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The deduction of desorption parameters using an exponential tempering function
Vacuum 57 (2000) 399}403
The deduction of desorption parameters using an exponential tempering function G. Carter Joule Physics Laboratory, School of Sciences, University of Salford, Salford M5 4WT, UK Accepted 27 March 2000
Abstract It is shown how, using the exponential tempering function ¹"¹ exp r t, the parameters of initial surface adatom coverage n , attempt frequency factor l , and activation energy for desorption E, when desorption is described by "rst-order reaction kinetics and the activation energy is single valued, can all be measured to an accuracy of about 3% in a "rst approximation by evaluating the adatom desorption rate as a function of increasing temperature only up to the maximum in the desorption rate. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Controlled heating or tempering continues to be used to evaluate the reaction process parameters in a number of areas including the desorption of adatoms from surfaces and thermally stimulated current analysis of charge detrapping in materials [1] when the reaction species are thermally activated to be released from their trapping centres with a unique and single-valued activation energy E. The most common tempering functions can be described by the power law form d¹/dt"r ¹G where i is integer and, of these, the linear schedule with i"0 and the hyperbolic G schedule with i"2 are most frequently employed. A number of publications [2}11] have analysed how the reaction rate (desorption rate in the case of adatom evolution) varies as a function of increasing time or temperature for these schedules and, as a consequence, how measurement of this rate can lead to evaluation of the process parameters of initial species density (adatom coverage n ), attempt frequency l and activation energy E. Some of these methods require variation of the tempering rate parameter r [5], measurement of the width of a desorption rate}temperature G pro"le [3,4,6}9], measurement of the rates at a number of di!erent temperatures below that at which the maximum rate occurs [10,11] and comparison of experimental rate pro"les with numerically simulated data based upon solution of the de"ning equations for desorption rate [1]. 0042-207X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 0 0 ) 0 0 2 2 0 - 7
400
G. Carter / Vacuum 57 (2000) 399}403
A complicating factor in all these approaches is that the desorption rate is a somewhat complex function of temperature and the relationship between the temperature at which maximum desorption rate occurs and the activation energy for desorption is only approximately linear. In several earlier publications the present author and colleagues [12}15] have demonstrated the advantage of employing the exponential tempering schedule in which i"1 and ¹"¹ exp r t where ¹ is the initial temperature since it leads to an exactly linear relation between the temperature at which maximum desorption rate occurs and the desorption activation energy. It is the purpose of the present short communication to further consider the properties of this tempering schedule and to demonstrate how measurement of the desorption rate as a function of temperature up to the temperature at which maximum desorption rate occurs can lead to direct evaluation of the three process parameters de"ned above in a more straightforward manner than any of the other tempering schedules allows. The estimate of the poorest accuracy of evaluation of these parameters is about 3% when only a "rst-order theoretical approximation is made but this can be considerably improved in higher-order approximations.
2. Theory of desorption using the exponential tempering function Although the theory of desorption from either sites of single valued or distributed activation energies under the in#uence of the exponential tempering function has been developed in detail elsewhere [12}15] it is helpful to brie#y summarize the approach and the main results relevant to the present work. It is assumed that adatoms are adsorbed, with initial density n , in sites with single-valued energy of activation for desorption E, and that, as the system is exposed to the exponential tempering function de"ned above, adatoms desorb according to a "rst-order rate reaction process so that the instantaneous rate of desorption is given by dn n o,! " , q dt
(1)
where n is the adatom density at time t and temperature ¹ and q is the corresponding adatom residence time. The adatom residence time is, in turn, de"ned through the relation q" l\ exp(E/k¹) where k is the Boltzmann constant. The formal solution to Eq. (1) is readily obtained as o,!dn/dt"(n /q) exp(!2 dt/q) and it is further easily shown [3,4,12] that the maximum 2 desorption rate occurs when "dq/dt" "!1. This relationship can be solved for any arbitrary K tempering schedule and for a power-law dependence the result is E Er ¹G "1, (2) l\ G K exp k¹ k¹ K K where ¹ is the temperature at which maximum desorption rate occurs. This result indicates K immediately that if it is required that the temperature at maximum desorption rate is to be linearly related to the activation energy, i.e. E"K¹ , then the index i should be equal to unity and the K tempering schedule is of the exponential form and it is evident that in this case
!K k exp . r "l K k
(3)
G. Carter / Vacuum 57 (2000) 399}403
401
This function can now be employed in obtaining speci"c solutions for the time dependences of the desorption rate and the residual adatom density and it is easily demonstrated that the latter is given by
n"n exp !l
R
!E k¹
dt exp
,n exp(l I)
where
W 1 e\W 1 2 1 W e\W dy "! 1! # 2 I"! y r y y y r W W and in which y"E/k¹ and the su$xes on y refer to its values at t"0 and t. It is readily arranged that the initial temperature is much lower than that at which maximum desorption rate occurs so that the lower limit in the preceding equation can be neglected while, at the rate maximum, y is always [16] of the order of 30 so, to a "rst approximation, only the "rst term in the expansion need be considered. Consequently, using the linear relationship E"K¹ and the identity for r , at the K temperature ¹ ; K l k (4) n "n exp ! e\) I "n e\ K r K and
K o "n l exp ! 1# K k
K "n e\r . k
(5)
Eqs. (3)}(5) now form the necessary set for the evaluation, from experimental data, of the three unknown parameters n , l and E. It may be "rst observed that if the desorption rate o is measured continuously as a function of time up to the rate maximum then the total desorbed adatom density is n "RK dt o and this must be equal to the di!erence between the initial adatom density and the B residual density at the rate maximum which, from Eq. (4) leads to the relation n "n (1!e\) or B n "n (1!e\)\. (6) B Consequently, measurement of the desorption rate at all times up to that at which rate maximum occurs and time integration of this rate leads, immediately, to deduction of n . Once this value has been deduced, and since the tempering rate constant r is speci"ed from experimental conditions, then if the desorption rate maximum o is measured Eq. (5) can be used directly to evaluate the K constant K. The activation energy E can now be evaluated from measurement of the temperature ¹ at which the maximum desorption rate occurs. Finally, since K has been determined, Eq. (3) is K used to evaluate l thus completing the determination of the three process unknowns. Eqs. (4) and (5) are, of course, "rst-order approximations since only the leading term in the expansion for I was employed. Eq. (4) is readily di!erentiated to show that the fractional error in n, if an error dI in I is assumed, is given by dn /n "!l dI and if the second term in the expansion K K for I is assumed to constitute this error then dn /n "!(l /r )(e\WK/y ). At the desorption rate K K K maximum the factor !(l /r )(e\WK /y ) in this relation is, from Eq. (2), equal to unity and, as K indicated above, y is about 30. Consequently, the error in the estimation of n is about 3%. Since K
402
G. Carter / Vacuum 57 (2000) 399}403
the estimates of the activation energy and the attempt frequency both depend linearly on the initial evaluation of n the magnitudes of the errors in the estimates of these former values are the same as that for n . The estimation accuracy of the three parameters can be improved by using higher order terms in the expansion series for I. Thus, if the second term in the expansion is also employed and it is recalled that y +30 then n is given by n "n e\\ and so n can be estimated more K K K accurately. The residual error in estimation of n now derives from neglect of third- and higher order terms in the the expansion for I and scales with increasing powers of 1/y . Consequently, use K of the second term in the expansion and the above modi"ed relation for n decreases the fractional K error estimate for n , l and E by a factor of . 3. Discussion and conclusions Although, at the time of its proposal [12], it was suggested that it would be advantageous to implement the exponential tempering schedule in experimental studies of desorption this appears not to have been undertaken and the largely linear schedule remains the preferred approach [1}11]. It is unfortunate, therefore, that the present analysis cannot be applied to published experimental results. It is very clear from the above analysis, however, that the exponential function leads to extremely simple relationships for the deduction of the desorption process parameters and that, even in the lowest-order approximation, the prediction accuracy of these is quite reasonable (3%) and can be improved substantially by using higher-order approximations. This simplicity of deduction is in contrast to the methods which must be used to obtain parameter values of similar accuracy when either the linear or hyperbolic tempering functions are employed in which cases the temperature at which maximum desorption rate occurs is only an approximately linear function of the activation energy and the relationships between ¹ and E also involve l which must, thereK fore, be estimated initially. It is again, consequently, advocated that the exponential tempering function should be realised experimentally which, with contemporary control systems, should not pose experimental di$culty. As discussed elsewhere [12}15], this function is also particularly advantageous and its application remains equally simple if the desorption activation energy is not single valued but is distributed or if desorption occurs following de-trapping and di!usion from a distribution, including a single-valued source, from di!erent depths below the surface. A further important property of the exponential tempering function, not shared with either the linear or hyperbolic functions, is that it can be employed, straightforwardly, to deduce if the assumed desorption kinetics is indeed "rst order. If the desorption rate is measured at temperatures ¹ and ¹ below and above ¹ respectively where the instantaneous desorption rate is o e\ C C K K then, since at these temperatures
!E !l exp(!E/k¹ ) C exp o e\"n l\ exp K k¹ r E/k¹ C C
use of Eq. (5) reveals that the temperatures ¹ must satisfy the equation (l /r )(1/xe )# C V x"2#K/k where x"E/k¹ . Since l and K are deducible by the methods described above and C r is speci"ed experimentally the measured values of both the ¹ temperature values can be C
G. Carter / Vacuum 57 (2000) 399}403
403
inserted into this equation and if, and only if, they satisfy the equation can the kinetics be properly described as "rst order. Other tempering schedules do not allow this simplicity of determination. Finally, it should be observed that while the analysis presented here shows how, on theoretical grounds, desorption parameters can be deduced, accurately, from measurements of desorption rate as a function of temperature up to the maximum rate, the determination of these parameters will also depend upon the precision of the experimental measurements. Thus, from Eqs. (3)}(5) the fractional accuracy in the determination of the parameters will scale linearly with the fractional accuracy in the measurement of desorption rates, temperature and speci"cation of the tempering exponent r . The "rst of these will depend upon the accuracy of instantaneous pressure measure ment and this will be true for all forms of tempering schedule adopted and the other experimentally determined values will also be identically conditioned by the measurement precision. The overall accuracy of the parameter evaluations will, therefore, be largely independent of which tempering schedule is employed and will be most in#uenced by the measurement precisions. Nevertheless, and in conclusion, it is reiterated that the exponential tempering schedule possesses considerable advantage in the ability to deduce, simply and directly, desorption process parameters from desorption rate measurements and it is proposed that experimental studies could bene"t from using this approach.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
Wang X, Lewin PL, Davies AE, Sutton SJ. IEE Proc Sci Meas Technol 2000;147:2. Lavergne C, Lacabanne C. IEEE Electr Insul Magn 1993;9:5. Redhead PA. Vacuum 1962;12:203. Carter G. Vacuum 1962;12:245. Booth H. Canad J Chem 1954;32:214. Garlick GFJ, Gibson AF. Proc Phys Soc A 1948;60:574. Grossweiner LL. J Appl Phys 1953;24:1306. Keating PN. Proc Phys Soc 1961;78:1408. Halperin A, Branner AA. Phys Rev 1960;117:408. Meata S, Sakaguchi K. Jpn J Appl Phys 1979;18:1983. Meata S, Sakaguchi K. Jpn J Appl Phys 1980;19:597. Carter G, Grant WA, Farell G, Colligon JS. Vacuum 1968;18:263. Carter G, Bailey P, Armour DG. Vacuum 1984;34:797. Carter G, Bailey P, Armour DG. Vacuum 1984;34:801. Carter G. Vacuum 1996;47:401. Grant WA, Carter G. Vacuum 1965;15:13.
The deduction of desorption parameters using an exponential tempering function G. Carter Joule Physics Laboratory, School of Sciences, University of Salford, Salford M5 4WT, UK Accepted 27 March 2000
Abstract It is shown how, using the exponential tempering function ¹"¹ exp r t, the parameters of initial surface adatom coverage n , attempt frequency factor l , and activation energy for desorption E, when desorption is described by "rst-order reaction kinetics and the activation energy is single valued, can all be measured to an accuracy of about 3% in a "rst approximation by evaluating the adatom desorption rate as a function of increasing temperature only up to the maximum in the desorption rate. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction Controlled heating or tempering continues to be used to evaluate the reaction process parameters in a number of areas including the desorption of adatoms from surfaces and thermally stimulated current analysis of charge detrapping in materials [1] when the reaction species are thermally activated to be released from their trapping centres with a unique and single-valued activation energy E. The most common tempering functions can be described by the power law form d¹/dt"r ¹G where i is integer and, of these, the linear schedule with i"0 and the hyperbolic G schedule with i"2 are most frequently employed. A number of publications [2}11] have analysed how the reaction rate (desorption rate in the case of adatom evolution) varies as a function of increasing time or temperature for these schedules and, as a consequence, how measurement of this rate can lead to evaluation of the process parameters of initial species density (adatom coverage n ), attempt frequency l and activation energy E. Some of these methods require variation of the tempering rate parameter r [5], measurement of the width of a desorption rate}temperature G pro"le [3,4,6}9], measurement of the rates at a number of di!erent temperatures below that at which the maximum rate occurs [10,11] and comparison of experimental rate pro"les with numerically simulated data based upon solution of the de"ning equations for desorption rate [1]. 0042-207X/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 2 - 2 0 7 X ( 0 0 ) 0 0 2 2 0 - 7
400
G. Carter / Vacuum 57 (2000) 399}403
A complicating factor in all these approaches is that the desorption rate is a somewhat complex function of temperature and the relationship between the temperature at which maximum desorption rate occurs and the activation energy for desorption is only approximately linear. In several earlier publications the present author and colleagues [12}15] have demonstrated the advantage of employing the exponential tempering schedule in which i"1 and ¹"¹ exp r t where ¹ is the initial temperature since it leads to an exactly linear relation between the temperature at which maximum desorption rate occurs and the desorption activation energy. It is the purpose of the present short communication to further consider the properties of this tempering schedule and to demonstrate how measurement of the desorption rate as a function of temperature up to the temperature at which maximum desorption rate occurs can lead to direct evaluation of the three process parameters de"ned above in a more straightforward manner than any of the other tempering schedules allows. The estimate of the poorest accuracy of evaluation of these parameters is about 3% when only a "rst-order theoretical approximation is made but this can be considerably improved in higher-order approximations.
2. Theory of desorption using the exponential tempering function Although the theory of desorption from either sites of single valued or distributed activation energies under the in#uence of the exponential tempering function has been developed in detail elsewhere [12}15] it is helpful to brie#y summarize the approach and the main results relevant to the present work. It is assumed that adatoms are adsorbed, with initial density n , in sites with single-valued energy of activation for desorption E, and that, as the system is exposed to the exponential tempering function de"ned above, adatoms desorb according to a "rst-order rate reaction process so that the instantaneous rate of desorption is given by dn n o,! " , q dt
(1)
where n is the adatom density at time t and temperature ¹ and q is the corresponding adatom residence time. The adatom residence time is, in turn, de"ned through the relation q" l\ exp(E/k¹) where k is the Boltzmann constant. The formal solution to Eq. (1) is readily obtained as o,!dn/dt"(n /q) exp(!2 dt/q) and it is further easily shown [3,4,12] that the maximum 2 desorption rate occurs when "dq/dt" "!1. This relationship can be solved for any arbitrary K tempering schedule and for a power-law dependence the result is E Er ¹G "1, (2) l\ G K exp k¹ k¹ K K where ¹ is the temperature at which maximum desorption rate occurs. This result indicates K immediately that if it is required that the temperature at maximum desorption rate is to be linearly related to the activation energy, i.e. E"K¹ , then the index i should be equal to unity and the K tempering schedule is of the exponential form and it is evident that in this case
!K k exp . r "l K k
(3)
G. Carter / Vacuum 57 (2000) 399}403
401
This function can now be employed in obtaining speci"c solutions for the time dependences of the desorption rate and the residual adatom density and it is easily demonstrated that the latter is given by
n"n exp !l
R
!E k¹
dt exp
,n exp(l I)
where
W 1 e\W 1 2 1 W e\W dy "! 1! # 2 I"! y r y y y r W W and in which y"E/k¹ and the su$xes on y refer to its values at t"0 and t. It is readily arranged that the initial temperature is much lower than that at which maximum desorption rate occurs so that the lower limit in the preceding equation can be neglected while, at the rate maximum, y is always [16] of the order of 30 so, to a "rst approximation, only the "rst term in the expansion need be considered. Consequently, using the linear relationship E"K¹ and the identity for r , at the K temperature ¹ ; K l k (4) n "n exp ! e\) I "n e\ K r K and
K o "n l exp ! 1# K k
K "n e\r . k
(5)
Eqs. (3)}(5) now form the necessary set for the evaluation, from experimental data, of the three unknown parameters n , l and E. It may be "rst observed that if the desorption rate o is measured continuously as a function of time up to the rate maximum then the total desorbed adatom density is n "RK dt o and this must be equal to the di!erence between the initial adatom density and the B residual density at the rate maximum which, from Eq. (4) leads to the relation n "n (1!e\) or B n "n (1!e\)\. (6) B Consequently, measurement of the desorption rate at all times up to that at which rate maximum occurs and time integration of this rate leads, immediately, to deduction of n . Once this value has been deduced, and since the tempering rate constant r is speci"ed from experimental conditions, then if the desorption rate maximum o is measured Eq. (5) can be used directly to evaluate the K constant K. The activation energy E can now be evaluated from measurement of the temperature ¹ at which the maximum desorption rate occurs. Finally, since K has been determined, Eq. (3) is K used to evaluate l thus completing the determination of the three process unknowns. Eqs. (4) and (5) are, of course, "rst-order approximations since only the leading term in the expansion for I was employed. Eq. (4) is readily di!erentiated to show that the fractional error in n, if an error dI in I is assumed, is given by dn /n "!l dI and if the second term in the expansion K K for I is assumed to constitute this error then dn /n "!(l /r )(e\WK/y ). At the desorption rate K K K maximum the factor !(l /r )(e\WK /y ) in this relation is, from Eq. (2), equal to unity and, as K indicated above, y is about 30. Consequently, the error in the estimation of n is about 3%. Since K
402
G. Carter / Vacuum 57 (2000) 399}403
the estimates of the activation energy and the attempt frequency both depend linearly on the initial evaluation of n the magnitudes of the errors in the estimates of these former values are the same as that for n . The estimation accuracy of the three parameters can be improved by using higher order terms in the expansion series for I. Thus, if the second term in the expansion is also employed and it is recalled that y +30 then n is given by n "n e\\ and so n can be estimated more K K K accurately. The residual error in estimation of n now derives from neglect of third- and higher order terms in the the expansion for I and scales with increasing powers of 1/y . Consequently, use K of the second term in the expansion and the above modi"ed relation for n decreases the fractional K error estimate for n , l and E by a factor of . 3. Discussion and conclusions Although, at the time of its proposal [12], it was suggested that it would be advantageous to implement the exponential tempering schedule in experimental studies of desorption this appears not to have been undertaken and the largely linear schedule remains the preferred approach [1}11]. It is unfortunate, therefore, that the present analysis cannot be applied to published experimental results. It is very clear from the above analysis, however, that the exponential function leads to extremely simple relationships for the deduction of the desorption process parameters and that, even in the lowest-order approximation, the prediction accuracy of these is quite reasonable (3%) and can be improved substantially by using higher-order approximations. This simplicity of deduction is in contrast to the methods which must be used to obtain parameter values of similar accuracy when either the linear or hyperbolic tempering functions are employed in which cases the temperature at which maximum desorption rate occurs is only an approximately linear function of the activation energy and the relationships between ¹ and E also involve l which must, thereK fore, be estimated initially. It is again, consequently, advocated that the exponential tempering function should be realised experimentally which, with contemporary control systems, should not pose experimental di$culty. As discussed elsewhere [12}15], this function is also particularly advantageous and its application remains equally simple if the desorption activation energy is not single valued but is distributed or if desorption occurs following de-trapping and di!usion from a distribution, including a single-valued source, from di!erent depths below the surface. A further important property of the exponential tempering function, not shared with either the linear or hyperbolic functions, is that it can be employed, straightforwardly, to deduce if the assumed desorption kinetics is indeed "rst order. If the desorption rate is measured at temperatures ¹ and ¹ below and above ¹ respectively where the instantaneous desorption rate is o e\ C C K K then, since at these temperatures
!E !l exp(!E/k¹ ) C exp o e\"n l\ exp K k¹ r E/k¹ C C
use of Eq. (5) reveals that the temperatures ¹ must satisfy the equation (l /r )(1/xe )# C V x"2#K/k where x"E/k¹ . Since l and K are deducible by the methods described above and C r is speci"ed experimentally the measured values of both the ¹ temperature values can be C
G. Carter / Vacuum 57 (2000) 399}403
403
inserted into this equation and if, and only if, they satisfy the equation can the kinetics be properly described as "rst order. Other tempering schedules do not allow this simplicity of determination. Finally, it should be observed that while the analysis presented here shows how, on theoretical grounds, desorption parameters can be deduced, accurately, from measurements of desorption rate as a function of temperature up to the maximum rate, the determination of these parameters will also depend upon the precision of the experimental measurements. Thus, from Eqs. (3)}(5) the fractional accuracy in the determination of the parameters will scale linearly with the fractional accuracy in the measurement of desorption rates, temperature and speci"cation of the tempering exponent r . The "rst of these will depend upon the accuracy of instantaneous pressure measure ment and this will be true for all forms of tempering schedule adopted and the other experimentally determined values will also be identically conditioned by the measurement precision. The overall accuracy of the parameter evaluations will, therefore, be largely independent of which tempering schedule is employed and will be most in#uenced by the measurement precisions. Nevertheless, and in conclusion, it is reiterated that the exponential tempering schedule possesses considerable advantage in the ability to deduce, simply and directly, desorption process parameters from desorption rate measurements and it is proposed that experimental studies could bene"t from using this approach.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
Wang X, Lewin PL, Davies AE, Sutton SJ. IEE Proc Sci Meas Technol 2000;147:2. Lavergne C, Lacabanne C. IEEE Electr Insul Magn 1993;9:5. Redhead PA. Vacuum 1962;12:203. Carter G. Vacuum 1962;12:245. Booth H. Canad J Chem 1954;32:214. Garlick GFJ, Gibson AF. Proc Phys Soc A 1948;60:574. Grossweiner LL. J Appl Phys 1953;24:1306. Keating PN. Proc Phys Soc 1961;78:1408. Halperin A, Branner AA. Phys Rev 1960;117:408. Meata S, Sakaguchi K. Jpn J Appl Phys 1979;18:1983. Meata S, Sakaguchi K. Jpn J Appl Phys 1980;19:597. Carter G, Grant WA, Farell G, Colligon JS. Vacuum 1968;18:263. Carter G, Bailey P, Armour DG. Vacuum 1984;34:797. Carter G, Bailey P, Armour DG. Vacuum 1984;34:801. Carter G. Vacuum 1996;47:401. Grant WA, Carter G. Vacuum 1965;15:13.