Journal of Siberian Federal University. Engineering & Technologies 1 (2014 7) 20-31 ~ ~ ~

УДК 004.7

Control of the Shape of 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- 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 . 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 -based optical method for determining the current diameter of the crystal is shown in Fig. 1. Under source control the camera in the single of diameter d, with a pulling speed Vз and the rotating crystal (seed) Wз, and the molten metal in the crucible with an inner diameter D rotates with angular velocity Wт. Computer-controlled crystal pulling speed Vз, crystal rotation Wз, crucible rotation Wт through the appropriate drive. Control

© Siberian Federal University. All rights reserved * Corresponding author E-mail address: [email protected] – 20 – Sergey P. Sahanskiy. Control of the Shape of Semiconductor Crystals when Growing in Czochralski Method

W W 7 з з 1

V V з з 2

X ; Δ 11 з з

13 3 12 4

17

18 14 d

D 6 Tз D

15 5

19 16 Wт W т 8 V т Xт; Δт Vт 9 10

Fig. 1. The control Figuresystem 1. in The growing control single system crystals in grow ofing germanium: single crystals 1 – rotational of germanium: drive seed; 2 – move the seed drive; 3 – optical system;1 - rotational 4 – image drive converterseed; 2 - movemeniscus; the seed 5 – Temperaturedrive; 3 - op ticalsensor; system; 6 – temperature 4 - image converter control; 7 – computers; 8 – drivemeniscus; crucible 5 rotation; - Temperature 9 – stepper sensor; motor; 6 - 10 temper – stepperature control;motor control 7 – co unit;mputers; 11 – encoder 8 - drive seed; 12 – camera; 13 – bar;crucible 14 – rotation;molten metal; 9 - stepper 15 – crucible; motor; 10 16 - – stepper heater; motor17 – pyrometer control unit; to measure 11 - encoder the axial seed; gradient 12 in the solid crystal;– 18 camera; – a digital 13 –computer bar; 14 axial - molten gradient; metal; 19 15 – –crystal crucible; solidification 16 – heater; front 17 - pyrometer to measure the axial gradient in the solid crystal; 18 - a digital computer axial gradient; 19 - crystal solidification front

The shape of the grown single crystal germanium (90 mm diameter), and management of key of the temperature of the melt is based on the issuance of the job temperature Тз of the computer on the temperaturegrowth control. parameters on the installation drawing for temperature and speed, with the mapping of the Ascontrol the feedback signal, a variationsensor is of used the currentfor temperature diameter of radiation the set, in pyrometer relative nume aimedrical unitsat the shown lateral in surfaceFig. of the graphite2-4. 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 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 – 21 – Sergey P. Sahanskiy. Control of the Shape of Semiconductor Crystals when Growing in Czochralski Method 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 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 Vзп(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, Vзп (x), L(x), x), (1) where T(x) – the average temperature of the melt in the zone of the crystallization front, Vзп(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, , and 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 – 22 –

Fig. 2. Bar d = 90 mm Figure. 2. Bar d = 90 mm

Figure. 2. Bar d = 90 mm

Fig. 3. Graph of the temperature of the heater ТЗ sites: 1 – growing area right cone; 2 – growing area of the cylin- der; 3 – land cultivationFigure. inverted 3. Graph cone; 4of – thesection temperature annealing of the heater ТЗ sites: 1 - growing area right cone; 2 - growing area of the cylinder;

3 - land cultivation inverted cone; 4 - section ingot annealing Figure. 3. Graph of the temperature of the heater ТЗ sites: 1 - growing area right cone; 2 - growing area of the cylinder; 3 - land cultivation inverted cone; 4 - section ingot annealing

Figure. 4. Plot of the rate of withdrawal and seed V control signal Δy at sites: Fig. 4. Plot of the rate of withdrawal and seed V control signal ∆y at Зsites: 1 – growing area right cone; 2 – growing 1 - growing area right cone; 2 З- growing area of the cylinder; 3 - growing area inverted area of the cylinder; 3 – growing area inverted cone; 4 – section ingot annealing cone; 4 - section ingot annealing

In general, the management of the main parameters of growing single crystal of germanium is shown in Fig. 5

dз(x), Т, Vзп, L L V d зп Т з

0 x x x x 1 2 3 Figure 5. Building process parameters of growing single crystals of germanium: dз - job diameter grown single crystal; Т - software job law of temperature change; L - job axial gradient, x -moving crystal; x1 - coordinate completion of the formation of the crystal right cone; x2 - coordinate completion of the formation of the cylindrical part of the crystal; x3 - coordinate the completion of the formation of inverted cone crystal; Vзп - software job law change the rate of withdrawal

Figure. 4. Plot of the rate of withdrawal and seed VЗ control signal Δy at sites: 1 - growing area right cone; 2 - growing area of the cylinder; 3 - growing area inverted cone; 4 - section ingot annealing

In general, the management of the main parameters of growing single crystal of germanium is Sergeyshown P. Sahanskiy. in Fig. 5 Control of the Shape of Semiconductor Crystals when Growing in Czochralski Method

dз(x), Т, Vзп, L L V d зп Т з

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 Vзп(x), which allow you to automate the data entry process

parameters in the control system. 0 x1 x2 x3 x Management model temperature and velocity (Fig. 6) during crystal growth can be represented Figure 5. Building process parameters of growing single crystals of germanium: Fig. 5. Buildingby the process expression parameters (1): of growing single crystals of germanium: dз – job diameter grown single dз - job diameter grown single crystal; Т - software job law of temperature change; crystal; Т – software job law of temperature change; L – job axial gradient, x -moving crystal; x1 – coordinate L - job axial gradient, x -moving crystal;T(x) = xF1( -Z coordinate, Y, Vзп (x), completionL(x), x), of the formation of the (1) completion of the formation of the crystal right cone; x – coordinate completion of the formation of the cylindri- crystal right cone; x2 - coordinate completion2 of the formation of the cylindrical part of the where T(x)- the average temperature of the melt in the zone of the crystallization front, Vзп(x) - the cal part of the crystal;crystal; xx33 – - coordinate coordinate the completion of the formation formation of inverted cone crystal; VVзпзп – - software job law change thesoftware softwarerate of the withdrawal jobspeed law crystal change pulling; the rate x -of coordinate withdrawal 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.

Y

ρт ρж σ λж λтв E Τк

d0

d1 Z x1 MANAGEMENT MODEL DURING CRYSTAL GROWTH

x2 T(x) T(x) = F( Z, Y, Vзп(x), L(x), x) x3

Vзп(x) When growing all major semiconductor crystals (germanium, silicon, and ) their crystallization front of the crystal,L( xwhich) 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 Fig. 6. Management model for crystal Figure 6. Management model for crystal 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 the dependence [2] crystalthe dependence diameter [2]d of crystal withdrawal diameter speed d of Vs withdrawal and temperature speed Vs T and melt temperature in the form T ofmelt an in the form of expression (2): an expression (2):

[L − CV ⋅Vз ] d = Ct ⋅ , (2) (2) []T −Tк

E σ ⋅λтв where CV = ρж ⋅ ; Ct = 4⋅ ; – 24 – λтв ρж ⋅λж ⋅ g

Vз - crystal pulling speed; Τк - the crystallization temperature of the material; Τ - the average temperature of the melt in the zone of the crystallization front; L - linear axial gradient in the solid crystal; E - latent heat of fusion of the material; λж - thermal conductivity of the melt; λтв - factor thermal conductivity of the crystal; g - acceleration of gravity; σ - the surface tension of the melt; ρж - u. the density of the liquid material; d - diameter of the crystal grown. To set the average temperature of the melt expression (2) can be written as (3):

[L − CV ⋅Vзп (x)] T (x) = Tк + Ct , (3) dз (x) where dз(x)- the program goals of the grown crystal diameter; Vзп(x) - the software the speed crystal pulling; x - coordinate movement of the crystal. In B. M. Turowski, B. A. Sakharov [4] that the curvature of the crystallization front is defined by the ratio of the axial and radial gradients in the grown crystal, which in turn depend on the crystal for a given diameter of the thermal fields in the crystal and the melt and the rate of withdrawal. With the increase in the rate of withdrawal of the axial gradient in the crystal increases (increasing heat flux caused by the release of latent heat of crystallization) and the crystallization front bends upward. Speed control drawing of germanium crystals in a closed heat snap allows you to create a flat crystallization front during the growth of the crystal right cone and a cylindrical part that is needed for many brands of the germanium crystal and minimizes dislocation grown crystal. Equation (2) can be seen as (4) to determine the appropriate rate of withdrawal of the crystal: [L − C ⋅V ] λ = C V з , (4) t d where λ = []T −Tк - the value of the average heat melt relative to the temperature of crystallization of the material оС.

When growing all major semiconductor crystals (germanium, silicon, and gallium arsenide) their crystallization frontWhen of the growing crystal, all which major sepa semiconductorrates the liquid crystals from the (germanium, solid part silicon,of the melt and is gallium arsenide) When growing all major semiconductor crystals (germanium, silicon, and gallium arsenide) raised above the theirsurface crystallization of the melt onfront the ofvalue the ofcrystal, 1 - 5 cm. which separates the liquid from the solid part of the melt is their crystallization front of the crystal, which separates the liquid from the solid part of the melt is If we equateraised the above weight the surface of the of lifted the melt weight on theof thevalue column of 1 - 5 of cm. liquid melt (to the front of raised above the surface of the melt on the value of 1 - 5 cm. crystallization), the surfaceIf we tension equate fo rces the acting weight on of the the circumference lifted weight andof height the column of the ofbar liquid to take melt into (to the front of If we equate the weight of the lifted weight of the column of liquid melt (to the front of account the expressioncrystallization), of the melt the thr surfaceough thetension heat fobalancerces acting at the on interf the circumferenceace, it is possible and toheight obtain of thethe bar to take into crystallization), the surface tension forces acting on the circumference and height of the bar to take into dependence [2] accountcrystal diam the expressioneter d of withdrawal of the melt speed through Vs theand heat temperature balance atT themelt interf in theace, form it is ofpossible an to obtain the account the expression of the melt through the heat balance at the interface, it is possible to obtain the expression (2): dependence [2] crystal diameter d of withdrawal speed Vs and temperature T melt in the form of an dependence [2] crystal diameter d of withdrawal speed Vs and temperature T melt in the form of an expression (2): [L − C ⋅V ] expression (2):d = C ⋅ V з , (2) t − []T Tк [L − C ⋅V ] d = C ⋅ V [Lз −, C ⋅ V ] (2) t d = C ⋅ V з []T −tTк , (2) E σ ⋅λтв − where C = ρ ⋅ ; C = 4⋅ Sergey P. Sahanskiy.; Control of the Shape of Semiconductor[]T Tк Crystals when Growing in Czochralski Method V ж λ t ρ ⋅λ ⋅ тв ж Eж g σ ⋅λ where C = ρ ⋅ ; C =E4⋅ тв σ; ⋅λ whereV ж Cλ = ρ t⋅ ; ρC ⋅=λ4⋅ тв ; Vз - crystal pulling speed; Τк - V theтв crystallizationж жt ж temperatureg of the material; Τ - the average λтв ρж ⋅λж ⋅ g temperature of theVз melt - crystal in the pulling zone of speed; the crysta Τк - llization the crystallization front; L - linear temperature axial gradient of the in material; the solid Τ - the average Vз - V crystalз – crystal pulling pulling speed; speed; Τк Τ - к the– the crystallization crystallization temperature temperature of of thethe material;material; Τ – - the the average average crystal; E - latenttemperature heat of fusion of the of melt the material;in the zone λ of- thermalthe crysta conductivityllization front; of the L -melt; linear λ axial - factor gradient in the solid temperaturetemperature ofof the meltmelt inin theжthe zonezone ofof the the crystallization crystallization front; front; L L – тв -linear linear axial axial gradient gradient in in the the solid solid thermal conductivitycrystal; of theEcrystal; - crystal;latent E heat–g latent- acceleration of fusionheat of of fusionof the gravity; material; of the σ material; - λtheж - surfacethermal λж – tension thermalconductivity of conductivity the melt;of the ρ жmelt;of the λ melt;тв - factor λтв – factor crystal; E - latent heat of fusion of the material; λж - thermal conductivity of the melt; λтв - factor - u. the density ofthermal the liquid conductivitythermal material; conductivity dof - thediameter crystal; of the of crystal;g the - acceleration crystal g – accelerationgrown. of gravity; of gravity; σ - the σ surface – the surface tension tension of the ofmelt; the melt;ρж ρж – thermal conductivity of the crystal; g - acceleration of gravity; σ - the surface tension of the melt; ρ u. the density of the liquid material; d – diameter of the crystal grown. ж To set the- averageu. the density temperature of the liquidof the material;melt expression d - diameter (2) can of bethe written crystal as grown. (3): - u. theTo densityset the averageof the liquid temperature material; of d the - diameter melt expression of the crystal (2) can grown. be written as (3): To set the average temperature[L − CV ⋅Vзп of(x )the] melt expression (2) can be written as (3): T (xTo) = setTк the+ C averaget temperature, of the me lt expression (2) can be written (3) as (3): dз (x) [L − CV ⋅Vзп (x)] T (x) = Tк + Ct [L − CV ,⋅ V зп ( x ) ] (3) (3) T (x) = Tк +dC(t x) , (3) where dз(x)- the program goals of the grown crystal diameter; Vзпз (x) - the software the speed crystal dз (x) pulling; x - coordinatewhere movemedзwhere(x)- the ntd з (programofx) the– the crystal. programgoals of goalsthe grown of the crystalgrown crystaldiameter; diameter; Vзп(x) V- зпthe(x) software– the software the speed the speed crystal crystal where dз(x)- the program goals of the grown crystal diameter; Vзп(x) - the software the speed crystal In B. M. pulling;Turowski, xpulling; - coordinateB. A. x Sakharov – coordinate moveme [4] ntmovementthat of the curvaturecrystal. of the crystal. of the crystallization front is defined pulling; x - coordinate movement of the crystal. In B. M. Turowski, B. A. Sakharov [4] that the curvature of the crystallization front is defined by by the ratio of the axialIn and B. radialM. Turowski, gradients B. in A. th eSakharov grown crystal, [4] that which the curvature in turn depend of the crystallizationon the crystal front is defined the ratioIn of B. the M. axial Turowski, and radial B. A.gradients Sakharov in the [4] grown that the crystal, curvature which of in the turn crystallization depend on the front crystal is defined for for a given diameterby the of ratio the thermalof the axial fields and in radialthe cr ystalgradients and thein thmelte grown and the crystal, rate ofwhich withdrawal. in turn dependWith on the crystal bya given the ratio diameter of the of axial the thermal and radial fields gradients in the crystalin the grownand the crystal, melt and which the rate in turnof withdrawal.depend on theWith crystal the the increase in thefor rate a given of withdrawal diameter ofof thethe thermalaxial gr adientfields in the crystalcrystal increasesand the melt (increasing and the heatrate offlux withdrawal. With forincrease a given in thediameter rate of of withdrawal the thermal of fields the axial in the gradient crystal in and the the crystal melt increasesand the rate (increasing of withdrawal. heat flux With caused by the releasethe increase of latentcaused in heat theby theofrate crystallization) release of withdrawal of latent and ofheat thethe of axialcr crystallization)ystallization gradient infront andthe bendsthecrystal crystallization upward. increases (increasing front bends heat upward. flux the increase in the rate of withdrawal of the axial gradient in the crystal increases (increasing heat flux Speed controlcaused drawing by theSpeed releaseof germaniu control of latentm drawing crystals heat of of crystallization)in germanium a closed heat crystals and sn apthe inallows cr ystallizationa closed you heatto create front snap bends aallows flat upward. you to create a flat caused by the release of latent heat of crystallization) and the crystallization front bends upward. crystallization front duringSpeedcrystallization the control growth drawing offront the during crystal of germaniu the right growth conem crystals of and the a crystal cy inlindrical a closed right partcone heat that and sn ap isa cylindricalneededallows youfor part to createthat is neededa flat for many brandsSpeed ofcontrol the germanium drawing ofcrystal germaniu and minimizesm crystals dislocationin a closed grown heat sn crystal.ap allows you to create a flat many brands of thecrystallization germanium frontcrystal during and minimizes the growth dislocation of the crystal grown right crystal. cone and a cylindrical part that is needed for crystallizationEquation (2)front can during be seen the as growth (4) to determine of the crystal the appropriateright cone andrate aof cy withdrawallindrical part of thethat crystal: is needed for Equation many(2) can brands be seen of theas (4) germanium to determine crystal the andappropriate minimizes rate dislocation of withdrawal grown of crystal.the crystal: many brands of the germanium crystal and minimizes dislocation grown crystal.

Equation (2) can be seen as (4)[L to− CdetermineV ⋅Vз ] the appropriate rate of withdrawal of the crystal: Equation (2) canλ = beCt seen as (4) to, determine the appropriate rate of wit (4)hdrawal of the crystal:(4) d [L − C ⋅V ] = V [Lз− C ⋅V ] where λ = []T −T - the value of the average heat melt relativeλ toC thet temperature, Vof crystallization з of (4) In general,к accordingwhere λ to= [theT – equationsTк] – the value overheating of the average λ is a heatfunctionλ = meltdCt ofrelative the axial to the, gradient temperature in the of crystallization of (4) d оIn general, according toо the equations overheating λ is a function of the axial gradient in the thecrystal material L С and. where given λ the the= [] T material rate−Tк of- the withdrawalС. value of theVз average crystal. heat The melt overheating relative λto canthe temperature be determined of crystallization by of where λ = []T −Tк - the value of the average heat melt relative to the temperature of crystallization of crystal L and given the rate of withdrawal Vз crystal. The overheating λ can be determined by measuring thermocouplesthe material о СIn area. general, adjacent according to the to the front equations of the overheating crystallization λ is a offunction the material of the axial when gradient in the crystal the material оС. measuring thermocouplesL and given area the adjacent rate of towithdrawal the front V of crystal. the crystallization The overheating of λ the can material be determined when by measuring developing specific technology of crystal growth. з developing specificthermocouples technology of crystalarea adjacent growth. to the front of the crystallization of the material when developing specific Overheating of the melt value of the average λ for the material, germanium (Ge) is in the range technology of crystal growth. о Overheating of the melt value of the average λ for the material, germanium (Ge) is in the range of 0.1-2 С. Overheating of the melt value of the average λ for the material, germanium (Ge) is in the range of 0.1-2 оС. Using a linearof approximation 0.1-2 оС. of the pulling rate on the key areas of crystal growth (straight Using a linear approximation of the pulling rate on the key areas of crystal growth (straight cone, the cylindrical part andUsing reve a linearrse cone) approximation can obtain of expressions the pulling ratefor onthe the determination key areas of crystalof the growthspeed (straight cone, the cone, the cylindrical part and reverse cone) can obtain expressions for the determination of the speed command to pull cone,cylindrical the cylindrical part and reverse part of cone) the can crystal obtain and expressions the formation for the ofdetermination inverted cone, of the speed command command to pull to cone, pull cone, the cylindrical the cylindrical part part of of th ethe crystal crystal andand the formation formation of ofinverted inverted cone, cone, respectively: respectively: respectively: x[V −V ] = − 0 1 Vзп (x) V0 − ; xx[V0 V1 ] Vзп (x) = V0 − 1 ; (x − x )[Vx−1 V ] = − 1 1 2 Vзп (x) V1 − − ; (()xx −x1x)[V1 V2 ] Vзп (x) = V1 − 2 1 ; ()x2 − x1 (x − x2 )[V3 −V2 ] Vзп (x) = V2 + , (x − x2 )[V3 −V2 ] V (x) = V + ()x3 − x2 , зп 2 ()x − x – 25 – where Vз(x) = Vзп(x) - software job pulling speed drills; V0- initial3 2crystal pulling speed when switched where Vз(x) = Vзп(x) - software job pulling speed drills; V0- initial crystal pulling speed when switched on automatic; V1 - pulling speed of the crystal at the end of the formation of inverted cone; V2 - pulling on automatic; V1 - pulling speed of the crystal at the end of the formation of inverted cone; V2 - pulling speed formation of the crystal at the end of the cylinder; V3 - pulling speed of the crystal at the end of speed formation of the crystal at the end of the cylinder; V3 - pulling speed of the crystal at the end of the formation of inverted cone; x1 - coordinate completion of the formation of the crystal right cone; x2 - the formation of inverted cone; x1 - coordinate completion of the formation of the crystal right cone; x2 - coordinate completion of the formation of the cylindrical part of the crystal; x3 - coordinate the coordinate completion of the formation of the cylindrical part of the crystal; x - coordinate the completion of the formation of inverted cone crystal; x - coordinate-axis of the crystal. 3 completion of the formation of inverted cone crystal; x - coordinate-axis of the crystal.

In order to determine the coordinates of the rate of withdrawal at the nodal points In order to determine the coordinates of the rate of withdrawal at the nodal points (V0, V1, V2, V3) transform (4) to (5): (V0, V1, V2, V3) transform (4) to (5): ⎡ λ ⋅ d ⎤ β = − i Vз (d) ⎢L⎡ λ ⋅⎥ ⎤ β, (5) C dC i Vз (d)⎣= ⎢L − t ⎦ ⎥V , (5) C C where Vз(d) - crystal pulling speed; d - crystal diameter; βi ⎣- technologicalt ⎦ V droop rate (0,95-0,25). where V (d) - crystal pulling speed; d - crystal diameter; β - technological droop rate (0,95-0,25). з i

From (5) for the rate of withdrawal of nodes obtain expressions: From (5) for the rate of withdrawal of nodes obtain expressions: ⎡ λ ⋅ d ⎤ β = − 0 0 V0 ⎢L0⎡ λ ⋅⎥ ⎤ β; C d0C 0 V0⎣= ⎢L0 − t ⎦ ⎥V ; ⎣ Ct ⎦ CV ⎡ λ ⋅ d ⎤ β = − 1 1 V1 ⎢L0⎡ λ ⋅⎥ ⎤ β; C dC1 1 V1⎣= ⎢L0 − t ⎦ ⎥V ; ⎣ Ct ⎦ CV ⎡ λ ⋅ d ⎤ β = − 1 2 V2 ⎢L1⎡ λ ⋅⎥ ⎤ β; C dC1 2 V2⎣= ⎢L1 − t ⎦ ⎥V ; ⎣ Ct ⎦ CV

In general, according to the equations overheating λ is a function of the axial gradient in the In general,In general,according according to the equations to the equations overheating overheating λ is a function λ is a function of the axial of the gradient axial gradient in the in the crystal L and given the rate of withdrawal Vз crystal. The overheating λ can be determined by crystal L and given the rate of withdrawal Vз crystal. The overheating λ can be determined by crystal Lcrystal and givenL and the given rate the of rate withdrawal of withdrawal Vз crystal. Vз crystal. The overheating The overheating λ can beλ can determined be determined by by measuring thermocouples area adjacent to the front of the crystallization of the material when measuringmeasuring thermocouples thermocouples area adjacent area adjacent to the front to the of front the crystallization of the crystallization of the material of the material when when developing specific technology of crystal growth. developingdeveloping specific specifictechnology technol of crystalogy of growth. crystal growth. Overheating of the melt value of the average λ for the material, germanium (Ge) is in the range Overheatingо Overheating of the melt of the value melt of value the average of the averageλ for the λ material, for the material, germanium germanium (Ge) is in(Ge) the isrange in the range of 0.1-2 оС. of 0.1-2 С. о of 0.1-2 ofоС 0.1-2. С. Using a linear approximation of the pulling rate on the key areas of crystal growth (straight Using a Using linear a approximation linear approximation of the pulling of the ratepulling on ratethe keyon the areas key of areas crystal of growth crystal (straightgrowth (straight cone, the cylindrical part and reverse cone) can obtain expressions for the determination of the speed cone, thecone, cylindrical the cylindrical part and part reve andrse cone)reverse can cone) obtain can expressions obtain expressions for the determinationfor the determination of the speed of the speed command to pull cone, the cylindrical part of the crystal and the formation of inverted cone, commandcommand to pull to cone, pull the cone, cylindrical the cylindrical part of part the crystalof the andcrystal the and formation the formation of inverted of inverted cone, cone, respectively: respectively:respectively: x[V −V ] x[V0 −V1 ] Vзп (x) = V0 − 0 1 ; Vзп (x) = V0 [− − x][V0 −;V 1 ] x V0 Vx11 V (x) =VVзп (−x) = V0 −x1 ; ; зп 0 1 x (x −x1x )[V − 1V ] (x − x1 )[V1 −V2 ] Sergey P. Sahanskiy.Vзп (x) =ControlV1 − of the Shape1 1 of Semiconductor2 ; Crystals when Growing in Czochralski Method Vзп (x) = V(1 −− ) (x − x1 )[V1 −;V 2 ] x x1 [()Vx12 −Vx21] V (x) =VVзп (−x) = V1 −()x2 − x1 ; ; зп 1 2 ()x 1 − x ()(xx2−−xx1)[V 2 −V 1] (x − x2 )[V3 −V2 ] Vзп (x) = V2 + 2 3 2 , Vзп (x) = V(2x+− x )[V(x −−Vx2 )][V3 −,V 2 ] 2 ()x33 − x22 V (x) =VVзп (+x) = V2 +()x3 − x2 , , зп 2 3 ()x 2 − x where Vз(x) = Vзп(x) - software job pulling speed drills;()x3 V−0x- 2initial3 crystal2 pulling speed when switched where Vз(x) = Vзп(x) - software job pulling speed drills; V0- initial crystal pulling speed when switched where Vwhereз(x) = VVзпз((xx)) =- Vsoftwareзп(x) - software job pulling job speedpulling drills; speed V 0drills;- initial V0 crystal- initial pulling crystal speedpulling when speed switched when switched on automatic; V1 - pulling speed of the crystal at the end of the formation of inverted cone; V2 - pulling on automatic; V1 - pullingwhere speed Vз( ofx) the = V crystalзп(x) – at software the end jobof the pulling formation speed of drills;inverted V0 -cone; initial V2 crystal- pulling pulling speed when on automatic;on automatic; V1 - pulling V1 - speedpulling of speed the crystal of the at crystal the end at ofthe the end formation of the formation of inverted of inverted cone; V 2cone; - pulling V2 - pulling speed formation of the crystal at the end of the cylinder; V3 - pulling speed of the crystal at the end of speed formation of the switchedcrystal at on the automatic; end of the V 1cylinder; – pulling V speed3 - pulling of the speed crystal of atthe the crystal end of at the the formation end of of inverted cone; speed formationspeed formation of the crystal of the atcrystal the end at theof the end cylinder; of the cylinder; V3 - pulling V3 - speedpulling of speed the crystal of the atcrystal the end at theof end of the formation of invertedV2 cone;– pulling x1 - speedcoordinate formation completion of the crystalof the fo atrmation the end of of the the crystalcylinder; right V3 cone;– pulling x2 - speed of the crystal the formation of inverted cone; x1 - coordinate completion of the formation of the crystal right cone; x2 - the formationthe formation of inverted of inverted cone;at the x1 cone;end- coordinate of x 1the - coordinate formation completion completion of ofinverted the fo ofrmation cone; the fo xrmationof –the coordinate crystal of the right crystal completion cone; right x2 cone; -of the x 2 formation- of the coordinate completion of the formation of the cylindrical part of the1 crystal; x3 - coordinate the coordinate completion of the formation of the cylindrical part of the crystal; x3 - coordinate the coordinatecoordinate completion completion of crystal the formation of right the formationcone; of thex2 – cylindricalcoordinate of the cylindrical partcompletion of part the of crystal; of the the formation crystal; x3 - coordinate of x 3the - coordinatecylindrical the part the of the crystal; completion of the formation of inverted cone crystal; x - coordinate-axis of the crystal. completioncompletion of the formation of the formationx 3of – invertedcoordinate of inverted cone the crystal; completion cone xcrystal; - coordinate-axis of thex - coordinate-axisformation of the of invertedcrystal. of the crystal. cone crystal; x – coordinate-axis of the

crystal. In order to determine the coordinates of the rate of withdrawal at the nodal points In orderIn to order determine to determine In the order coordinates to the determine coordinates of the the coordinates of rate the of rate withdrawalof the of rate withdrawal of at withdrawal the at nodal the at the points nodal nodal points points ( V 0 , V1, V2, V3) (V0, V1, V2, V3) transform (4) to (5): (V0, V1, V2, V3) transformtransform (4) to (5): (4) to (5): (V0, V1, V(V2,0 ,V V3)1 ,transform V2, V3) transform (4) to (5): (4) to (5): ⎡ λ ⋅ d ⎤ β ⎡ λ ⋅ d ⎤ βi Vз (d) = ⎢L − ⎥ i , (5) Vз (d⎡) = ⎢Lλ −⋅ ⎡ ⎤ βλ⎥ ⋅ d ⎤, β i (5) з ⎣⎢ d Ct i⎦⎥ CV V (d) = VLз (−d⎣) = ⎢LC−t ⎦,C V ⎥ , (5) (5) (5) з ⎢ ⎣ ⎥ t ⎦C V C where V (d) - crystal pulling speed; d - crystal diameter;⎣ βC -t ⎣technological⎦ CV t ⎦ V droop rate (0,95-0,25). where Vз(d) - crystal pulling speed; d - crystal diameter; βi - technological droop rate (0,95-0,25). whereз V (d) - crystal pulling speed; d - crystal diameter;i β - technological droop rate (0,95-0,25). where Vз(d) - crystalз pullingwhere speed; V (d)d - – crystal crystal diameter; pulling speed; βi - technological d – icrystal diameter; droop rate β (0,95-0,25).– technological droop rate (0,95-0,25). з i From (5) for the rate of withdrawal of nodes obtain expressions: From (5) for the rate of withdrawal of nodes obtain expressions: From (5)From for the (5) rate for ofthe withdr rate ofawal withdr of nodesawal of obtain nodes expressions: obtain expressions: ⎡ λ ⋅ d ⎤ β = ⎡ − λ ⋅ d0 ⎤ β0 V0 = ⎢L0 −⎡ 0 ⎥ 0 ⎤; V⎡0 = ⎢Lλ0 −⋅ d ⎤ βλ⎥⋅ d0 ; β0 V⎣ = L0 C−t 0⎦ CV ; V0 = ⎢L0 −⎣0 ⎢ 0C⎥ t ⎦;C V ⎥ C⎣ C Ct ⎦ CV ⎣ ⎡ t λ⎦⋅ d V⎤ β = ⎡ − λ ⋅ d1 ⎤ β 1 V1 = ⎢L0 −⎡ 1 ⎥ 1⎤; V⎡1 = ⎢Lλ0 −⋅ d ⎤ βλ⎥⋅ d1 ; β 1 V⎣ = L1 C−t 1⎦ CV ; V1 = ⎢L0 −⎣1 ⎢ 0C⎥ t ⎦;C V ⎥ C⎣ C Ct ⎦ CV ⎣ ⎡ t λ⎦⋅ dV⎤ β ⎡ λ ⋅ d1 ⎤ β2 V2 = ⎢L1 − 1 ⎥ 2 ; V⎡2 = ⎢Lλ1 −⋅⎡ ⎤ βλ⎥⋅ d1 ⎤; β2 2 ⎣⎢ 1 d1 Ct 2⎦⎥ CV ⎡ λ ⋅d ⎤ β V = L V−⎣2 = ⎢L1C−t ⎦;C V ⎥ ; 0 3 2 ⎢ 1 ⎣ ⎥ t ⎦ V V3 = ⎢L1 − ⎥ , C⎣ C Ct ⎦ C V ⎣ t ⎦ V ⎣ Ct ⎦ CV ⎡ λ ⋅d ⎤ β where L0 - axial gradient at the beginning0 3 of the cylindrical part of the crystal; L1 -axial gradient at the V3 = ⎢L1 − ⎥ ,

⎣ Ct ⎦ CV end of the cylindrical part of the crystal; V0, V1, V2, V3 - nodes pulling rate; d0 - craned neck diameter of where L - axial gradient at the beginning of the cylindrical part of the crystal; L -axial gradient at the 0 where L – axial gradient at the beginning of the cylindrical1 part of the crystal; L -axial gradient at the the crystal0 when the automatic mode; d1 - diameter of the cylindrical part crystal.1 end of the cylindrical part of the crystal; V0, V1, V2, V3 - nodes pulling rate; d0 - craned neck diameter of end of the cylindrical part of the crystal; V0, V1, V2, V3 – nodes pulling rate; d0 – craned neck diameter Technological adjustment coefficients βi are introduced to the possibility of adjusting the rate of the crystal when the automaticof the crystal mode; when d1 - diameter the automatic of the mode; cylindrical d1 – diameter part crystal. of the cylindrical part crystal. withdrawal on the basis of technical requirements (eg, uniform of the crystal along its length, Technological adjustmentTechnological coefficients adjustment βi are introduced coefficients to the βi are possibili introducedty of adjustingto the possibility the rate ofof adjusting the rate of assuming a certain amount of deflection of the crystallization front in the direction of the melt on the withdrawal on the basiswithdrawal of technical on requirementsthe basis of technical (eg, uniform requirements doping of(eg, the uniform crystal dopingalong itsof thelength, crystal along its length, cylindricalassuming a part certain of the amount growing of deflectioncrystal.) of the crystallization front in the direction of the melt on the assuming a certain amount of deflection of the crystallization front in the direction of the melt on the cylindricalCosine part law of thefor growingthe continuous crystal.) calcula tion of main controller in the plant growth is controlled by cylindrical part of the growing crystal.) the conesCosine in alaw continued for the continuous fraction Jacobi calculation [5] standard of main accuracy controller of the in theexpression: plant growth is controlled by Cosine law for thethe continuous cones in a continuedcalculation fraction of main Jacobi controller [5] standard in the plant accuracy growth of theis controlled expression: by the cones in a continued fraction Jacobi [5] standard accuracy of the expression: ⎡ ⎤

⎢ K2 ⎥ cos(x) = ()π / 2 − x ⋅ ⎢K1+ ⎥ , ⎡ ⎢ 2 ⎤ K4 ⎥ ⎢ ()π / 2 − x +⎥ K3+ K⎢2 π − 2 + ⎥ cos(x) = ()π / 2 − x ⋅ ⎢K1+ ⎣ ⎥ , ()/ 2 x K5 ⎦ 2 K4 ⎢ ()π / 2 − x + K3+ ⎥ where K1 = 6,63550098; K2 = - 729,384055; 2 K3 = 52,9056381; K4 = 1212.885446; where K1 = 6,63550098;⎣⎢ K2 = – 729,384055;()π /K32 − =x 52,9056381;+ K5 ⎦⎥ K4 = 1212.885446; K5 = 15,8503569 K5 = 15,8503569 – 26 – where K1 = 6,63550098; K2 = - 729,384055; K3 = 52,9056381; K4 = 1212.885446;

K5 = 15,8503569 Using a linear approximation of the parameters on the remaining sections of the crystal growth,

it is possible to obtain an expression for calculating the rate of the control program and the temperature Using a linear approximation of the parameters on the remaining sections of the crystal growth, at all sites, with the linear law of the job of the axial gradient, based on the installation of measuring the it is possible to obtain an expression for calculating the rate of the control program and the temperature results of the previous drawing of the crystal. at all sites, with the linear law of the job of the axial gradient, based on the installation of measuring the Expression on orders diameter dз(x) and temperature Т(x) on the right cone crystals take the results of the previous drawing of the crystal. form: Expression on orders diameter dз(x) and temperature Т(x) on the right cone crystals take the (d − d ) ⎛ d − d ⎞ ⎛ π ⎞ form: = + 1 0 − 1 0 ⋅ ⎜ ⋅ ⎟ dз (x) d0 ⎜ ⎟ cos⎜ x⎟ ; 2 ⎝ 2 ⎠ ⎝ x1 ⎠ (d − d ) ⎛ d − d ⎞ ⎛ π ⎞ = + 1 0 − 1 0 ⋅ ⎜ ⋅ ⎟ dз (x) d0 ⎜ ⎟ cos⎜ x⎟ ; − ⋅ 2 ⎝ 2 ⎠ x [L0 CV Vзп (x)] Tзп (x) = Tк + Ct ⋅ ⎝ 1 ⎠ , ⎡ ()d − d ⎛ d − d ⎞ ⎛ π ⎞⎤ + 1 0 − 1 0 ⋅ ⎜ ⋅ ⎟ − ⋅ ⎢d0 ⎜ ⎟ cos⎜ x⎟⎥ [L0 CV Vзп (x)] 2 ⎝ 2 ⎠ x Tзп (x) = Tк + Ct ⋅ ⎣ , ⎝ 1 ⎠⎦ ⎡ ()d − d ⎛ d − d ⎞ ⎛ π ⎞⎤ d + 1 0 − ⎜ 1 0 ⎟ ⋅ cos⎜ ⋅ x⎟ where x1 - coordinate ⎢completion0 of the formation of ⎜the crystal⎟⎥ right cone; L0 - the axial gradient in the ⎣ 2 ⎝ 2 ⎠ ⎝ x1 ⎠⎦ crystal into the conical part; d0 - craned neck diameter of the crystal; d1 - diameter of the cylindrical where x1 - coordinate completion of the formation of the crystal right cone; L0 - the axial gradient in the part of the crystal. crystal into the conical part; d0 - craned neck diameter of the crystal; d1 - diameter of the cylindrical Expression on orders diameter dз(x) of the crystal and the temperature Т(x) on the cylindrical part of the crystal. part of the crystal are as follows: Expression on orders diameter dз(x) of the crystal and the temperature Т(x) on the cylindrical part of the crystal are as follows:

⎡ λ ⋅d ⎤ β = − 0 3 V3 ⎢L1 ⎡ λ⎥⋅d ⎤, β C C 0 3 V⎣3 = ⎢L1 −t ⎦ V ⎥ , ⎣ Ct ⎦ CV where L0 - axial gradient at the beginning of the cylindrical part of the crystal; L1 -axial gradient at the where L0 - axial gradient at the beginning of the cylindrical part of the crystal; L1 -axial gradient at the end of the cylindrical part of the crystal; V0, V1, V2, V3 - nodes pulling rate; d0 - craned neck diameter of end of the cylindrical part of the crystal; V0, V1, V2, V3 - nodes pulling rate; d0 - craned neck diameter of the crystal when the automatic mode; d1 - diameter of the cylindrical part crystal. the crystal when the automatic mode; d1 - diameter of the cylindrical part crystal. Technological adjustment coefficients βi are introduced to the possibility of adjusting the rate of Technological adjustment coefficients βi are introduced to the possibility of adjusting the rate of withdrawal on the basis of technical requirements (eg, uniform doping of the crystal along its length, withdrawal on the basis of technical requirements (eg, uniform doping of the crystal along its length, assuming a certain amount of deflection of the crystallization front in the direction of the melt on the assuming a certain amount of deflection of the crystallization front in the direction of the melt on the cylindrical part of the growing crystal.) cylindrical part of the growing crystal.) Cosine law for the continuous calculation of main controller in the plant growth is controlled by Cosine law for the continuous calculation of main controller in the plant growth is controlled by the cones in a continued fraction Jacobi [5] standard accuracy of the expression: the cones in a continued fraction Jacobi [5] standard accuracy of the expression:

⎡ ⎤ ⎢ ⎡ K2 ⎥ ⎤ cos(x) = ()π / 2 − x ⋅ ⎢K1+⎢ K2 ⎥ , ⎥ 2 K4 cos(x) = ()π / 2 −⎢ x ⋅ ⎢K()π1+/ 2 − x + K3+ ⎥ ⎥ , ⎢ ⎢ 2 π − 2 +K4 ⎥ ⎥ ⎣ ()π / 2 − x + K()3+/ 2 x K25 ⎦ ⎣⎢ ()π / 2 − x + K5 ⎦⎥ where K1 = 6,63550098; K2 = - 729,384055; K3 = 52,9056381; K4 = 1212.885446; where K1 = 6,63550098; K2 = - 729,384055; K3 = 52,9056381; K4 = 1212.885446; K5 = 15,8503569 K5 = 15,8503569

Using a linearSergey approximation P. Sahanskiy. Control of th eof parametersthe Shape of Semiconductor on the remaining Crystals sections when Growing of the in crystalCzochralski growth, Method Using a linear approximation of the parameters on the remaining sections of the crystal growth, it is possible to obtain an expression for calculating the rate of the control program and the temperature it is possibleUsing to obtain a linear an expressiapproximationon for calculatingof the parameters the rate on of the the remaining control program sections and of the the crystal temperature growth, it at all sites, with the linear law of the job of the axial gradient, based on the installation of measuring the at all sites,is possible with the to linearobtain law an expressionof the job of for the calculating axial gradient, the rate based of the on control the installation program ofand measuring the temperature the results of theat previous all sites, drawingwith the oflinear the crystal.law of the job of the axial gradient, based on the installation of measuring results of the previous drawing of the crystal. Expressionthe results on oforders the previous diameter drawing dз(x) andof the temperature crystal. Т(x) on the right cone crystals take the Expression on orders diameter dз(x) and temperature Т(x) on the right cone crystals take the form: Expression on orders diameter dз(x) and temperature Т(x) on the right cone crystals take the form: form: (d − d ) ⎛ d − d ⎞ ⎛ π ⎞ = + 1 0 − 1 0 ⋅ ⎜ ⋅ ⎟ dз (x) d0 (d − d⎜ ) ⎛ d ⎟− dcos⎞⎜ ⎛xπ⎟ ; ⎞ = +2 1 ⎝0 −2 1 ⎠ 0 ⋅ x ⎜ ⋅ ⎟ dз (x) d0 ⎜ ⎟⎝ cos1 ⎜ ⎠ x⎟ ; 2 ⎝ 2 ⎠ ⎝ x1 ⎠

[L0 − CV ⋅Vзп (x)] Tзп (x) = Tк + Ct ⋅ − ⋅ , ⎡ [L0 CV Vзп (x)] ⎤ Tзп (x) = Tк + Ct ⋅ ()d1 − d0 ⎛ d1 − d0 ⎞ ⎛ π ⎞ , ⎢d0 +⎡ − ⎜ ⎟ ⋅ cos⎜ ⋅ x⎟⎥ ⎤ ()d1 − d0 ⎛ d1 − d0 ⎞⎜ ⎛ π⎟ ⎞ ⎣ ⎢d +2 ⎝ −2⎜ ⎠ ⎟⎝⋅ cosx1 ⎜ ⎠⎦⋅ x⎟⎥ 0 2 2 ⎜ x ⎟ ⎣ ⎝ ⎠ ⎝ 1 ⎠⎦ where x1 - coordinate completion of the formation of the crystal right cone; L0 - the axial gradient in the where xwhere1 - coordinate x1 – coordinate completion completion of the formationof the formation of the ofcrystal the crystal right cone;right cone;L0 - the L0 –axial the axial gradient gradient in the in the crystal into the conical part; d - craned neck diameter of the crystal; d - diameter of the cylindrical crystal into the conical0 part; d – craned neck diameter of the 1crystal; d – diameter of the cylindrical crystal into the conical part; d0 - craned0 neck diameter of the crystal; d1 - diameter1 of the cylindrical part of the crystal.part of the crystal. d (x) = d part of the crystal. з 1 ; d з (x) = d1 ; ExpressionExpression on orders on diameter orders diameter dз(x) of dtheз(x) crystalof the crystal and the and temperature the temperature Т(x) onТ(x the) on cylindricalthe cylindrical part Expression on orders diameter⎡ dз(x) (of −thed crystal) (x) = dand; the⎤ temperature Т(x) on the cylindrical L1 L0 з d1 (x) = d part of the crystalof the crystalare as follows: are ⎡as follows: ⎢L0 + []x(−L x−1 L⋅ ) − CV ⋅⎤Vзпз (x)⎥ 1 ; 1 0 − part of the crystal are as ⎢follows:L0 + []x⎣ − x1 ⋅ ()x2− CxV1 ⋅Vзп (x)⎥ ⎦ T(x) = Tк + Ct ⋅ ()x − x ⎡ (L − L, ) ⎤ ⎣ 2 1 + −⎡ ⎦⋅ 1 0 − −⋅ ⎤ T(x) = Tк + Ct ⋅ d (x) = d ; ⎢dL10 []x x1 , (LC1 V LV0зп) (x)⎥ з 1 d з (x) = dL1 ;+ []x − x ⋅ − C ⋅V (x) d1 ⎣ ⎢ 0 ()x2 − x11 ⎦V зп ⎥ T(x) = T + C ⋅ ⎣ ()x2 − x1 , ⎦ where x - coordinate completion of the formation к ofT (x the)t = cylindricalT + C ⋅ part of the crystal; , 2 ⎡ (L − L⎡ ) к t ⎤ d1 ⎤ where x2 - coordinate completion of the formation of the 1 cylindrical0 part of(L 1 the− L 0 crystal;) d 1 ⎢L0 + []x − x1 ⋅ L −+C[]Vx ⋅−Vxзп (⋅x)⎥ − C ⋅V (x) L1 - axial gradient in the crystal at the end of the cylindrical()x − x ⎢part.0 1 V зп ⎥ where x2 - coordinate ⎣ completion of 2 the form1 ation of () the⎦x2 − cylindricalx1 part of the crystal; L1 - axial gradient in the crystalT at(x the) = Tend+ Cof ⋅the cylindrical part. ⎣ , ⎦ where xк 2 - t coordinateT(x) completion= Tк + Ct ⋅ of the formation of the cylindrical, part of the crystal; Expression specifying diameter dз(x) of the crystald1 and the temperatured Т(x) on the L1 - axial gradient in the crystal at the end of the cylindrical part.1 Expression specifying diameterL - axial d gradientз(x) of the in the crystal crystal and at thethe temperatureend of the cylindrical Т(x) on thepart. opposite cone takes the following1 form: where x2 - coordinatewhere completion x2 Expression - coordinate of the specifying formcompletionation diameter of of the the cylindrical d formз(x) ofation the part of crystal the of cylindrical the and crystal; the temperature part of the Т ( crystal;x) on the opposite cone takes the followingwhere form: x2 Expression– coordinate specifying completion diameter of the formationd (x) of theof the crystal cylindrical and the part temperature of the crystal; Т(x) onL1 the– axial з L1 - axial gradient inL the1 opposite- axialcrystal gradientgradient coneat the takes inendin thethe ofthe crystal crystalthe following cylindrical atat thethe form: endend part. of of thethe cylindricalcylindrical part.part. opposite cone takes the following form:

Expression specifyingExpression diameterExpression specifyingdз(x) specifying of the diameter crystal diameter d andз(x) thed ofз( x the) temperature of crystalthe crystal and Т( andx the) on the temperature the temperature Т( xТ)( x on) on the the opposite cone opposite cone takes the followingtakes form: the following (d form:− d ) ⎛ d − d ⎞ ⎡ π ⎤ opposite cone takesd (x) the= d following+ 1 0form:+ ⎜ 1 0 ⎟⋅cos ⋅()x − x з (d1 0− d0 ) ⎛ d1 − d0 ⎞ ⎡ π ⎢ ⎤ 2 ⎥ ; dз (x) = d0 + +2⎜ ⎝ ⎟⋅cos2 ⎢⎠ ⎣ x⋅3()x−−x2x2 ⎥ ; ⎦ 2 ⎝ 2 ⎠ (d −x d− )x ⎛ d − d ⎞ ⎡ π ⎤ d (x) = d + 1⎣ 3 0 +2(d⎜ −1d ) 0⎦⎟⎛⋅dcos− d ⎞ ⋅⎡()x −πx ; ⎤ з d0 (x) = d + 1 0 + ⎜ 1 ⎢ 0 ⎟⋅cos 2 ⎥⋅()x − x [L1з − CV 2⋅V0зп (x)⎝] 2 ⎠ ⎣ x3 − x2 ⎢ ⎦ 2 ⎥ ; Tзп (x) = Tк + Ct ⋅ [L − C ⋅V (x)] 2 ⎝ 2 ⎠, ⎣ x3 − x2 ⎦ T (x) = T + C ⋅ ⎡ − 1 V −зп ⎡ π ⎤⎤ зп к t (d − d ())d1 ⎛ dd0− d ⎛⎞d1 d⎡0 ⎞ π ⎤ ⎡, ⎤ ⎡ ⎢1d −+0 1 −+ 0⎜ (d1 − d⎡0⎟)⋅ cosπ⎛ d1 [−Ld−0 ⎞C⋅ ()⎤x⋅V⎤− x(xπ)]⎥ dз (x) = d0 + ()d1 0 d0d+(x⎜⎛)d=1 d d+0⎟⎞⋅cos⎢ + ⎜ ⎢⋅()x 1− x⎟2 ⋅V⎥cos; зп 2 ⎥ ⋅()x − x d + Tзп (x) = зT+к2⎜+ Ct ⋅0 ⎝ ⎟ ⋅2cos ⎠ x⋅ ()x−−x x [L⎢1 − CV ⋅Vзп (x)]2 ⎥ ; , ⎢ 0 ⎣ 2 ⎝ 2= ⎠ + ⋅⎢x3 − x2⎣ 3 2 2 ⎥⎥ −⎦⎦ Tзп (x) ⎡Tк C()2t ⎣ − −⎝ ⎛2 −⎠ ⎦ ⎞ ⎣ x3 ⎡ x2π ⎦ ⎤⎤ , ⎣ 2 ⎝ 2 ⎠ d1⎣⎡x3d0 x2 d1 d0 ⎦⎦ ⎡ ⎤⎤ where x3 - coordinate the completion of the formation⎢d 0of+ inverted cone+()d⎜1 − crystal.d0 ⎟⎛⋅ dcos1 −⎢d0 ⎞ ⋅ ()x −πx2 ⎥⎥ ⎢2d0 + ⎝ 2 +⎠⎜ x ⎟−⋅ cosx ⎢ ⋅ ()x − x2 ⎥⎥ where x3 - coordinate the completion of the formation[L of1 − invertedC⎣V ⋅Vзп ( conex)] crystal.− ⋅ ⎣ 3 2 ⎦⎦ = + ⋅ [L1 2CV Vзп⎝(x)]2 ⎠ ⎣ x3 − x2 ⎦ Tзп (x) Tк Ct T (x) = T + C ⋅ ⎣ , , ⎦ where x3 - coordinate⎡ зп the compleк t tion of the formation⎡ of inverted⎤⎤ cone crystal. ()d1 − d0 ⎛ d1⎡− d0 ⎞()d − d π ⎛ d − d ⎞ ⎡ π ⎤⎤ where⎢d x3+ - coordinate+ ⎜ the comple⎟ ⋅ cos1tion0 of the 1⋅formation()x −0 x ⎥ of inverted cone crystal. In turn, the expressionwhere x 3for 0– coordinatethe linear plotsthe completion⎢ dof0 +growing of⎢ jobsthe+ ⎜formation right cone⎟2⋅ cosof⎥ crystal inverted⎢ and cone⋅ ()itsx − cylindricalcrystal.x2 ⎥⎥ 2 ⎝ 2 ⎠ ⎣ x3 − x2 ⎦ − In turn, the expression for the⎣ linear plots of growing⎣ jobs2 right ⎝cone2 crystal⎠ ⎦ and⎣ x3 its xcylindrical2 ⎦ ⎦ part and reverse cone will lookIn turn, like: the expression for the linear plots of growing jobs right cone crystal and its cylindrical where x3 - coordinatewhere the comple x3 -In coordinate turn,tion ofthe the theexpression formation comple tionfor of invertedtheof thelin earformation cone plots crystal. of of growing inverted jobs cone right crystal. cone crystal and its cylindrical part and reverse cone will lookpart like: and reverseIn turn, cone the expressionwill look like: for the linear plots of growing jobs right cone crystal and its cylindrical part and reverse cone will look like: part and reverse cone will look like: In turn, the expression for the linear plots of growing jobsx[d right− d cone] crystal and its cylindrical In turn, the expression dfor( xthe) = lind ear+ plots1 of0 growing jobs right cone crystal and its cylindrical з x[d01 − d 0 ] ; d (x) = d + ;x part and reverse conepart will and look reverse like: cone willз look like:0 1 x1 x[d1 − d 0 ] d (x) = d + x[;d 1 − d 0 ] d з (x) = d1 ;з 0 d з (x) = xd 0 + ; d з (x) = d1 ; 1 x1 (x − x2 )[d1 − d 0 ] d (x) = d−x−[d − d−] d з (x.) = d1 ; з (x 1 x21)[d1 0 d 0 ] x[d1 − d 0 ] = d (dxз)(=x)d= −d 0 + ()x ;− x= + d з (x) d1 ; з 1 d з3(x) .2 d 0 – 27 – ; ()x x−1 x (x − x )[d − d ] 3 2 = − x12 (1 − 0 )[ − ] d з (x) d1 x x2 .d 1 d 0 = d з (x) =()dx1 −− x . d з (x) d1 ; d (x) = d ; 3 2 Simulation speed and temperature grown single crystals з of germanium-based1 ()x3 − modelx2 Simulation speed and temperature grown single(x − crystalsx )[d − ofd germanium-based] model 2 1 0 (x − x2 )[d1 − d 0 ] and the reduced thermal constant ofd з (thex) =materiald1 − [6] isd shown(x) = d in. − Fig. 7-10 for the linear and Simulation speed and temperature()x −зx grown1 single crystals of. germanium-based model and the reduced thermal constant of theSimulation material [6] speed is shown and3 temperature in2 Fig. 7-10 grown for()x 3the − six nglelinear2 crystals and of germanium-based model for the referenceand cosine the inversereduced cone thermal type constant crystal (cosine of the lawmaterial isolated [6] solidis shown line). in Fig. 7-10 for the linear and for the reference cosine inverse andcone the type reduced crystal thermal (cosine constantlaw isolated of the solid material line). [6] is shown in Fig. 7-10 for the linear and Simulation speedfor andtheSimulation reference temperature speedcosine grown andinverse temperature single cone crystals type grown crystal of germanium-based si (cosinengle crystals law isolated of model germanium-based solid line). model for the reference cosine inverse cone type crystal (cosine law isolated solid line). and the reduced thermaland the constant reduced of thermalthe material constant [6] isof shown the material in Fig. [6] 7-10 is shownfor the inlinear Fig. and7-10 for the linear and for the reference cosinefor the inverse reference cone cosinetype crystal inverse (cosine cone typelaw isolated crystal (cosinesolid line). law isolated solid line).

d з (x) = d1 ;

⎡ (L1 − L0 ) ⎤ ⎢L0 + []x − x1 ⋅ − CV ⋅Vзп (x)⎥ ⎣ ()x2 − x1 ⎦ T(x) = Tк + Ct ⋅ , d1 where x2 - coordinate completion of the formation of the cylindrical part of the crystal;

L1 - axial gradient in the crystal at the end of the cylindrical part.

Expression specifying diameter dз(x) of the crystal and the temperature Т(x) on the opposite cone takes the following form:

(d1 − d0 ) ⎛ d1 − d0 ⎞ ⎡ π ⎤ dз (x) = d0 + + ⎜ ⎟⋅cos⎢ ⋅()x − x2 ⎥ ; 2 ⎝ 2 ⎠ ⎣ x3 − x2 ⎦

[L − C ⋅V (x)] T (x) = T + C ⋅ 1 V зп , зп к t ⎡ ⎤ ()d1 − d0 ⎛ d1 − d0 ⎞ ⎡ π ⎤ ⎢d0 + + ⎜ ⎟ ⋅ cos⎢ ⋅ ()x − x2 ⎥⎥ ⎣ 2 ⎝ 2 ⎠ ⎣ x3 − 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 like:

x[d1 − d 0 ] d з (x) = d 0 + ; Sergey P. Sahanskiy. Control of the Shape ofx1 Semiconductor Crystals when Growing in Czochralski Method

d з (x) = d1 ; (x − x )[d − d ] d (x) = d − 2 1 0 . з 1 − ()x3 x2

Simulation speedSimulation and temperature speed grownand temperature single crystals grown of single germanium-based crystals of germanium-based model model and the and the reduced thermalreduced constant thermal of constantthe material of the [6] material is shown [6] in is Fig.shown 7-10 in forFig. the 7-10 linear for the and linear and for the reference cosine inverse cone type crystal (cosine law isolated solid line). 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).

dз(x), cm 12

10

dз(x), cm 812

10 6

8 4

6 2

4 0 0 50 100 150 200 250 2 x, mm Figure 7. Setting the diameter of the crystal germanium: Fig. 7. Setting the diameter of the crystal germanium: x = 50 mm; x = 210 mm; x = 260 mm; d = 0,5 cm; x01 = 50 mm; x2 = 210 mm; x3 = 260 mm;1 d0 = 0,5 cm;2 d1 = 11 cm; 3 0 d = 11 cm 0 50 100 150 200 250 1 x, mm Vз, mm Figure/ min 7. Setting the diameter of the crystal germanium: 1.2 x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d0 = 0,5 cm; d1 = 11 cm;

Vз, mm / min 1.2 1

1

0.8

0.8

0.6

0.6

0.4 0 50 100 150 200 250 0.4 0 50 100 150 200 250 x, mm Figure 8. Setting the rate of withdrawal of germanium at: x, mm d0 Figure= 0,5 cm;8. Setting x1 = 50 the mm; rate ofx2 withdrawal= 210 mm; of x 3germanium = 260 mm; at: d 1 = 11 cm; λ = 0,4 ° C; Fig. 8. Setting the β1d =0rate =0,8; 0,5 of βcm; 2withdrawal = x0,8;1 = 50 β3 mm; = 0,4;of x 2germanium =β 4210 = 0,6; mm; L x0 3at:= = 20260 d 0° =mm;C 0,5/ cm; d 1cm; = L 111 x= cm;1 40= 50°λ C= mm; 0,4/ cm ° C;x2 = 210 mm; x3 = 260 mm; d1 = 11 cm; λ = 0,4 β 1° = C; 0,8; β 1β =2 = 0,8; 0,8; β β23 == 0,8;0,4; ββ43 = = 0,6; 0,4; L 0β =4 =20 0,6; ° C L/ cm;0 = 20L1 =° 40C /° cm;C / cm L1 = 40 ° C / cm

– 28 –

о T, С 955 о T, С 955

950

950 945

945 940

940 935 0 50 100 150 200 250 x, mm Figure 9. Temperature setting pulling germanium: 935d0 = 0,5 cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm; λ = 0,4 ° C; 0 50 100 150 200 250 Fig. 9. Temperatureβ setting1 = 0,8; pullingβ2 = 0,8; germanium:β3 = 0,4; β4 = 0,6;d = L0,50 = cm;20 ° xC /= cm; 50 Lmm;1 = 40 x ° =C 210/ cm mm; x = 260 mm; d = 11 cm; 0 1 2 x3, mm 1 λ = 0,4 ° C; β1 = 0,8; β2 = 0,8; β3 = 0,4; β4 = 0,6; L0 = 20 °C / cm; L1 = 40 ° C/cm Figure dз( x9.), Temperaturecm setting pulling germanium: d0 = 0,512 cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm; λ = 0,4 ° C; β1 = 0,8; β2 = 0,8; β3 = 0,4; β4 = 0,6; L0 = 20 ° C / cm; L1 = 40 ° C / cm

dз(x), cm10 12

8 10

6

8

4

6

2

4

0 0 10 20 30 40 50

2 x, mm Figure 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

0 0 10 20 30 40 50 x, mm Figure 10. Setting the diameter of the germanium crystal with a straight cone: d0 Fig. 10. Setting the diameter= 0,5 of cm; the x 1germanium = 50 mm; x 2crystal = 210 mm;with xa3 straight = 260 mm; cone: d1 d=0 11 = 0,5cm cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm

ModelingModeling the velocity the velocity and temperature and temperature of the single of the crystal single silicon crystal grown silicon for the grown same for crystal the same crystal formform as germanium, as germanium, but using but permanentusing permanent thermal silithermalcon material, silicon shownmaterial, in Fig. shown 11-12 in (with Fig. a11-12 separate (with a separate Sergey P. Sahanskiy. Control of the Shape of Semiconductor Crystals when Growing in Czochralski Method cosinecosine law oflaw formation of formation and re verseand re conesverse crystal cones solid crystal line). solid line).

Vз, mm / min Vз, mm / min 9 9

8 8

7

7

6

6

5

5 4 0 50 100 150 200 250 x, mm 4 Figure 11. Setting0 the drawing50 speed of 100silicon: 150 200 250 Fig. 11. Setting thed drawing= 0,5 cm; speed x = 50 of mm;silicon: x = d 2100 = 0,5mm; cm; x =x1 260 = 50 mm; mm; d x =2 =11 210 cm; mm; x3 = 260 mm; d1 = 11 cm; 0 1 2 3 1 x, mm λ = 0,4 °; β1 = 0,8;λ β =2 =0,4 0,8; °; ββ13 == 0,8;0,4; ββ24 == 0,8;0,6; βL30 == 0,4;20 ° β C4 =/ cm;0,6; LL10 == 4020 °C/cm° C / cm; L1 = 40 ° C / cm о Figure 11. Setting the drawing speed of silicon: T, С d0 = 0,5 cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm; 1470 λ = 0,4 °; β1 = 0,8; β2 = 0,8; β3 = 0,4; β4 = 0,6; L0 = 20 ° C / cm; L1 = 40 ° C / cm о T, С 1460 1470

1450 1460

1440 1450 1430

1440 1420

1430 1410 0 50 100 150 200 250 1420 x, mm Figure 12. Setting the draw temperature of silicon at: d0 = 0,5 cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm; λ = 0,41410 °; β = 0,8; β = 0,8; β = 0,4; β = 0,6; L = 20 ° C / cm; L = 40 ° C / cm 01 2 50 3 1004 0 150 1200 250

x, mm Fig. 12. Setting the draw temperature of silicon at: d0 = 0,5 cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; Figure 12. Setting the draw temperature of silicon at: d1 = 11 cm; λ = 0,4 °; β1 = 0,8; β2 = 0,8; β3 = 0,4; β4 = 0,6; L0 = 20 °C/cm; L1 = 40 °C/cm d0 = 0,5 cm; x1 = 50 mm; x2 = 210 mm; x3 = 260 mm; d1 = 11 cm;

λ = 0,4 °; β1 = 0,8; β2 = 0,8; β3 = 0,4; β4 = 0,6; L0 = 20 ° C / cm; L1 = 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 – Sergey P. Sahanskiy. Control of the Shape of Semiconductor Crystals when Growing in Czochralski Method 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 c.

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.