Phenomenological Representation of the Conduction of ZnO Varistors

(Overall representation of I / V characteristics from pre-breakdown,

through breakdown, to up-turn region)

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　May. 5 2007

rev..May.20 2007

　　　　　　　　　　　　　　　　　　　　　　　　　 Incisive Keen Engineering Inc.

                                                    e -mail; ys-ikeda@amber.plala.or.jp

                                              Y. Ikeda

                                               All Rights Reserved

PREFACE:

Current and voltage (I / V) characteristics of ZnO varistors have three regions.

1. Pre-breakdown region at low current;

         This region is temperature dependent expressed as I = Io exp(-E_G/kT) at constant voltage, where E_G is an activation energy (about 0.8~0.9 eV at room temperature), k and T are Boltzman constant and the absolute temperature, respectively.

      2. Breakdown region at middle current;

         This region is small temperature dependent, and has the typical varistor characteristics described empirically as I1 / I2 = (V1 / V2)^αwith nonlinearity exponent α.

      3. Up-turn region at high current;

         This region has no longer varistor characteristics, but is dominated only by bulk ZnO resistivity, i.e. ohmic.

PURPOSE:

   This paper is intended to figure out the overall representation of I / V characteristics

of ZnO varistors, from pre-breakdown, through breakdown, to up-turn region.

MODEL:

   Extension of the thermionic emission model supposed to be dominant in pre-breakdown region to breakdown region

   Electronic barrier height is assumed to decrease on applied voltage exponentially.

REPRESENTATION:

Representation for single grain boundary

   Thermionic emission model (pre-breakdown region);

       J = Jo exp(-eE_B/kT)[1-exp(-eV/kT)]                                      -(0)

          Jo = A* T² exp(-E_F/kT)

           J;   current density

           A*;  Richardson’s constant

           E_F;  Fermi level

           E_B;　potential barrier height

           V;   applied voltage

           e;   electronic charge

           k;   Boltzman constant

           T;   absolute temperature

   Ikeda’s model (pre-breakdown region, through breakdown region to up-turn region);

       J = Φ(V) [(V / R*-1)Φ(V) + 1]                                         -(1)

           J;   current density (A/cm²)

           R^*;  constant of 5 derived from the empirical data of resistivity 0.2 ohm at up-turn

Φ(V); transmission probability factor of carriers through barrier

V;    applied voltage (V)

Φ(V) = exp(-eE_B/kT)                                                  -(2)

           E_B;  electronic barrier height (eV)

           e;   electronic charge (1.602×10^-19 C)

           k;   Boltzman constant (1.38×10^-23 J / K)

           T;   absolute temperature

                                             -(3)

           Eo;  0.9 eV supposed empirically

           Vo;  parameter relevant to breakdown voltage (breakdown voltage is defined as voltage at maximum α)

           N;   parameter relevant to Nth-order non-linearity

                     -(4)

RESULTS:

   Current vs Voltage characteristics;

       Fig.1  I vs V, parameter; temperature ℃

             Vo = 3, N = 5

   Dependence of nonlinearity exponent α on Voltage (V);

       Fig.2  α vs V, parameter; temperature ℃

             Vo = 3, N = 5

   Dependence of maximum nonlinearity exponent αmax on absolute temperature T

       Fig.3  αmax vs 1 / T

             Vo = 3, N = 5

CONCLUSIONS:

    1.  As shown by Fig,1, equation (1) expresses the phenomenological behavior of the overall I / V characteristics of ZnO varistors, from temperature-dependent pre-breakdown region through nonlinear breakdown region, to ohmic up-down region.

    2.  Equation (1) is based on the assumption that the barrier height depend on applied electric force, and the boundary height is strongly affected by electric force as Nth-order exponential formula of equation (3).

  Here, N = 5 mean 5th-order nonlinearity.

       Nth-order exponential formula equation (3) suggests that the origin of the nonlinearity of ZnO varistors would be quantitatively difficult to explore, and the control of the nonlinearity exponent in manufacturing would be tremendously difficult a fortiori.

    3. From Fig.2, nonlinearity exponent α can be recognized as temperature- dependent, and from Fig.3, maximum nonlinearity exponent αmax correlate linearly with 1 / T as experimentally demonstrated.        

    4.  The validity of equation (1) and (3) should be followed experimentally.

　　5.  As Schottky barrier problem,

ZnO barristers are accepted to be composed of back to back Schottky diodes.

      The breakdown voltage of ZnO varistors is around 3 V irrespective of ingredients and manufacturing processes, that is, the Fermi level at interface is ‘pinned’.

      The height of barrier (the transmission probability of electron) of ZnO varistors is voltage-sensitive, and the dependence of Schottky barrier height (the transmission probability of electron) on applied voltage is exclusive example so far as I know, and the control of Schottky barrier height (the transmission probability of electron) on applied voltage is impossible in general, that is the reason why the manufacturing of ZnO varistors is unstable.  

　　6.  For quest for the origin of varistor characteristics, equation (3) will be of help (physical meaning?).

　　

　　NB1

        For pre-break and breakdown region, equation (1) is expressed as

              J = Φ(V)

    NB2

        Critical voltage (threshold voltage) V_C to up-turn region is defined as

Φ(V_C) ≒ 1