THE APPLICATION OF COMBINED CONDUCTIVITY AND SEEBECK-EFFECT PLOTS FOR THE ANALYSIS OF SEMICONDUCTOR PROPERTIES

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1 R657. Philips Res. Repts 23, , 1968 THE APPLICATION OF COMBINED CONDUCTIITY AND SEEBECK-EFFECT PLOTS FOR THE ANALYSIS OF SEMICONDUCTOR PROPERTIES by G. H. JONKER Abstract A semiconductor can be characterized at a certain temperature by a curve representing the Seebeck coefficient as a function of the conductivity. Such a pear-shaped curve can be constructed from measurements 'on a number of samples having p-type, /I-type and mixed conduction, respectively. Quantitative relations between transport properties can be derived from such curves. 1. Introduetion This paper discusses a new method of analysing the transport properties of semiconductors by using a combined plot of conductivity and Seebeck coefficient. The method is applicable for the study of semiconductors attemperatures where extrinsic as well as mixed p-type and n-type conduction can be measured. At room temperature these possibilities exist for semiconductors with forbidden energy gaps Eg lower than 0 5 e. The method is especially suitable for the estimation of the energy gap if the value cannot be obtained by optical measurements. Usually the conductivity and the Seebeck effect of a series of samples with varying dope concentrations are studied separately. In each series of measurements, however, irregularities occur as a result of deviations from the expected chemical composition, impurities, defects, etc. As both quantities depend on the actual concentrations of charge carriers, these irregularities disappear in a plot of the Seebeck coefficient Ct as a function of the conductivity a. Such plots, especially of Ct as a function of log a, have been used earlier for extrinsic semiconductors, e.g. in the study of thermoelectric elements for heat converters or for Peltier cooling 1). It will be shown that more information can be obtained if the regions of mixed conduction are also included. 2. Mathematics of the IX - log 0' plot. In order to give a general impression of the method we shall start with the simple case of a "symmetric" semiconductor, i.e. a semiconductor with equal parameters N, I' and A (density of states, mobility and transport coefficient, respectively) for both types of charge carriers with the assumption that these

2 132 G. H. JONKER parameters are independent of the dope concentration. Moreover, we shall, restrict the calculations first to the non-degenerate state, where Boltzmann statistics can be applied. In the second part the more general asymmetrie case will be considered and a few remarks on the extension to high charge-carrier concentrations will be made Symmetric case by with The intrinsic, partial and total conductivities are described in the usual way (1) I1j = N exp (-Eo/kT), (2) 1 a: = nep, = -l1jep" a a = a+ + a: = (a + ~}iep, = rta + ~)ai' (3a) (3b) (4) The usual expressions for the partial and total Seebeck coefficients can be transformed in the following way: k (N ) k Ne A k (Eg ) a+ = - In - + A = -In _<_ = - - -I- 2A -In a2, epe ani 2e kt (Sa) k (N ) k ane A k (Eg ) «, = -- ln- + A = --In-- = A + In a 2,(Sb) e 11 e n, 2e kt a+a+ + a:«: k {a2-1 (Eo ) } a = = A -In a2 a+ + a_ 2e a kt (6a) With eq. (4) the expression of a can be further transformed into a = ± ~ (Eg <+ 2A) (1- ai2)1/2_~ In [!!._ {I ± (1- a j2 )1/2}J. (6b) 2e kt a 2 e a, a2 Equation (6b) shows the interesting result that the shape and dimensions of any plot of/l(a) versus/ 2 (a/ai) depend onthe parameter (Eg/kT+ 2A) only. This equation can be used, for example, to calculate curves of a as a function of log (a/ai) for a number of values of (Eo/kT + 2A). An example of such a curve is shown in fig. 1. An advantage of this type of curve is that the extrinsic parts are represented by straight lines.

3 CONDUCTIITY AND SEEBECK EFFECT IN SEMICONDUCTORS D a: 600 (!l/lok) 500 ' JO 0 o C [é A r _ IT -bt ' 6 Ol' t"-r-.. 1\ '\? c-b ~ P ~ '" R a n2 10 JO _JL. Ui Fig. 1. Calculated curve of the Seebeck coefficient Cl: as a function of the relative conductivitya/a" for Eg/kT + 2A = 14. The numbers 2, 4 etc. denote the values of a in p = all, or 11 = a 11,. Dashed-dotted curve: calculated deviation for A = 2. On the curve the following points are of interest: (a) The intrinsic composition is represented by point A, with a = 1, C( = 0 and a/a, = 1. (b) A second zero of C( is found at B where a/a, = Ne A /2n,. (c) Point C is chosen at a/a, = t. The value of Eg/kT + 2Acan be found from the distance between the points Band C according to Eg + 2A= 4.6 {log(!! ) - log (!! ) }. kt. a, B a, c (d) Point C represents the values of the partial contributions of the p-type and n-type conduction to the intrinsic conductivity. Thus in points D and E the C(values corresponding to C(+and C(_of the intrinsic semiconductor can be found. From eqs (5a) and (5b) it follows, with a = 1, that k (Eg ) e kt (C(+- C(_)c= A. (e) The points F and G represent the extreme values of a. By differentiating eq. (6a) it is found.that the following relations hold at these.points: (7) (8) (al + 1)2 Eg ---=-+2A, 2a 2 kt (9)

4 134 G.H.JONKER k {Eg (Eg)} «(I(+-(I(-)cxtr =- -+2A-I-ln2 -+2A-I, ekt kt a) (Eg ) 1/2 (- = -+A a, extr 2kT. (10) (11) Thus, if extreme values are known, Eg/kT + 2Acan be calculated. For the presentation of experimental data instead of a plot of a-log (a/a,) generally plots of «-log a or ce-iog e willbe used. This gives no change in the calculations using the points A, Band C General case In most semiconductors the parameters of the n-type and p-type conduction are not equal. Therefore we have to introduce different densities of states N+ and N_, mobilities f-l+ and I-L, and transport constants A+ and A_. In this asymmetrie case our first interest is to see whether the general shape of the (I(-log(a/a,) plot is changed. To this end we have to rewrite eqs (I) to (6): a, = n,e(f-l+ + f-l_) with n, = (N+N_)1/2 exp (-Eg/kT), a+ = an,ef-l+, o: = (I/a) n,ef-l_, a = n.e {af-l+ + (I/a) f-l_}. The minimum of the conductivity is now different from a, and is found to occur for a+ = c: or at Thus, with f-l+/f-l_ = b, (12) a, = ~(b + _I_)amlR> (14) 2 b a = ~(ab + ~)amln. (15) 2. a vb Further, introducing the quantity e = N+e A + /N_e A -, and after some calculation, the total Seebeck coefficient is found to be equal to k {a 2 b - 1 (Eg ) } (I(= A+ + A_ - In (a 2 b) + In (be) 2e a 2 b + 1 kt (l6a)

5 CONDUCTIITY AND SEEBECK EFFECT IN SEMICONDUCTORS 135 or k (E a 2 )1/2 ct= ± o + A+ + A_) (1 - ~ + 2e kt a 2 k [a { ( a 2 ) 1/2}] k --I~ - I ± l-~ +-ln(bc). e alliin a, 2e (I6b) The conclusion of these transformations is again that shape and dimensions of an ct-log (a/a,) plot depend on one parameter only: (Eo/kT + A+ + A_). An 800 D r t::::-_,._ r b- I-F 2 3 slh., Ik J 10Oe A ID I'--- t--- 1'\ v./ l..---"' '\.m L3- r;:::; r-, r----. t--- r-; / Fig. 2. Curve.calculated for an asymmetrie case with #+/I-I_ = 10. Point i denotes the intrinsic composition. example of such a curve is shown in fig. 2. The difference is that amin replaces a, and that all ct values are increased by k k N+e A +f.1-+ -ln(bc)=-ln. 2e 2e N_eLfh_ This means that the symmetry axis of the curve is shifted with respect to the ct = 0 axis. This shift reflects the total "asymmetry" of the semiconductor parameters. The value of (Eg/kT + A+ + A_).can be calculated in the same two ways as in the symmetric case, using now the symmetry axis as a base. We now have two intersection points B+ and B_ of the extrinsic parts of the curve

6 136 G. H.JONKER with the a.= 0 axis. These points represent In (N+eA+ef-l+laml n ) and In (N_eA-ef-l_laml n )' respectively. With these results the possibilities of obtaining direct information are exhausted. Other measurements are necessary to complete the analysis: (a) If the intrinsic conductivity is known, the value of b = f-l+lf-l- can be calculated from eq. (14). Inserting b in the value of k]e In (be) leads then to the value ofe = N+eA+IN_e A -. (b) If the «-Iog a plots can be constructed for two temperatures Tl and T 2 the values of Eg and (A+ + A_) can be calculated. One must be aware, however, that Eg is generally temperature-dependent: Eg = s, -f3t. In such a case only the values of Eo and (A+ + A_ - f3lk) can be found. For band semiconductors the products NeAef-l at B are only weakly temperature-dependent, as the temperature coefficients of Nand f-l are of the same order of magnitude, but opposite in sign. Therefore the straight extrinsic parts of the curves shift only slightly if the temperature is changed. The presence of a thermally' activated hopping process can be detected from a stronger shift of these straight parts of the curve. The value of the activation energy of the mobility can then be fairly accurately determined from the temperature dependence of ab = NeAef-l, with an uncertainty of R::I kt, as in this case the temperature dependence of N is not known. (c) If also measurements of the Hall effect are available, the pand n values are known and these give in combination with a the mobility values f-l+ and f-l_. Then also the values of n., N+e A + and N_e A - can be calculated. The values of pand n, calculated from the Hall coefficients RH can also be used for a plot of a. as a function of logp and log n. Such a plot resembles the «-Iog a plot in the extrinsic parts, but is simpler, because the asymmetry due to b = f-l+lf-l- is absent. Such a plot, however, cannot be continued into the region of mixed conduction as RH is a fairly complicated function of p, n, f-l+ and f-l_ (see next section). The points B are not really measurable points, but are only found by extrapolation from measured points at low charge-carrier concentrations. For calculations at high concentrations the complete Fermi integrals must be used. In fig. 1 an example is shown, calculated for A = 2. An uncertaintyin the construction of the a.-log a plots is caused by the possibility of impurity conduction. This effect, which depends on the degree of compensation of the donor or acceptor centres, often occurs at higher dope concentrations and at lower temperatures 2). 3. PbS As an illustration in fig. 3 an a.-log 11 curve for PbS at room temperature is

7 CONDUCTIITY AND SEEBECK EFFECT IN SEMICONDUCTORS ei 600 (p! C) 500 t ~OO o <; I,<, -Lh [7 F r-..., ~ I ~ " f--t;-., A f , ~, \, v,'...!..s, : -400 \ '"»> G r:::f & Io Charge-carrier density(cm-3) Fig. 3. a-log n plot for PbS at room temperature constructed from measurements of Bloem 3). shown. For the construction of this curve in the first place measurements of a and n of extrinsic p-type and n-type samples have been used 3). The extrinsic lines show already that the total curve will be symmetric with respect to the a = 0 axis, which means equality of N+e A + and N_e A -. Moreover, the maximum values of a (+ and -515 [LrC) were known, so that, using eq. (10), the value of (Eg/kT + A+ + A_) was calculated to be This means, at room temperature, that Eg + (A+ + A_)kT = e. With this value the total curve was calculated and the points representing the maximum values of a were inserted (points F and G). From optical measurements it is known that Eg = 0 30 e, so that A+ + A_ = 4 5. Further it can be read from.fig.3 that 2nl = cm- 3 (point A) and that N+e A + = N_e L = cm=" (pointb). Assuming equality of A+ and A_leads to N+= N_ = cm=". In this way a reasonable basis for further calculations or speculations is obtained, giving, for example, a value of m* = 0 22 m for the effectivemass of the charge carriers (compare ref. 3). It is remarkable that the measured points.fitto the straight parts of the curve up to much higher carrier concentrations than corresponds to the density of states. It may be that this is due to a slight contribution of impurity conduction. 4. Conclusion In principle the data obtained from the oe-log a plots can also be calculated from the oeand a measurements separately. The advantage, however,

8 138 G.H.JONKER of a combined plot is in the first place that it gives a check on the correctness of the measurements. Further all available measurements contribute to the analysis, as from a series of calculated curves that one is chosen which fits in the,best way to the measured points. Eindhoven, October 1967 REFERENCES 1) J. D. Wasscher, W. Albers and C. Haas, Solid State Electron. 6, ) A. J. Bosman and C. Crevecoeur, Phys. Rev. 144, , , J. Bloem, Philips Res. Repts 11, ,