μA current. The whole circuit was powered by a += baVT (2) constant voltage power supply (Elenco Precision, Model XP-660), keeping the 27 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 7, NO. 1 Figure 1. (a) The MPS2222AG Transistor and (b) the 1N914 Diode are at variable temperature. The MPF102 transistor was used as a constant current source. The circuit for constant current source is shown in dotted box above. The gate-source bias voltage was adjusted by a 100-kΩ variable resistor to obtain a constant current of 10 μA. voltage at approximately 9 V. The voltage across the pn junction in the temperature drop across the pn-junction was measured range of our interest, except at the higher by a digital voltmeter (Heath, Model end of temperature (see the Discussion). SM2320), while the temperature was We also used a 1N914 silicon diode in place measured by a carefully calibrated AD-590 of the transistor (MPS2222AG) and temperature sensor. Both diode and repeated the experiment. The −VT data temperature sensors, were wrapped for the silicon diode (1N914) is shown in together in a thin plastic foil to insulate them Figure 2(b). The values of constants a and electrically. The whole system was then b of Eqn. (2) were obtained by the least- placed in a de-ionized ice-water mixture (the squares fit of the data from 275 K to 330 K ice was also made from de-ionized water) to obtain the temperature of 273 K. temperature. The value of E g was then The voltage readings were then taken calculated from Eqn. (3). for various temperatures while the system The uncertainties in results were was heated slowly. The measurements were found in the following way. Since the repeated several times to reduce the calculation of E depends on two statistical errors. g variables, a and b , the uncertainty in E g is III. RESULTS ⎛ ∂Eg ⎞ ⎛ ∂Eg ⎞ written as ⎜ ⎟ ⎜ ⎟ . dEg = ⎜ ⎟db + ⎜ ⎟da ⎝ ∂b ⎠ ⎝ ∂a ⎠ The −VT curve for the silicon pn The maximum fractional error is therefore junction of transistor (MPS2222AG) in the temperature range of 273 K to 335 K is dEg db da ε +== , where da and db shown in Figure 2(a). The data exhibits a E b a fairly good linear relationship between g absolute temperature and the voltage drop are standard deviations of the slope, a and 28 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 7, NO. 1 340 330 T (K) = - 449.2 V + 541.25 K 320 310 300 Temperature (K) Temperature 290 Silicon transistor MPS2222AG 280 270 0.45 0.47 0.49 0.51 0.53 0.55 0.57 0.59 Voltage (V) 330 320 T (K) = - 386.46 V + 451.78 K 310 300 Silicon Diode 1N914 Temperature (K) 290 280 270 0.320 0.340 0.360 0.380 0.400 0.420 0.440 0.460 Voltage (V) Figure 2. Temperature vs voltage graphs for the silicon: (a, at top) pn junction of npn transistor (MPS2222AG). The T-V curve becomes nonlinear above 330 K. (b, at bottom) diode (1N914). Solid line is the least-squares fit to the data. y-intercept, b of straight line fit to −VT 547.40 K, and db = ± 4.12 K, Therefore in data, respectively. the temperature range of 273 K to 330 K, For the pn junction of the the value of E g is 1.20 eV with uncertainty MPS2222AG transistor [see Figure 2 (a)], a of ± 2.47%. Similarly for the 1N914 diode = - 462.17 K/V, da = ± 7.95 K/V, b = 29 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 7, NO. 1 E (eV) E (eV) E (eV) g g a (K/V) b (K) g 288 K to 383 K 300 K 273 K to 330 K [Ref. 4] [Ref. 6] -462.17 547.40 1.20 MPS2222AG ± 7.95 ± 4.12 ± 0.03 1.23 1.12 -393.54 454.18 1.17 1N915 ± 6.38 ± 2.39 ± 0.03 Table 1. Linear regression coefficients a and b obtained from measured TV− data, and Eg calculated from them. Last two columns are Eg values at given temperature from Ref. 4 and Ref. 6, for comparison. [see Figure 2(b)], a = - 393.538 K/V, da = also observed in Germanium above 313 K ± 6.3828 K/V, b = 454.18 K, and db = ± [4]. When deriving equation (2) using the Ideal Diode Equation [4, 6]: 2.388 K. Hence in the temperature range of 278 K to 323 K, the value of is 1.17 eV E g (eV ) kT (4) with uncertainty of about ± 2.15 %. The I= I0 [ e − 1] results are summarized in Table 1 and compared with known values of energy gap where I 0 is reverse saturation current, we in silicon from different references. assumed e (/)eV kT >> 1 to obtain For silicon the value of E g is known within ± 5% uncertainty and varies from 1.13 eV ( kT ) ± 0.057 eV at 273 K to 1.11 ± 0.056 eV at I= I0 e (5) 330 K [6,8]. Therefore our results are in good agreement to the known values in For a forward bias voltage of 0.4 V, taken at literature [4,6,8], although the values room temperature, we get obtained in here tend to be slightly higher (see Discussion). eV = 4.0 = 16 , We have further analyzed our data to kT 0.025 investigate the temperature dependence of E g . The straight line was obtained by the thus ev least-squares fit of the TV− data in small ekT ≈8.9 E + 6 . temperature intervals of ± 2.5 K, assuming that E g of silicon remains constant in small Therefore, the approximation is generally temperature intervals. The results are shown justified at room temperature and below. But in Figure 3. as temperature increases the exponential term decreases. This may cause a deviation from the linear behavior of TV− data. IV. DISCUSSION The reverse current I 0 depends strongly on temperature. It is proportional [6] a. Temperature – Voltage Curve to the Boltzmann factor and exp(−Eg / kT ) As shown in Figure 2 (a) TV− to T 3+γ / 2 , where γ is a constant. curve deviates from a straight line towards Therefore, higher temperature side (>330 K). Such non-linear behavior in TV− curve was 30 AMERICAN JOURNAL OF UNDERGRADUATE RESEARCH VOL. 7, NO. 1 1.3 1.25 1.2 1.15 (eV) g E 1.1 1.05 1 273 283 293 303 313 323 333 Temperature (K) Figure 3. The temperature dependence of the band gap energy in silicon: ◊ the pn junction of MPS2222AG npn transistor, ∆ 1N914 diode, and solid line represents the universal function taken from Ref. 6 for comparison. 3+γ / 2 TV− graph will be smaller and hence the I0 = AT exp(−Eg / kT ) (6) value of E g obtained here is expected to be 3+γ / 2 slightly higher. Similar results are reported Here T dependence of I 0 is neglected for silicon as well as for germanium in Ref. 4 as compared to its exponential dependence (see table 1). Also the contribution of ln(T ) on T and thus term will be larger towards higher temperature and this may also contribute in I= Bexp( − E / kT ) (7) 0 g non-linear behavior of TV− data at higher temperature.
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