CC BY
15Journal of Siberian Federal University. Engineering & Technologies 1 (2014 7) 20-31
УДК 004.7
Control of the Shape of Semiconductor Crystals when Growing in Czochralski Method
Sergey P. Sahanskiy*
Siberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia
Received 16.10.2013, received in revised form 14.12.2013, accepted 12.01.2014
A model of the formation temperature and the rate of withdrawal of semiconductor crystals when grown by the method of "Czochralski", which allows you to control the shape of the crystals, providing a flat solidification front and getting a quality finished product.
Keywords: model, growing, semiconductor chips.
Introduction
Base extraction methods monocrystals from a melt process "Czochralski" is that the small single-crystal seed is introduced into the melt shallow and is then slowly pulled from the melt, the melt temperature and controlling the pulling speed of the single crystal. In the process of pulling the right cone shape of the crystal, its cylindrical part and reverse cone, the control system is programmed by the software changes the drawing speed and temperature of the crystal. The process of pulling crystals from the melt requires compliance with a number of conditions that deliver quality material specified geometry. Changes in single crystal pulling rate, and degree of cooling of the melt temperature to a predetermined geometry affect crystal and largely determine the number of defects in the crystal lattice [1-3]. Therefore, for the growth of perfect single crystals brands use automated systems management, control and direction in the growth temperature, current speed and diameter of the single crystal.
Basis by pulling single crystals from the melt by the method of "Czochralski" consists in the fact that a small single-crystal seed shallow injected into the melt and then slowly pull it out of the melt by controlling the melt temperature and the rate of extraction of a single crystal. In the process of pulling the right cone shape of the crystal, its cylindrical part and reverse cone, the control system is programmed by software changes the pulling rate and the temperature of the crystal.
The control system in growing single crystals of germanium-based optical method for determining the current diameter of the crystal is shown in Fig. 1. Under source control the camera in the single crystal growth of diameter d, with a pulling speed V3 and the rotating crystal (seed) W3, and the molten metal in the crucible with an inner diameter D rotates with angular velocity WT. Computer-controlled crystal pulling speed V3, crystal rotation W3, crucible rotation WT through the appropriate drive. Control
© Siberian Federal University. All rights reserved
* Corresponding author E-mail address: [email protected]
Fig. 1. The control system in growing single crystals of germanium: 1 - rotational drive seed; 2 - move the seed drive; 3 - optical system; 4 - image converter meniscus; 5 - Temperature sensor; 6 - temperature control; 7 -computers; 8 - drive crucible rotation; 9 - stepper mo tor; 10 - stepper motor contro l ucit; 11 - enco der se ed; 12 - camera; 13 - bar; 14 - molten metal; 15 -5 ccucible; 16 - heater; 17 - pyrometer to measure rhe axial g radient in the solid crystal; 18 -a digital c omputer axial gisdient; 19 - crystal solidification front
of the temperature of ihe melt is based on the issuance of the jote te mperature T3 of the compute r (in the temperature control.
As the feedback sensor is used for temperature radiaiion pyrometer aimed at the lateral surface of the graphite heater. Information about the grown crystal diameter optical system comes with the transmitter, based on what the system is determined by the current position of the brightness of the halo of the meniscus of the crystal and the calculation of the control signal is proportional to the deviation of the diameter of the crystal grown from the set of the program. Linear axial gradient in the solid part of the growing crystal is calculated by the continuous measurement of the grown crystals additional pyrometer, a distance of 1 - 2 cm from the crystallization front of the crystal.
If the system control growing crystal from the melt by a method " Czochralski " in accordance with a predetermined shape ( geometry ) growing a single crystal pulling form rate control and the
temperature of the crystal, while ensuring crystal solidification front is close to the flat, during the whole process, it gives the ability to provide high quality of the single crystals. The closer the speed and temperature control are obtained close to the shape of the crystal for a given thermal conditions, the smaller the correction with the influence on the rate and temperature at the current deviation from the predetermined diameter to the cylindrical portion of the crystal.
To maintain a stable single-crystal crystal growth requires further smooth transition from the initial stage of pulling the seed crystal to the crystal growing direct cone, as well as to complete the transition from the direct cultivation of a cone on the cylindrical portion of the crystal. These conditions must be observed during the formation of inverted cone of the crystal.
Condition of smooth changes in the shape of crystals grown in these areas is necessary to ensure the continued growth of single-crystal germanium single crystals of large diameter (150 mm), with the provision in the final crystal minimum dislocation and lack of low-angle boundaries. Violation of a smooth transition when pulling single crystals of germanium can lead to failures of single crystal growth and the inability to obtain this type of finished product.
The shape of the grown single crystal germanium (90 mm diameter), and management of key growth parameters on the installation drawing for temperature and speed, with the mapping of the control signal, a variation of the current diameter of the set, in relative numerical units shown in Fig. 2-4.
In general, the management of the main parameters of growing single crystal of germanium is shown in Fig. 5
The form of the forward and reverse cone crystal grown in Fig. 5 has the form kosinousoidalnyh continuous lines in the areas of direct and inverted cone, with zero initial and final coupling angle to the surface of the crystal grown. This ensures a smooth transition and stability of single-crystal growth of a crystal in the transition area.
The following are the mathematical expressions for the formation of the program goals of the temperature T(x), and the rate of withdrawal Vm(x), which allow you to automate the data entry process parameters in the control system.
Management model temperature and velocity (Fig. 6) during crystal growth can be represented by the expression (1):
T(x) = F(Z, Y, Vm (x), L(x), x), (1)
where T(x) - the average temperature of the melt in the zone of the crystallization front, Vm(x) - the software the speed crystal pulling; x - coordinate movement along the axis of the crystal; L(x)- a linear axial gradient in the solid crystal; Z - vector geometry grown crystal; Y - vector thermo physical material parameters.
When growing all major semiconductor crystals (germanium, silicon, and gallium arsenide) their crystallization front of the crystal, which separates the liquid from the solid part of the melt is raised above the surface of the melt on the value of 1 - 5 cm.
If we equate the weight of the lifted weight of the column of liquid melt (to the front of crystallization), the surface tension forces acting on the circumference and height of the bar to take into account the expression of the melt through the heat balance at the interface, it is possible to obtain
Fig. 2. Bar d = 90 mm
Tsmii ( Lj)
t>-
k
in _L_L
is lO'M 0"^
uiv 05-i J0r.i))i
li. io.'ia o? JO
Fig. 3. Graph of the temperature of the heater T3 sites: 1 - growing area right cone; 2 - growing area of the cylinder; 3 - land cultivation inverted cone; 4 - section ingot annealing
409tkjQ
lOJfc.G
a,o
1014,0
VH fl Diaimn-ati'')
"f-
sAA 'jn J til
if \ V A'V J v \ iu i \
V \ \
1 - "t 11 ] ] ] 1 "T V
2«■■ io ^K-
19-lO.^H Oi JO
Fig. 4. Plot of the rate ofwithdrawal and seed V3 control signal Ay at sites: 1 - growing area right cone; 22 - growing area of the cylinder; 3 - growing area inverted cone; 4 - section ingot annealing
d,(x), T, V3n, TL
Fig. 5. Building; process parameters of growing; single crystals oil germanium: d - job diameter grown single crystal; T - software job law of Semperature change; L - job axial gradient, x -moving crystal; x1 - coordinate completion of the formation of the crystal right gene; xg - coordigare completion of the; formation of the cylindrical part of the crystal; x3 - coordinate tha completion of the formation of inverted cone crystal; V^ - software job law change the rate of withdrawaj
fh
Px
Âx
T
do
d\ _
Xi MANAGEMENT MODEL DURING CRYSTAL GROWTH
X2 T(x) = F( Z, j V3n(x), L(x), x)
X3
V3n(x)
LX
T(x)
E
a
Fig. 6. Management model for crystal
the dependence [2] crystal diameter d of withdrawal speed Vs and temperature T melt in the form of an expression (2):
d = Ct , (2)
' [t - T ] ' w
Sergey7 P. Sahanskiy. Control of the Shape of Semiconductor Crystals when Growing in Czochralski Method
where Cv=px- ; Ct = 4 •- CT
^XB P- -¿X-g
V - crystal pulling speed; TK - the crystallization temperature of the material; T - the average temperature of the melt in the zone of the crystallization front; L - linear axial gradient in the solid crystal; E - laaent heaa of fusion of the maOerial; ^ - thermal cooductivity o. the melt; /lTB - factor tlae rmat conductivity of" the crystal; g - acceleration of gravity; c - the suifaca tension of the melt; px -u. thhe density rf tOr liouid material; f - diameter of the crystal grown.
To set the average temperature of the melt expression (2) can be written as (3):
T(x) = Tt+ c=, (3)
d3( x)
where d3(x) - the program goals ol the grown crysfal diameter; V3n(x) - the software the speed crystal pulling; x- comtdinate movement of the crystal.
InB. M. Turowski, B. A. Sakharov [4] that the curvature of the crystallization front is defined by the ratio of the axial awd radial gradknts in the grown crystal, which in turn depend on the crystal for a given diameter of the thermal fields in the crysthl and the melt and the hate oU withdrawan With rhe increase in the raOr af withdrawe of the axial gradi-nt in the crystal inceeases (increasing heat flux caused ley- the retease of lrtent heaO of crystallization) and tha crystrliization front bends upward.
Speed control drawing of farmanium crystals inf closrd heat -nap allowh you io create a flat crystallization foot duringehe growth of the crystal right cene and a cylintticrl part lhal is needed for many brands of the germanium crystal and minimrzes dislo cation grow n crystal.
Equation (2) can be seen a s (4) tn determine rhe appropriete rare of withdrawal of the c rystal:
^JtadeSJ, (4)
d
where X = [T - 7q] - the value of the average heat melt relative to the te mperature of c ry stall izatio n of thh material °C.
In general , according to the equatio ns overheating X is a Ounction of the ax ial gradient in the crystal L ead given the rate of" withdrawal V3 crystal. The overheating X ean be determined by measuring thermocouples area adjaceat to the front of the crystallization of the material when developing specific technology of crystal growth.
Overheating of the melt value of the average X for the material, germanium (Ge) is in the range
oe 0.1-2 "a
Usi ng a line; ar approaimatten of the pulli eg rate on the key areasae crystal growth (straight cone, the cylindrical part and reverse cone) can obtain expressions foe the diterminftion of the speed command to pull cone, the cylindrical part oh the crystal and the formation of inverted cone, respectively:
V, (x) = ,0 - ^;
xn (s) = v2 +
(x -x2) -Vj ]
(f3 — X2 )
where F,(a) = Fn(x) - software job pulling speed drills;; F0- initial c; ryyst-ai^ pulling speed when switched on automatic; Fa - pulling speed of the crystal at the end of the formalion of inverted cone; F, - pulling speed formation of xhe crystal at the end of thecylinder; F, - pulling speed of the crystal at the end of the formation of inverted cone; xn - coordinate completion of thx formation of the ciystal might aone; x2 - rooadinate cempletion ol the aormation of tha cvlindrical part of the crystal; x3 - caordidale the completion nf the formation of inveated cone (crystal; x - coordinate-axis of the crystal.
In order no ttetermine the cootdinates oJt the rrsita; of withdrawal at tfe nodai points ( F0 , Fb F2, F3) transfoom (¿1) to (5):
x (d) =
L-
A-d
A
Cxx
(5)
where F3(d) - crystad pulling speed; d- crystal diameter; /,■ - technological droop rate (0,95-0,25). From (5) for the rate of withdrawal of nodesobtain expressions:
V =
V =
V =
2- d0
A -LA-
dj
dl
2 • da ~c7
A.
cV'
A;
cv '
A. A
C'
where L0 - axial gradient at ^lie heginning of the cylindrical part of the crystal; Li -axial gradient at the end of the cylindrical port of the crystal; Fr Fb F^ F3 - nodes pulling rate; b0 - craned neck diametea of the cryrtal when the automatic mode; d - diameter of the cylindrical part crystal.
Technological adj ustme nc coefficients /f are introduced fo the possibility of adjusting the rate of withdrawal on the basis of techniccl requirements leg, unifotm doping of tho caystal along its length, assumi ng a cerpai it amount of deflection op tine c rystallization front in the dicectlon of the melt on the cylindricai part oW the growing crystal.)
Cosine law for the continuoua calculation of main controller in the plant growth is controlled by the cones in a continued fraction Jacobi [5] standazd afcuracy of the expressirn:
ooe(x) = (n/2 - x )•
K1 + -
K 2
(n/2 - x)2 + K 3 +
K 4
(n/2 - x)2 + K 5 _
where K1 = 6,63530098; K2 = - 729,384055; 123 = 52,3056381; K4 = 1212.88544(1; K5 = 15,8503569
Using a linear approximation of the parameters on the remaining sections of the crystal growth, it is possible to obtarn an expression for calculating tire rate of the control progrom ann the temperature ot all sites, with the lrnear law of the job of the axial gradient, based on the installation of measuring the results of the previous drawing of tho crystal.
Expression on orders diameter d3(x) and temperature T(x) on the right cone crystals take the form:
d3 ( x) = d0 +
(dl - d0 ) f dl - d(
•cos
f „ ^ — x
Tm (x) = T + ct •
[L, -Cy V3n(x)]
d0 +
-f dL-A). xo 1 f nn.x
where x1 - coordinate; completion of tne formation of the cryatal tight cone; Z0 the aaral gradient in the crystal into the conical part; d0 - craned neck diameter of the crystal; di - diameter of the cySindrical past oh the c rystal.
Expression on orders diameter d3(x) of the ceystal and the temperature T(x) on the cylindrical part of thee; erystal ore as follows:
d 3 ( x) = d{.
h+V-xù-^eJÀ-Cr^ (x) . \X2 X1 ' ,
dl
where x2 - coordinate completion of the formation of the cylindrical part of the crystal; Li - axial gradient in the crystal at the end of thn cylindrical part.
Expression specifying diameter fx) of the crystal and the temperature T(x) on the opposite cone takes thf fotlowing form:
d3 ( x) = dd +
(di-do ) + (dkzdL
cos
n
• (x - x2 )
Tn( x)=T-K+Ct-
dn +
((-d0 ) . ( dl-d<.
+ 1 -
• cos
n
-•(x - x2 )
where x3 - coordinate the completion of the formation of inverted cone crystal.
In turn, the expression for the linear plots of growing jobs right cone crystal and its cylindrical part and reverse cone will look: likn:
f3 (x)=d1 +
x[Sj - d0 ] o
d 3 ( x) = d{,
X3 X2
x3 x2
d3(x) = dx -{X-d°]
VX3 x2 /
Simulation speed and temperature grown single crystals of germanium-based model and the reduced thermnl constant of the material [6] is shown in Fig;. 7-10 for the linear and for the reference cosine inverse cone type crystal (cosine law isolated solid line).
Modeling the velocity and temperature of the single crystal silicon grown for the same crystal form as germanium, but using permanent thermal silicon material, shown in Fig. 11-12 (with a separate cosine law of formation and reverse cones crystal solid line).
cm
x, mm
Fig. 7. Setting the diameter of thx crystal germanium: xx = 500 mm; x2 = 2210 mm; x3 = 260 mm; d0 = 0,5 cm; d1 = 11 cm
x, mm
Fig. 8. Setting the nate of withdrawxl of germanium n2: d0 = 0,5 cm; n = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm; I = 0,4 ° C; f = 0,8; f = 0,8; ff = 0,4; f4 = 0,(5; L0 = 20 ° C / cm; U = 40 ° C / cm
T, 0C
Fig. 9. Temperature setting pulling germanium: d0 = 0 ,5 cm; jc1 = 5 0 mm; x2 = 2 10 mm; x3 = 260 mm; d1 = 11 cm; X = 0,4 ° C ; f 1 = 0, 8 ; f = 0,8; & = 0 ,4; f = 0,6; /,0 = 20° C / cm; / = 40 ° C/cm
x, mm
Fig. 10. Setting the diameter of the germanium crystal with a straight cone: d0 = 0,5 cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm
V3, mm / min
x, mm
Fig. 11. Setting the drawing speed of silicon: d0 = 0,5 cm; = = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm; I = 0,4 °; fa = 0 , 8; fa2 = 0 , 8 ; fa = 0 ,4; fa = 0 ,6; L0 = 22 0 ° C = cm ; L 1 = 4 0 ° C/cm
Fig. 12. Setting the draw temperature of silicon at: d0 = 0,5 cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d = 11 cm; I = 0,4 °; fa = 0,8; fa = 0,8; fa = 0,4^ fa = 0,6; L0 = 20 °Ci/cm; h = 40 °C/cm
Findings
A model of formation temperature and the rate of withdrawal of semiconductor crystals with a cosine law, the formation of cones in the crystal growth process of the " Czochralski ", which allows you to enter this control in managing the installation drawing, providing a flat crystallization front of the crystal and obtaining high-quality finished products.
The proposed mathematical model of the process control growing semiconductor crystals can be successfully applied to the extraction plants such as semiconductor crystal silicon, germanium,
- 30 -
gallium arsenide, and the algorithm can easily programmable and operates in real time under the current master controllers with floating-point operations per system commands.
References
[1] Саханский С.П. // Мехатроника. Автоматизация. Управление. 2008. № 1. C. 42-46.
[2] Саханский С.П. Управление процессом выращивания монокристаллов германия: монография. Красноярск: Сиб. гос. аэрокосмич. ун-т, 2008. 104 с.
[3] Саханский С.П. // Нано- и микросистемная техника. 2012. № 6. C. 2-5.
[4] Туровский Б.М., Сахаров Б.А. // Научные труды Гиредмета. М.: Отдел научно-технической информации подотрасли, 1969. Т. 25. С. 94-103.
[5] Благовещенский Ю.В., Тестер П.С. Вычисление элементарных функций на ЭВМ. Киев: Техника, 1977. 207 с.
[6] Бабичев А.П., Бабушкина Н.А., Братковский А.М. Физические величины: справочник / ред. И.С. Григорьев, Е.З. Мелихов. М.: Энергоатомиздат, 1991. 1232 с.
Управление формой полупроводниковых кристаллов при выращивании по способу чохральского
С.П. Саханский
Сибирский федеральный университет Россия, 660041, Красноярск, пр. Свободный, 79
Предложена модель формирования температуры и скорости вытягивания полупроводниковых кристаллов при выращивании по способу Чохральского, которая позволяет управлять формой кристаллов, обеспечивая плоский фронт кристаллизации и получение качественной готовой продукции.
Ключевые слова: модель, выращивание, полупроводниковые кристаллы.
