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On neutrality and equilibrium in semiconductors
Solid-Stat< Ektmnic~ Vol. 22, pp. 215-216 @JPaganon Press Ltd., 1979. Pliakd in Great Britaill
NOTES
ON NEUTRALITY
AND EQUILIBRIUM IN SEMICONDUCTORS
(Received 6 May 1978;in revised In device modeling it is necessary to set forth clearly the conditions prevailing within the structure being analyzed, region by region. For example, the concept of quasineutrality, or the condition n -p = no-p0 is often a useful one; it has been extensively used since van Roosbroeck treated ambipolar transport in semiconductors[l], using certain principles developed for ionic transport in gases. In an analogous way one can sometimes profitably employ the concept of quasiequilibrium. Three equations specify strict equilibrium: CONSTANT = I$~= 4,, = CONSTANT. The quasi-Fermi levels for holes and electrons (i.e. their electrochemical potentials) must coincide and must both be constant throughout the region in question. Equivalent to the “middle” equilibrium equation is, of course, pn = nf. Equivalent to the first and third are J. = 0 and JP =O, where electrical current rather than particle currents are intended. Let us take as the defining condition for quasiequilibrium the case where the electrochemical potentials are equal to each other but nonconstant-the familiar and important case of ohmic conduction. Accepting this definition, we then have three degrees of neutrality and also of equilibrium. Hence we can construct the matrix of Table I, where all nine possible combinations are displayed. To lend concreteness to the exhibit, we have intro-
form
IO August
1978)
duced an example for each condition. All examples are steadystate and isothermal. In cases where joule heating exists, we consider that heat flows from the semiconductor to adjacent sinks through negligible temperature gradients. In the quasiequilibrium column, the noncpnstant quasi-Fermi levels require finite electron and hole currents, and clearly the total current J is also finite. In Ihe nonequilibrium column, defined by I$“# &,, nothing is specified concerning these three currents. This matter bears closer scrutiny because the most important device-modeling problems are non-equilibrium cases set apart from one another by conditions on these currents. Therefore, let us expand the third column of Table I into another 3 x 3 matrix for a closer examination of such conditions. The result is shown in Table 2. Across the top are three sets of conditions on the three currents, J,,, J,, and J. First of all, the blanks near the top of the chart suggest that it is difficult to find nonequilibrium examples wherein strict neutrality coexists with a nonzero current. The center-column examples wherein total current I can vanish while the individual net currents J. and f,, remain finite (equal and opposite) are taken from a recent analysis of the bipolar junction transistor[2]. The examples of Table 2 are specific (as indeed are those of
Table I. A 3 x 3 neutrality-equilibrium matrix. The shaded symbols represent ohmic contacts, or contacts that preserve equilibrium carrier densities (no and pO) where they touch the semconductor, with or without the passage of current. The term isolated implies protection from radiation of all kinds, as well as from electrical disturbances
EQU*LISRIU”
samples)
(isolated
QLmSIE*“ILIBRIU”
(ohmic ‘$,lX.Y.Z)
NONEQUILIBRIUM
conduction) =
u?p~x,Y,z)
e”n(x.Y.z)
,AMPLE WIT" ‘IN‘?
-
p
=
“o-
PO
DOPED PENETRA-
RADIATION
lNIFQRMLY ”
$p(X’Y.rl
UNIFORMLY
‘WIN, NEUTRALITY
#
INCIDENT
llJJl LN E=O
REGION OF OPEN:IRC"ITED SOLAR CELL FNDER IRRADlATION IASE
NONNEUTRALITY
”
-
P #
no-
NONUNIFORM WITH HIGH P,
f+-p E#O
NET DOPING GRADIENTS
iONUNIFORM UT” HIGH
NET DOPING GRADIENTS
IPACE-CHARGE REGION ,F NP-JUNCTION ;I\MPLE UNDER BIAS
216
Notes Table
2. A matrix for nonequilibrium conditions, wherein three neutrality conditions are specified as well as three sets of conditions on the net electron current J., the net hole current J,, and the total current J
J”
=
0
Jn
# 0
J”
# 0
Jp
=
0
Jp
# 0
Jp
# 0
J
=o
J
=o
J
to
NE”TRRLlTY ”
-
p
no-
=
P,
Nn - emno-
COLLECTOR REGION OF SJT OPERATING AT THE BOUNDARY OF FOriW*RD y REVERSE SATURATION
P,
“+l-$p2 = E#O
NONNEUTRALITY ”
-
P #
“o-
CoLLECTp ABOVE N
PN
<“pION N SJT
OF
P,
:w
pi+-&&”
=
T
E#O
E#O
Table I) and do not constitute an exhaustive listing of the possibilities. It is instructive to consider variations. For example. examine a special case of neutrality with all currents zero (upper left example in Table 2). In the general case of this example &# I&, and neither equals 4, the equilibrium Fermi level. But there is a low-level special case wherein I&+,+= I$ and hnorityi
6
Consider also the quasineutrality example in the right column of Table 2. The end-region conditions presented here are the general conditions of the Rittner-Fletcher[3,4] analysis. wherein significant values of electric field and significant amounts of conductivity modulation are permitted. Fletcher employed the term quasiequilibrium for these conditions in the work just cited. Accepting present definitions, however, a better term is nonequilibrium quasineurrality for the end regions of PN junction biased at intermediate or high levels. By contrast to this rather general case. the low-level special case treated by Shockley[5] has E= 0. Further, this case has CONSTANT= &(x)# &(x)# CONSTANT on one side of the sample, and CONSTANT # &.(x) # &,(x) = CONSTANT on the other side. Both the Shockley and the Rittner-Fletcher analyses in effect assumed constant values for 4. and bD in the space-charge region, a valid assumption for quite a wide range of practical situations, as shown by Hauser[6] and van der Ziel[7]. Closely related to Shockley’s treatment of the junction problem under low-level bias is the classical treatment of the junction transistor’s base region given simultaneously and independently by WebsterI and Rittner[3]. wherein the minority diffusivity is effectively doubled by the presence of an electric field. Crucially important to their analysis is the fact that the majority-carrier quasi-Fermi level is very nearly constant, a fact that has been forgotten in recent treatments of the solar cell that assume a doubled minority diffusivity. In addition to narrowing the given examples, we may in some instances want to generalize them. Consider the open-circuited
s#alar cell at the right in Table I. Its base region is taken to have net doping. For this important special case, (dn/dx)= 1miform dpldx). On the other hand, if gradients of net doping are allowed n the base, then (dn/dx) # (dpldx), as van Vliet has emphasized[9]. Such subtle distinctions can be quite important analytically. The examples offered in Tables I and 2 are not in any sense a “complete set.” Nonetheless they may serve as a kind of framework against which other specific cases may be positioned for comparison and clarification in future device modeling work, in much the manner of the cases in the three paragraphs immediately above.
Acknowledgements-For extremely helpful debted to Mr. R. P. Jindal, Dr. B. L. Grung, Vliet.
comments I am inand Prof. K. M. van
Dept. of E/e&Cal Engineering Uniuersiry of Minnesota Mineapolis MN 55455, U.S.A.
R. M. WARNER,JR.
REFERENCES I. W. van Roosbroeck, Phys. Rev. 91,282 (1953). 2. B. L. Grung and R. M. Warner, Jr., Solid-St. Electron. 20, 753
(1977). 3. E. S. Rittner. 4. 5. 6. 7. 8. 9.
Phys. Rev. 94, 1161 (1954). N. H. Fletcher, J. Electron. 2, 609 (1957). W. Shockley, BSTJ 28,435 (1949). J. R. Hauser, Solid-9. Electron. 14, 133 (1971). A van der Ziel, Solid&. Electron. 16, 1509 (1973). W. M. Webster, Proc. IRE 42, 914 (1954). K. M. van Vliet, Solid-St. Electron. 9, I85 (1%6).
NOTES
ON NEUTRALITY
AND EQUILIBRIUM IN SEMICONDUCTORS
(Received 6 May 1978;in revised In device modeling it is necessary to set forth clearly the conditions prevailing within the structure being analyzed, region by region. For example, the concept of quasineutrality, or the condition n -p = no-p0 is often a useful one; it has been extensively used since van Roosbroeck treated ambipolar transport in semiconductors[l], using certain principles developed for ionic transport in gases. In an analogous way one can sometimes profitably employ the concept of quasiequilibrium. Three equations specify strict equilibrium: CONSTANT = I$~= 4,, = CONSTANT. The quasi-Fermi levels for holes and electrons (i.e. their electrochemical potentials) must coincide and must both be constant throughout the region in question. Equivalent to the “middle” equilibrium equation is, of course, pn = nf. Equivalent to the first and third are J. = 0 and JP =O, where electrical current rather than particle currents are intended. Let us take as the defining condition for quasiequilibrium the case where the electrochemical potentials are equal to each other but nonconstant-the familiar and important case of ohmic conduction. Accepting this definition, we then have three degrees of neutrality and also of equilibrium. Hence we can construct the matrix of Table I, where all nine possible combinations are displayed. To lend concreteness to the exhibit, we have intro-
form
IO August
1978)
duced an example for each condition. All examples are steadystate and isothermal. In cases where joule heating exists, we consider that heat flows from the semiconductor to adjacent sinks through negligible temperature gradients. In the quasiequilibrium column, the noncpnstant quasi-Fermi levels require finite electron and hole currents, and clearly the total current J is also finite. In Ihe nonequilibrium column, defined by I$“# &,, nothing is specified concerning these three currents. This matter bears closer scrutiny because the most important device-modeling problems are non-equilibrium cases set apart from one another by conditions on these currents. Therefore, let us expand the third column of Table I into another 3 x 3 matrix for a closer examination of such conditions. The result is shown in Table 2. Across the top are three sets of conditions on the three currents, J,,, J,, and J. First of all, the blanks near the top of the chart suggest that it is difficult to find nonequilibrium examples wherein strict neutrality coexists with a nonzero current. The center-column examples wherein total current I can vanish while the individual net currents J. and f,, remain finite (equal and opposite) are taken from a recent analysis of the bipolar junction transistor[2]. The examples of Table 2 are specific (as indeed are those of
Table I. A 3 x 3 neutrality-equilibrium matrix. The shaded symbols represent ohmic contacts, or contacts that preserve equilibrium carrier densities (no and pO) where they touch the semconductor, with or without the passage of current. The term isolated implies protection from radiation of all kinds, as well as from electrical disturbances
EQU*LISRIU”
samples)
(isolated
QLmSIE*“ILIBRIU”
(ohmic ‘$,lX.Y.Z)
NONEQUILIBRIUM
conduction) =
u?p~x,Y,z)
e”n(x.Y.z)
,AMPLE WIT" ‘IN‘?
-
p
=
“o-
PO
DOPED PENETRA-
RADIATION
lNIFQRMLY ”
$p(X’Y.rl
UNIFORMLY
‘WIN, NEUTRALITY
#
INCIDENT
llJJl LN E=O
REGION OF OPEN:IRC"ITED SOLAR CELL FNDER IRRADlATION IASE
NONNEUTRALITY
”
-
P #
no-
NONUNIFORM WITH HIGH P,
f+-p E#O
NET DOPING GRADIENTS
iONUNIFORM UT” HIGH
NET DOPING GRADIENTS
IPACE-CHARGE REGION ,F NP-JUNCTION ;I\MPLE UNDER BIAS
216
Notes Table
2. A matrix for nonequilibrium conditions, wherein three neutrality conditions are specified as well as three sets of conditions on the net electron current J., the net hole current J,, and the total current J
J”
=
0
Jn
# 0
J”
# 0
Jp
=
0
Jp
# 0
Jp
# 0
J
=o
J
=o
J
to
NE”TRRLlTY ”
-
p
no-
=
P,
Nn - emno-
COLLECTOR REGION OF SJT OPERATING AT THE BOUNDARY OF FOriW*RD y REVERSE SATURATION
P,
“+l-$p2 = E#O
NONNEUTRALITY ”
-
P #
“o-
CoLLECTp ABOVE N
PN
<“pION N SJT
OF
P,
:w
pi+-&&”
=
T
E#O
E#O
Table I) and do not constitute an exhaustive listing of the possibilities. It is instructive to consider variations. For example. examine a special case of neutrality with all currents zero (upper left example in Table 2). In the general case of this example &# I&, and neither equals 4, the equilibrium Fermi level. But there is a low-level special case wherein I&+,+= I$ and hnorityi
6
Consider also the quasineutrality example in the right column of Table 2. The end-region conditions presented here are the general conditions of the Rittner-Fletcher[3,4] analysis. wherein significant values of electric field and significant amounts of conductivity modulation are permitted. Fletcher employed the term quasiequilibrium for these conditions in the work just cited. Accepting present definitions, however, a better term is nonequilibrium quasineurrality for the end regions of PN junction biased at intermediate or high levels. By contrast to this rather general case. the low-level special case treated by Shockley[5] has E= 0. Further, this case has CONSTANT= &(x)# &(x)# CONSTANT on one side of the sample, and CONSTANT # &.(x) # &,(x) = CONSTANT on the other side. Both the Shockley and the Rittner-Fletcher analyses in effect assumed constant values for 4. and bD in the space-charge region, a valid assumption for quite a wide range of practical situations, as shown by Hauser[6] and van der Ziel[7]. Closely related to Shockley’s treatment of the junction problem under low-level bias is the classical treatment of the junction transistor’s base region given simultaneously and independently by WebsterI and Rittner[3]. wherein the minority diffusivity is effectively doubled by the presence of an electric field. Crucially important to their analysis is the fact that the majority-carrier quasi-Fermi level is very nearly constant, a fact that has been forgotten in recent treatments of the solar cell that assume a doubled minority diffusivity. In addition to narrowing the given examples, we may in some instances want to generalize them. Consider the open-circuited
s#alar cell at the right in Table I. Its base region is taken to have net doping. For this important special case, (dn/dx)= 1miform dpldx). On the other hand, if gradients of net doping are allowed n the base, then (dn/dx) # (dpldx), as van Vliet has emphasized[9]. Such subtle distinctions can be quite important analytically. The examples offered in Tables I and 2 are not in any sense a “complete set.” Nonetheless they may serve as a kind of framework against which other specific cases may be positioned for comparison and clarification in future device modeling work, in much the manner of the cases in the three paragraphs immediately above.
Acknowledgements-For extremely helpful debted to Mr. R. P. Jindal, Dr. B. L. Grung, Vliet.
comments I am inand Prof. K. M. van
Dept. of E/e&Cal Engineering Uniuersiry of Minnesota Mineapolis MN 55455, U.S.A.
R. M. WARNER,JR.
REFERENCES I. W. van Roosbroeck, Phys. Rev. 91,282 (1953). 2. B. L. Grung and R. M. Warner, Jr., Solid-St. Electron. 20, 753
(1977). 3. E. S. Rittner. 4. 5. 6. 7. 8. 9.
Phys. Rev. 94, 1161 (1954). N. H. Fletcher, J. Electron. 2, 609 (1957). W. Shockley, BSTJ 28,435 (1949). J. R. Hauser, Solid-9. Electron. 14, 133 (1971). A van der Ziel, Solid&. Electron. 16, 1509 (1973). W. M. Webster, Proc. IRE 42, 914 (1954). K. M. van Vliet, Solid-St. Electron. 9, I85 (1%6).