Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown
Effect of magnetic field (up to 14 T) on current-voltage characteristics of silicon n⁺ -p diodes which manifests hysteresis loops related with low-temperature impurity breakdown has been studied. With growth of magnetic field, the hysteresis loops are narrowed and decreased in amplitude and then...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2012
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| Цитувати: | Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown / A.B. Aleinikov, V.A. Berezovets, V.L. Borblik, M.M. Shwarts, Yu.M. Shwarts // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 288-293. — Бібліогр.: 19 назв. — англ. |
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Aleinikov, A.B. Berezovets, V.A. Borblik, V.L. Shwarts, M.M. Shwarts, Yu.M. 2017-05-29T18:05:18Z 2017-05-29T18:05:18Z 2012 Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown / A.B. Aleinikov, V.A. Berezovets, V.L. Borblik, M.M. Shwarts, Yu.M. Shwarts // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 288-293. — Бібліогр.: 19 назв. — англ. 1560-8034 PACS 85.30.Kk https://nasplib.isofts.kiev.ua/handle/123456789/118325 Effect of magnetic field (up to 14 T) on current-voltage characteristics of silicon n⁺ -p diodes which manifests hysteresis loops related with low-temperature impurity breakdown has been studied. With growth of magnetic field, the hysteresis loops are narrowed and decreased in amplitude and then disappear, but the breakdown continues in a soft form. Planar design of the diode has allowed separating the influence of magnetic field on mobility of the carriers executing impact ionization of the impurities and on the ionization energy itself. Theoretical analysis of the experimental data permitted us to determine the dependence of the ionization energy on the magnetic field. As in other investigated semiconductors, our results demonstrate the dependence of B¹/³ type. A model capable to explain qualitatively the mechanism of suppression of the hysteresis loops by magnetic field is proposed as well. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown Article published earlier |
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| title |
Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown |
| spellingShingle |
Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown Aleinikov, A.B. Berezovets, V.A. Borblik, V.L. Shwarts, M.M. Shwarts, Yu.M. |
| title_short |
Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown |
| title_full |
Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown |
| title_fullStr |
Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown |
| title_full_unstemmed |
Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown |
| title_sort |
effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown |
| author |
Aleinikov, A.B. Berezovets, V.A. Borblik, V.L. Shwarts, M.M. Shwarts, Yu.M. |
| author_facet |
Aleinikov, A.B. Berezovets, V.A. Borblik, V.L. Shwarts, M.M. Shwarts, Yu.M. |
| publishDate |
2012 |
| language |
English |
| container_title |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| format |
Article |
| description |
Effect of magnetic field (up to 14 T) on current-voltage characteristics of
silicon n⁺
-p diodes which manifests hysteresis loops related with low-temperature
impurity breakdown has been studied. With growth of magnetic field, the hysteresis
loops are narrowed and decreased in amplitude and then disappear, but the breakdown
continues in a soft form. Planar design of the diode has allowed separating the influence
of magnetic field on mobility of the carriers executing impact ionization of the impurities
and on the ionization energy itself. Theoretical analysis of the experimental data
permitted us to determine the dependence of the ionization energy on the magnetic field.
As in other investigated semiconductors, our results demonstrate the dependence of B¹/³
type. A model capable to explain qualitatively the mechanism of suppression of the
hysteresis loops by magnetic field is proposed as well.
|
| issn |
1560-8034 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118325 |
| citation_txt |
Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown / A.B. Aleinikov, V.A. Berezovets, V.L. Borblik, M.M. Shwarts, Yu.M. Shwarts // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 288-293. — Бібліогр.: 19 назв. — англ. |
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| first_indexed |
2025-11-24T16:30:18Z |
| last_indexed |
2025-11-24T16:30:18Z |
| _version_ |
1850486499603120128 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 288-293.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
288
PACS 85.30.Kk
Effect of magnetic field on hysteretic characteristics of silicon diodes
under conditions of low-temperature impurity breakdown
A.B. Aleinikov1,3, V.A. Berezovets2,3, V.L. Borblik1, M.M. Shwarts1, Yu.M. Shwarts1,3
1V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Nauky, 03028 Kyiv, Ukraine;
Phone/fax: +38 (044) 525-7463; e-mail: borblik@isp.kiev.ua
2Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 Saint-Petersburg, Russia
3International Laboratory of High Magnetic Fields and Low Temperatures,
53-421, Wroclaw, Poland
Abstract. Effect of magnetic field (up to 14 T) on current-voltage characteristics of
silicon n+-p diodes which manifests hysteresis loops related with low-temperature
impurity breakdown has been studied. With growth of magnetic field, the hysteresis
loops are narrowed and decreased in amplitude and then disappear, but the breakdown
continues in a soft form. Planar design of the diode has allowed separating the influence
of magnetic field on mobility of the carriers executing impact ionization of the impurities
and on the ionization energy itself. Theoretical analysis of the experimental data
permitted us to determine the dependence of the ionization energy on the magnetic field.
As in other investigated semiconductors, our results demonstrate the dependence of B1/3-
type. A model capable to explain qualitatively the mechanism of suppression of the
hysteresis loops by magnetic field is proposed as well.
Keywords: p–n diode, silicon, low temperature, impurity breakdown, hysteresis,
magnetic field.
Manuscript received 23.07.12; revised version received 27.08.12; accepted for
publication 10.09.12; published online 25.09.12.
1. Introduction
In silicon, as in a number of other semiconductors doped
with shallow impurities, the concentration of which is
below than the critical one for insulator-metal transition
(IMT), freezing-out of free current carriers takes place at
low temperatures. In relation with this fact, low-
temperature silicon conduction has hopping character
and in electric fields of a sufficient amplitude, impact
ionization of these impurities can occur, which is
accompanied, in a number of cases, by hysteresis loops
in the current-voltage characteristics (CVC). These
hysteresis loops also manifests themselves in devices
based on silicon: in diodes [1, 2], in field-effect
transistors [3].
In Ref. [1], effect of magnetic field on impurity
breakdown in commercial silicon diodes 1N4001-4005
was studied, where switching characteristics were
observed within the temperature range of 15 down to
7 K. The magnetic field (as low as 0.14 T) perpendicular
to the current suppressed the hysteresis loops.
Conclusive interpretation of the effect was absent; the
higher magnetic fields were not investigated.
Numerous investigations of magnetic field
influence on low-temperature impurity breakdown in p-
Ge and n-GaAs (see, for example, papers [4-6]) were
concentrated on problems of beginning spontaneous
current oscillations under these conditions and their
transition to chaos. It is talked, as a rule, about low
magnetic fields and slightly doped materials in the form
of resistors. Only in papers [7, 8], the effect of magnetic
field on hysteresis itself was described for n-GaAs.
In this paper, we report about measurements of
low-temperature characteristics of experimental silicon
n+-p diode that exhibits switching effect at temperatures
below 27 K (measurements were performed down to
1.7 K) in magnetic fields up to 14 Т. Design of the
investigated diode was planar, with circular geometry of
layers; the substrate was (111)-oriented Si. The doping
level of the diode base (by boron) was close to the
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 288-293.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
289
critical value for IMT but remained on its insulator side;
therefore at low temperatures, freezing-out of free
carriers into impurities took place. Theoretical analysis
of the experimental results has allowed determination of
the magnetic-field dependence of ionization energy of
boron impurity in silicon at high doping level. A
qualitative model for explanation of magnetic
suppression of hysteresis loops in the diode CVC is
given as well.
2. Experimental data
The results of measurements of the diode CVCs at 4.2 K
are represented in Fig. 1: for magnetic field oriented
perpendicular to the diode structure plane (a) and for its
orientation in parallel to the diode structure plane (b).
The lower parts of Fig. 1 illustrate a character of motion
for free electrons which are available in small number
even at low temperatures and execute the impact
ionization of the impurities. Note that under orientation
of magnetic field parallel to the structure plane, a portion
of the current lines is available, which coincides in
direction with the magnetic field and, therefore, do not
suffer action of the Lorentz force.
Under magnetic field orientation perpendicular to
the structure plane, hysteresis loops in CVC narrow with
growth of magnetic field decreasing in the amplitude and
then disappear. The breakdown, however, remains in a
soft form (corresponding thresholds are marked by bold
points on the curves), and later it disappears as well.
Under magnetic field orientation parallel to the structure
plane, character of its action is substantially different.
Disappearance of the hysteresis loops takes place earlier,
in lower magnetic fields, but the soft breakdown, in
contrast to previous case, remains up to the highest
reached fields (14 T).
As it is known, there are two main mechanisms of
magnetic field action on the effect of impact ionization:
1) curling trajectories of carriers executing impact
ionization that manifests itself in decreasing their
mobility and 2) compressing wave functions of carriers
localized on impurity center that manifests itself in the
increasing ionization energy of the center (effect of
magnetic freezing-out)1. It is obvious that in the case of
perpendicular orientation, the first mechanism
(decreasing electron mobility) dominates: in high
magnetic fields, this decreasing is so significant that the
breakdown disappears at all. In the case of parallel
orientation, electron mobility does not vary with
magnetic field; therefore the breakdown (in soft form)
remains up to the highest fields reached.
1 Note that, in contrast to the described situation, in semiconductors
with a very low ionization energy of impurities, magnetic field
resulting in freezing-out of free carriers creates preconditions for
appearance of impact ionization [9].
1600 2000 2400 2800 3200 3600
0,01
0,1
1
10
100
1000
3
4
6
I,
A
U, mV
GPI-153
4.2 K
0 - 14 T
0
1
2 5 14
a
B
1600 1800 2000 2200 2400 2600 2800 3000 3200
0,01
0,1
1
10
100
9108765432 1410.60.40
U, mV
I,
A
GPI-153
4.2 K
0 - 14 T
b
B
Fig. 1. Current-voltage characteristics of investigated diode at 4.2 K against magnetic induction as a parameter (numbers near the
curves are its magnitude in Tesla): (а) – magnetic field is perpendicular to the diode structure plane, (b) – magnetic field is
parallel to it; bold points on the curves indicate the thresholds of impact ionization.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 288-293.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
290
0 2 4 6 8 10 12 14
2,0
2,2
2,4
2,6
2,8
3,0
3,2
bU
b
,
V
B, T
a
Fig. 2. Dependences of the threshold breakdown voltages on
the magnetic field; (a), (b) are the same as in Fig. 1.
The values of breakdown voltages for both
magnetic field orientations are shown in Fig. 2 as a
function of magnetic induction B. A qualitatively similar
picture was observed in Ge samples (with negligible
hysteresis loops) [10].
3. Analysis of the experimental results
3.1. Determination of breakdown electric fields in the
diode base
The breakdown voltages represented in Fig. 2 are the
total diode voltages. For quantitative analysis of
ionization processes in the freezed-out base of the diode,
it is necessary to separate out (in the total voltage) a
portion of the voltage drop across the base. Thereto, we
used our previously developed method of extraction
from diode CVC of the series resistance generated under
freezing-out of the diode base [11, 12]. As it was shown
in [12] (where the same diode has been analyzed) the
voltage drop across the base under the breakdown at
4.2 K constituted 1.18 V; meanwhile, the total voltage
drop across the diode was 2.4 V. The difference of
1.22 V which falls on the n+-p junction was subtracted
from all the points in Fig. 2.
The breakdown voltages baseU of the base obtained
in this way (as a function of magnetic induction) were
used then for calculation of the electric field in the base
under breakdown. Due to circular symmetry of the diode
(Fig. 3), the electric field in the base is a function of
radius and its voltage drop 112
1
1
ln rrr
U
E base , when the
inner radius 1r is maximal. It was this value that was
used for analysis of impact ionization in the diode base.
Calculated dependences of the breakdown electric fields
Eb on magnetic induction B are shown for both magnetic
field orientations in Fig. 4 (as experimental points).
Fig. 3. Scheme of the diode structure in plan.
0 2 4 6 8 10 12 14
1600
1800
2000
2200
2400
2600
2800
3000
3200
E
B
,
V
/c
m
B, T
a
b
Fig. 4. Dependences of the breakdown electric fields in the
diode base on the magnetic field; (a), (b) are the same as in
Fig. 1.
3.2. Determination of impurity ionization energy
depending on magnetic field
Semi-empirical theory [13] for impact ionization of
shallow impurities in silicon gives the following
expression for the impact ionization coefficient:
xx
EAii
33.1
exp
1
const)(
2/3
, (1)
where E is the electric field,
ikTmEx 0
2221 , m – effective mass of
carriers executing impact ionization, )(E – their
mobility, k – the Boltzmann constant, 0T – lattice
temperature, i – ionization energy of the impurity
center depending on the magnetic field. The parameter
describes increase in the effective carrier temperature
effT at the expense of its drift motion in electric field:
2
0 deff vmTkTk .
Ubase
r1
n+
r2
p
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 288-293.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
291
At small values of x, the coefficient )(EAii is
exponentially small. Therefore, it is naturally to believe
that impact ionization takes place at x close to 1. In
presence of the magnetic field B, the condition x = 1
takes the form
)(
1
)(
2
1
022
2
2 BkT
B
mBE ib
, (2)
where )(BEb is the breakdown electric field and
influence of magnetic field on both electron mobility
and ionization energy i has been taken into account.
The phenomenological dependence of electron mobility
on the electric field was used in the form
satvE00 1 , where 0 is the mobility in low
electric fields, satv is the saturation drift velocity equal
to107 cm/s for Si [13].
Under perpendicular orientation of the magnetic
field, the effect of curling electron trajectories is the
main one. The obtained from (2) dependence of the
breakdown electric field Eb on the magnetic field
induction B (at εi = const = 0.025 eV2 and m = 0.26 m0)
agrees in the best way with experimental points at 0 =
1840 cm2/V·s and 4.28
2
1
(dashed curve in Fig. 4).
Under parallel orientation of magnetic field, the
electron mobility does not vary with magnetic field. For
this case, the equation (2) takes the form
)(
1
)(
2
1
0
2
0
02 BkT
vE
mBE i
satb
b
, (3)
wherefrom, using the experimental dependence )(BEb
for parallel orientation and abovementioned values of
0 and
2
1
, the dependence of )(Bi can be
determined. It is shown in Fig. 5.
It follows from Fig. 5 that the ionization energy of
boron impurity in silicon (at a given doping level)
increases in magnetic field of 14 T by the factor close to
1.54. Existing theoretical calculations of this dependence
has been carried out in the framework of the hydrogenic
impurity model with the scalar effective mass and
parabolic dispersion law of carriers in the bands. In
particular, results of these calculations have been
represented [14] in the form of the dependence of an
impurity center ground state on the dimensionless
parameter
R
c
2
that is the ratio of the characteristic
carrier energy in the lowest Landau band 2/c to the
2 Value of the ionization energy used here is equal to the activation
energy of Hall effect in silicon resistor identical completely (in its
properties) to the base layer of the investigated diode.
energy of effective Rydberg 224 2/ emR h for the
acceptor center. Here, c is the cyclotron frequency,
– Plank constant (divided by 2π), e – elementary
charge, – dielectric constant. In the given case, at B =
14 T = 0.34 (the light hole effective mass of 0.16m0 is
used here). According to the theory [14], increase in the
ground state energy by the factor 1.29 corresponds to
this value that is somewhat less than it is follows from
the analysis of the experimental data. The reasons of
disagreement are, obviously, both roughness of the
analysis and imperfection of the theory. Though the
results of analysis demonstrate the commonly observed
in similar cases [15-17] dependence on the magnetic
field of B1/3-type (the insert in Fig. 5).
0 2 4 6 8 10 12 14
0,022
0,024
0,026
0,028
0,030
0,032
0,034
0,036
0,038
0,040
1.0 1.5 2.0 2.5
0.024
0.026
0.028
0.030
0.032
0.034
0.036
0.038
0.040
i ,
e
V
B1/3, T1/3
i ,
e
V
B, T
Fig. 5. Calculated from the experimental data dependence of
the ionization energy of boron impurity in silicon (at given
doping level) on the magnetic field.
Fig. 6. Model for the density of states in the diode base which
explains a qualitatively possible mechanism providing
suppression of the hysteresis loops by magnetic field.
ε i2
N(ε)
Ev
ε
ε
N(ε)
εi
B≈ 1 T
44 meV
Ev
Г8
ε i1= 25 meV
12 meV
p-Si
Г6
B= 0
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 288-293.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
292
3.3. Interpretation of suppression of the hysteresis loops
by magnetic field
Generally accepted treatment of the hysteresis
mechanism in low-temperature impurity breakdown is
based on taking into account (in ionization and
recombination processes) another (one) middle energy
level – as a rule, of the first excited state inherent to the
impurity center. Impact ionization of the carriers
accumulated on this level takes place at a lower applied
voltage. This fact results in existence of holding electric
field.
However, in our case, due to a high enough doping
level (close to IMT) all the energy levels are
substantially fuzzy. Theoretical analysis of the
fluctuation potential amplitude was carried out within
the limits of light doping and heavy doping [18].
Concentration region near the IMT but on its insulator
side, as far as we know, has not been considered
theoretically. We used here, for assessment of the
fluctuation potential amplitude , more suitable for our
case, in our view, formulas of the heavy-doping limit
when 2/13
0
0
2
2 rN
r
e
A
and
ANe
Tk
r
2
2
0 4
. At
318 cm103 AN (doping level of the diode base), we
obtain meV12 that is of the same order of
magnitude as the ionization energy of the first excited
state of boron in silicon, which is assumed to be spread-
out as well. Therefore, there is a rather narrow gap in
this place of the density of states (Fig. 6, top picture)
which can be responsible for existence of the second
(lower) ionization energy and, as consequence, of the
holding voltage.
Under application of magnetic field, both states of
the first excited level and localized states in the tail of
the density of states in the valence band suffer Zeeman
effect, which, in principle, can close the abovementioned
gap (Fig. 6, bottom picture). Thereby, the lower
ionization energy is eliminated, and it also eliminates
hysteresis.
As one of arguments for this model, one can point
to results of the paper [19], where authors have
measured Zeeman effect in Si(:B) for different
orientations of magnetic field relatively to
crystallographic axes. When B||[111], the effect was
substantially less than for other field orientations. In our
case, the effect of magnetic field on the hysteresis loops
is less just under perpendicular orientation of the
magnetic field relatively to the (111)-oriented diode
structure, i.e. for B||[111] direction.
4. Conclusion
So, although silicon is the main material of
semiconductor electronics, dependence of the ionization
energy of boron impurity in it on magnetic field, as far
as we know, was not determined so far, all the more – at
high doping level (close to Mott transition). And the
suppression of the hysteresis loops in the diode CVC by
magnetic field can be qualitatively explained by Zeeman
effect on localized states taking into account a
fluctuation potential connected with heavy doping.
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InSb // Phys. Rev. Lett. 18(19), p. 773-775 (1967).
16. L.A. Kaufman and L.J. Neuringer, Magnetic
freeze-out and band tailing in n-InAs // Phys. Rev.
B, 2(6), p. 1840-1846 (1970).
17. T.O. Poehler, Magnetic freeze-out and impact
ionization in GaAs // Phys. Rev. B, 4(4), p. 1223-
1229 (1971).
18. B.I. Shklovskii and A.L. Efros, The Electronic
Properties of Doped Semiconductors. Nauka,
Moscow, 1979 (in Russian).
19. F. Merlet, B. Pajot, Ph. Arcas, and A.M. Jean-
Louis, Experimental study of the Zeeman splitting
of boron levels in silicon // Phys. Rev. B, 12(8),
p. 3297-3317 (1975).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 288-293.
PACS 85.30.Kk
Effect of magnetic field on hysteretic characteristics of silicon diodes under conditions of low-temperature impurity breakdown
A.B. Aleinikov1,3, V.A. Berezovets2,3, V.L. Borblik1, M.M. Shwarts1, Yu.M. Shwarts1,3
1V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Nauky, 03028 Kyiv, Ukraine;
Phone/fax: +38 (044) 525-7463; (e-mail: borblik@isp.kiev.ua
2Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 Saint-Petersburg, Russia
3International Laboratory of High Magnetic Fields and Low Temperatures,
53-421, Wroclaw, Poland
Abstract. Effect of magnetic field (up to 14 T) on current-voltage characteristics of silicon n+-p diodes which manifests hysteresis loops related with low-temperature impurity breakdown has been studied. With growth of magnetic field, the hysteresis loops are narrowed and decreased in amplitude and then disappear, but the breakdown continues in a soft form. Planar design of the diode has allowed separating the influence of magnetic field on mobility of the carriers executing impact ionization of the impurities and on the ionization energy itself. Theoretical analysis of the experimental data permitted us to determine the dependence of the ionization energy on the magnetic field. As in other investigated semiconductors, our results demonstrate the dependence of B1/3-type. A model capable to explain qualitatively the mechanism of suppression of the hysteresis loops by magnetic field is proposed as well.
Keywords: p–n diode, silicon, low temperature, impurity breakdown, hysteresis, magnetic field.
Manuscript received 23.07.12; revised version received 27.08.12; accepted for publication 10.09.12; published online 25.09.12.
1. Introduction
In silicon, as in a number of other semiconductors doped with shallow impurities, the concentration of which is below than the critical one for insulator-metal transition (IMT), freezing-out of free current carriers takes place at low temperatures. In relation with this fact, low-temperature silicon conduction has hopping character and in electric fields of a sufficient amplitude, impact ionization of these impurities can occur, which is accompanied, in a number of cases, by hysteresis loops in the current-voltage characteristics (CVC). These hysteresis loops also manifests themselves in devices based on silicon: in diodes [1, 2], in field-effect transistors [3].
In Ref. [1], effect of magnetic field on impurity breakdown in commercial silicon diodes 1N4001-4005 was studied, where switching characteristics were observed within the temperature range of 15 down to 7 K. The magnetic field (as low as 0.14 T) perpendicular to the current suppressed the hysteresis loops. Conclusive interpretation of the effect was absent; the higher magnetic fields were not investigated.
Numerous investigations of magnetic field influence on low-temperature impurity breakdown in p-Ge and n-GaAs (see, for example, papers [4-6]) were concentrated on problems of beginning spontaneous current oscillations under these conditions and their transition to chaos. It is talked, as a rule, about low magnetic fields and slightly doped materials in the form of resistors. Only in papers [7, 8], the effect of magnetic field on hysteresis itself was described for n-GaAs.
In this paper, we report about measurements of low-temperature characteristics of experimental silicon n+-p diode that exhibits switching effect at temperatures below 27 K (measurements were performed down to 1.7 K) in magnetic fields up to 14 Т. Design of the investigated diode was planar, with circular geometry of layers; the substrate was (111)-oriented Si. The doping level of the diode base (by boron) was close to the critical value for IMT but remained on its insulator side; therefore at low temperatures, freezing-out of free carriers into impurities took place. Theoretical analysis of the experimental results has allowed determination of the magnetic-field dependence of ionization energy of boron impurity in silicon at high doping level. A qualitative model for explanation of magnetic suppression of hysteresis loops in the diode CVC is given as well.
160020002400280032003600
0,01
0,1
1
10
100
1000
3
4
6
I,
A
U, mV
GPI-153
4.2 K
0 - 14 T
0
1
2
5
14
a
B
2. Experimental data
The results of measurements of the diode CVCs at 4.2 K are represented in Fig. 1: for magnetic field oriented perpendicular to the diode structure plane (a) and for its orientation in parallel to the diode structure plane (b). The lower parts of Fig. 1 illustrate a character of motion for free electrons which are available in small number even at low temperatures and execute the impact ionization of the impurities. Note that under orientation of magnetic field parallel to the structure plane, a portion of the current lines is available, which coincides in direction with the magnetic field and, therefore, do not suffer action of the Lorentz force.
Under magnetic field orientation perpendicular to the structure plane, hysteresis loops in CVC narrow with growth of magnetic field decreasing in the amplitude and then disappear. The breakdown, however, remains in a soft form (corresponding thresholds are marked by bold points on the curves), and later it disappears as well. Under magnetic field orientation parallel to the structure plane, character of its action is substantially different. Disappearance of the hysteresis loops takes place earlier, in lower magnetic fields, but the soft breakdown, in contrast to previous case, remains up to the highest reached fields (14 T).
As it is known, there are two main mechanisms of magnetic field action on the effect of impact ionization: 1) curling trajectories of carriers executing impact ionization that manifests itself in decreasing their mobility and 2) compressing wave functions of carriers localized on impurity center that manifests itself in the increasing ionization energy of the center (effect of magnetic freezing-out)
. It is obvious that in the case of perpendicular orientation, the first mechanism (decreasing electron mobility) dominates: in high magnetic fields, this decreasing is so significant that the breakdown disappears at all. In the case of parallel orientation, electron mobility does not vary with magnetic field; therefore the breakdown (in soft form) remains up to the highest fields reached.
0
2
4
6
8
10
12
14
2,0
2,2
2,4
2,6
2,8
3,0
3,2
b
U
b
, V
B, T
a
Fig. 2. Dependences of the threshold breakdown voltages on the magnetic field; (a), (b) are the same as in Fig. 1.
The values of breakdown voltages for both magnetic field orientations are shown in Fig. 2 as a function of magnetic induction B. A qualitatively similar picture was observed in Ge samples (with negligible hysteresis loops) [10].
3. Analysis of the experimental results
3.1. Determination of breakdown electric fields in the diode base
The breakdown voltages represented in Fig. 2 are the total diode voltages. For quantitative analysis of ionization processes in the freezed-out base of the diode, it is necessary to separate out (in the total voltage) a portion of the voltage drop across the base. Thereto, we used our previously developed method of extraction from diode CVC of the series resistance generated under freezing-out of the diode base [11, 12]. As it was shown in [12] (where the same diode has been analyzed) the voltage drop across the base under the breakdown at 4.2 K constituted 1.18 V; meanwhile, the total voltage drop across the diode was 2.4 V. The difference of 1.22 V which falls on the n+-p junction was subtracted from all the points in Fig. 2.
The breakdown voltages
base
U
of the base obtained in this way (as a function of magnetic induction) were used then for calculation of the electric field in the base under breakdown. Due to circular symmetry of the diode (Fig. 3), the electric field in the base is a function of radius and its voltage drop
(
)
1
1
2
1
1
ln
r
r
r
U
E
base
=
, when the inner radius
1
r
is maximal. It was this value that was used for analysis of impact ionization in the diode base. Calculated dependences of the breakdown electric fields Eb on magnetic induction B are shown for both magnetic field orientations in Fig. 4 (as experimental points).
Fig. 3. Scheme of the diode structure in plan.
0
2
4
6
8
10
12
14
1600
1800
2000
2200
2400
2600
2800
3000
3200
E
B
, V/cm
B, T
a
b
Fig. 4. Dependences of the breakdown electric fields in the diode base on the magnetic field; (a), (b) are the same as in Fig. 1.
3.2. Determination of impurity ionization energy depending on magnetic field
Semi-empirical theory [13] for impact ionization of shallow impurities in silicon gives the following expression for the impact ionization coefficient:
÷
ø
ö
ç
è
æ
-
=
x
x
E
A
ii
33
.
1
exp
1
const
)
(
2
/
3
,
(1)
where
E
is the electric field,
(
)
[
]
i
kT
m
E
x
e
+
m
+
a
=
0
2
2
2
1
,
m
– effective mass of carriers executing impact ionization,
)
(
E
m
– their mobility,
k
– the Boltzmann constant,
0
T
– lattice temperature,
i
e
– ionization energy of the impurity center depending on the magnetic field. The parameter
a
describes increase in the effective carrier temperature
eff
T
at the expense of its drift motion in electric field:
2
0
d
eff
v
m
T
k
T
k
a
+
=
.
At small values of x, the coefficient
)
(
E
A
ii
is exponentially small. Therefore, it is naturally to believe that impact ionization takes place at x close to 1. In presence of the magnetic field B, the condition x = 1 takes the form
(
)
[
]
)
(
1
)
(
2
1
0
2
2
2
2
B
kT
B
m
B
E
i
b
e
=
+
m
+
m
÷
ø
ö
ç
è
æ
+
a
,
(2)
where
)
(
B
E
b
is the breakdown electric field and influence of magnetic field on both electron mobility
m
and ionization energy
i
e
has been taken into account. The phenomenological dependence of electron mobility on the electric field was used in the form
(
)
sat
v
E
0
0
1
m
+
m
=
m
, where
0
m
is the mobility in low electric fields,
sat
v
is the saturation drift velocity equal to107 cm/s for Si [13].
Under perpendicular orientation of the magnetic field, the effect of curling electron trajectories is the main one. The obtained from (2) dependence of the breakdown electric field Eb on the magnetic field induction B (at εi = const = 0.025 eV
and m = 0.26 m0) agrees in the best way with experimental points at
0
m
= 1840 cm2/V·s and
4
.
28
2
1
=
+
a
(dashed curve in Fig. 4).
Under parallel orientation of magnetic field, the electron mobility does not vary with magnetic field. For this case, the equation (2) takes the form
)
(
1
)
(
2
1
0
2
0
0
2
B
kT
v
E
m
B
E
i
sat
b
b
e
=
+
÷
÷
ø
ö
ç
ç
è
æ
m
+
m
÷
ø
ö
ç
è
æ
+
a
,
(3)
wherefrom, using the experimental dependence
)
(
B
E
b
for parallel orientation and abovementioned values of
0
m
and
2
1
+
a
, the dependence of
)
(
B
i
e
can be determined. It is shown in Fig. 5.
It follows from Fig. 5 that the ionization energy of boron impurity in silicon (at a given doping level) increases in magnetic field of 14 T by the factor close to 1.54. Existing theoretical calculations of this dependence has been carried out in the framework of the hydrogenic impurity model with the scalar effective mass and parabolic dispersion law of carriers in the bands. In particular, results of these calculations have been represented [14] in the form of the dependence of an impurity center ground state on the dimensionless parameter
R
c
2
w
=
g
h
that is the ratio of the characteristic carrier energy in the lowest Landau band
2
/
c
w
h
to the energy of effective Rydberg
2
2
4
2
/
k
=
h
e
m
R
h
for the acceptor center. Here,
c
w
is the cyclotron frequency,
h
– Plank constant (divided by 2π),
e
– elementary charge,
k
– dielectric constant. In the given case, at B = 14 T
g
= 0.34 (the light hole effective mass of 0.16m0 is used here). According to the theory [14], increase in the ground state energy by the factor 1.29 corresponds to this
g
value that is somewhat less than it is follows from the analysis of the experimental data. The reasons of disagreement are, obviously, both roughness of the analysis and imperfection of the theory. Though the results of analysis demonstrate the commonly observed in similar cases [15-17] dependence on the magnetic field of B1/3-type (the insert in Fig. 5).
0
2
4
6
8
10
12
14
0,022
0,024
0,026
0,028
0,030
0,032
0,034
0,036
0,038
0,040
1.0
1.5
2.0
2.5
0.024
0.026
0.028
0.030
0.032
0.034
0.036
0.038
0.040
e
i
, eV
B
1/3
, T
1/3
e
i
, eV
B, T
Fig. 5. Calculated from the experimental data dependence of the ionization energy of boron impurity in silicon (at given doping level) on the magnetic field.
Fig. 6. Model for the density of states in the diode base which explains a qualitatively possible mechanism providing suppression of the hysteresis loops by magnetic field.
3.3. Interpretation of suppression of the hysteresis loops by magnetic field
Generally accepted treatment of the hysteresis mechanism in low-temperature impurity breakdown is based on taking into account (in ionization and recombination processes) another (one) middle energy level – as a rule, of the first excited state inherent to the impurity center. Impact ionization of the carriers accumulated on this level takes place at a lower applied voltage. This fact results in existence of holding electric field.
However, in our case, due to a high enough doping level (close to IMT) all the energy levels are substantially fuzzy. Theoretical analysis of the fluctuation potential amplitude was carried out within the limits of light doping and heavy doping [18]. Concentration region near the IMT but on its insulator side, as far as we know, has not been considered theoretically. We used here, for assessment of the fluctuation potential amplitude
G
, more suitable for our case, in our view, formulas of the heavy-doping limit when
(
)
2
/
1
3
0
0
2
2
r
N
r
e
A
k
p
=
G
and
A
N
e
T
k
r
2
2
0
4
p
k
=
. At
3
18
cm
10
3
-
×
=
A
N
(doping level of the diode base), we obtain
meV
12
»
G
that is of the same order of magnitude as the ionization energy of the first excited state of boron in silicon, which is assumed to be spread-out as well. Therefore, there is a rather narrow gap in this place of the density of states (Fig. 6, top picture) which can be responsible for existence of the second (lower) ionization energy and, as consequence, of the holding voltage.
Under application of magnetic field, both states of the first excited level and localized states in the tail of the density of states in the valence band suffer Zeeman effect, which, in principle, can close the abovementioned gap (Fig. 6, bottom picture). Thereby, the lower ionization energy is eliminated, and it also eliminates hysteresis.
As one of arguments for this model, one can point to results of the paper [19], where authors have measured Zeeman effect in Si(:B) for different orientations of magnetic field relatively to crystallographic axes. When B||[111], the effect was substantially less than for other field orientations. In our case, the effect of magnetic field on the hysteresis loops is less just under perpendicular orientation of the magnetic field relatively to the (111)-oriented diode structure, i.e. for B||[111] direction.
4. Conclusion
So, although silicon is the main material of semiconductor electronics, dependence of the ionization energy of boron impurity in it on magnetic field, as far as we know, was not determined so far, all the more – at high doping level (close to Mott transition). And the suppression of the hysteresis loops in the diode CVC by magnetic field can be qualitatively explained by Zeeman effect on localized states taking into account a fluctuation potential connected with heavy doping.
References
1.
R.V. Aldridge, K. Davis, and M. Holloway, An investigation of the effect of a magnetic field on the forward characteristics of some silicon diodes at low temperatures // J. Phys. D, 8(1), p. 64-68 (1975).
2.
E. Simoen, B. Dierickx, L. Deferm, and C. Claeys, The behavior of silicon p-n junction based devices at liquid helium temperatures // J. Appl. Phys. 70(2), p. 1016-1024 (1991).
3.
B. Dierickx, L. Warmerdam, E. Simoen, J. Wermeiren, and C. Claeys, Model for hysteresis and kink behavior of MOS transistors operating at 4.2 K // IEEE Trans. ED-35(7), p. 1120-1125 (1988).
4.
R. Richter, A. Kittel, G. Heinz, G. Flatgen, J. Peinke, and J. Parisi, Type-I intermittency in semiconductor breakdown: An experimental confirmation // Phys. Rev. B, 49(13), p. 8738-8746 (1994).
5.
J. Spangler, U. Margull, and W. Prettl, Regular and chaotic current oscillations in n-type GaAs in transverse and longitudinal magnetic fields // Phys. Rev. B, 45(20), p. 12137-12140 (1992).
6.
S-Y.T. Tzeng and Y. Tzeng, Two-level model of longitudinal magnetic field-induced current instability and chaos in n-GaAs // Phys. Rev. B, 72, 205201(1-7), (2005).
7.
K. Aoki, T. Kondo, and T. Watanabe, Cross-over instability and chaos of hysteretic I-V curve during impurity avalanche breakdown in n-GaAs under longitudinal magnetic field // Solid State Communs. 77(1), p. 91-94 (1991).
8.
V.A. Samuilov, V.K. Ksenevich, G. Remenyi, G. Kiss, and B. Podor, Impact ionization breakdown of n-GaAs in high magnetic field // Semicond. Sci. Technol. 14(12), p. 1084-1087 (1999).
9.
R.J. Phelan and W.F. Love, Negative resistance and impact ionization impurities in n-type indium antimonide // Phys. Rev. 133(4A), p. A1134-A1137 (1964).
10.
T.O. Poehler and J.R. Apel, Impurity ionization in germanium in strong magnetic fields // Phys. Rev. B, 1(8), p. 3240-3244 (1970).
11.
V.L. Borblik, Yu.M. Shwarts, M.M. Shwarts, A new method of extraction of a p-n diode series resistance from I-V characteristics and its application to analysis of low-temperature conduction of the diode base // Semiconductor Physics, Quantum Electronics & Optoelectronics, 12(3), p. 339-342 (2009).
12.
V.L. Borblik, Yu.M. Shwarts, M.M. Shwarts, and A.M. Fonkich, Concerning the nature of relaxation oscillations in silicon diodes in the cryogenic temperature region // Cryogenics. 50(6-7), p. 417-420 (2010).
13.
B. Dierickx, E. Simoen, and G. Declerck, Transient response of silicon devices at 4.2 K: I. Theory // Semicond. Sci. Technol. 6(9), p. 896-904 (1991).
14.
D.M. Larsen, Shallow donor levels of InSb in a magnetic field // J. Phys. Chem. Sol. 29(2), p. 271-280 (1968).
15.
O. Beckman, E. Hanamura, and L.J. Neuringer, Quantum limit galvanomagnetic phenomena in n-InSb // Phys. Rev. Lett. 18(19), p. 773-775 (1967).
16.
L.A. Kaufman and L.J. Neuringer, Magnetic freeze-out and band tailing in n-InAs // Phys. Rev. B, 2(6), p. 1840-1846 (1970).
17.
T.O. Poehler, Magnetic freeze-out and impact ionization in GaAs // Phys. Rev. B, 4(4), p. 1223-1229 (1971).
18.
B.I. Shklovskii and A.L. Efros, The Electronic Properties of Doped Semiconductors. Nauka, Moscow, 1979 (in Russian).
19.
F. Merlet, B. Pajot, Ph. Arcas, and A.M. Jean-Louis, Experimental study of the Zeeman splitting of boron levels in silicon // Phys. Rev. B, 12(8), p. 3297-3317 (1975).
� �
Fig. 1. Current-voltage characteristics of investigated diode at 4.2 K against magnetic induction as a parameter (numbers near the curves are its magnitude in Tesla): (а) – magnetic field is perpendicular to the diode structure plane, (b) – magnetic field is parallel to it; bold points on the curves indicate the thresholds of impact ionization.
Ubase
r1
n+
r2
p
ε i2
N(ε)
Ev
ε
ε
N(ε)
εi
B≈ 1 T
44 meV
Ev
Г8
ε i1= 25 meV мэВ мэВ
12 meV
p-Si
Г6
B= 0
� Note that, in contrast to the described situation, in semiconductors with a very low ionization energy of impurities, magnetic field resulting in freezing-out of free carriers creates preconditions for appearance of impact ionization [9].
� Value of the ionization energy used here is equal to the activation energy of Hall effect in silicon resistor identical completely (in its properties) to the base layer of the investigated diode.
© 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
288
160018002000220024002600280030003200
0,01
0,1
1
10
100
910
8
7
6
5
432
14
1
0.6
0.4
0
U, mV
I,
A
GPI-153
4.2 K
0 - 14 T
b
B
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