Analysis of amorphous-silicon deposition conditions
in plasma-silane mixtures at low pressure
S. V. PETROV, E. P. PROKOPIEVand E. M. SOKOLOV
Research Institute of Material Science and Technology - 103460 Moscow, K-460 Russia (ricevuto il 19 Novembre 1996; approvato il 27 Gennaio 1997)
Summary. — Approaching the mass transfer theory from the viewpoint of a gas mixture (SiH4-H2, SiH4-He) flow configuration in r.f.-glow discharge conditions at the extreme point, the formula of amorphous-silicon (a-Si » H) deposition rate on various substrates for an experimental reactor is given. The offered formula allows clearly to analyze the dependence Vd(h) upon the main technological parameters of the process and the geometrical parameters of a diode reactor cell and to have numerical evaluations of the deposition rate. Calculated and experimental values Vd(h) in the case of r.f.-glow discharge in the SiH4-H2 gas mixture for standard conditions of the process are in satisfactory agreement with each other, which confirms the real character of the offered model.
PACS 81.15 – Methods of deposition of films and coatings; film growth and epitaxy.
1. – Introduction
Amorphous-silicon (a-Si » H) films on various substrates have important significance in modern microelectronics [1-3]. The main method of such film deposition is r.f.-glow discharge in silane gas mixtures with various gas carriers (for example, SiH4-H2, SiH4-He). The main dependences of film growth on various substrates are
connected, hence, to the fundamental characteristics of low-temperature silane plasma—its composition, energy of its particles, hydrodynamics of gas mixture flow in a reaction chamber (a diode reactor cell). The composition of silane plasma has been the subject of many fundamental researches [1, 2]. It was found that the plasma contains, mainly, silane, hydrogen, neutral-silane fragments SixHy (x 41, 2, R, y4
0 , 1 , 2 , R), atomic hydrogen, positive ions SixHy1 and H1, H21, H31 and electrons.
Thus, in our case the r.f.-glow discharge represents a typical low-temperature plasma, in which the electrons, the ions and neutral-silane fragments are not in equilibrium with each other. Typical temperatures of ions and, apparentely, neutral-silane fragments are about hundreds of kelvin degrees, while the temperature of electrons changes from 1000 up to 10 000 K.
It is shown [1, 2, 4-9] that the main particles providing the a-Si » H deposition are neutral-silane fragments SiHn, in particular, SiH2 and SiH3. The flows of the other
fragments and ions are very small in comparison with SiHn, but they play an important
role in the physico-chemical processes on the growing surface (for example, treatment of the growing film by high-energy ions SixHy1and H1). In this work we deal with the
problem of the analysis of the conditions of the a-Si » H deposition by the interaction of SiH2and SiH3radicals with a growing surface. In the beginning we shall focus on the
features of the experiments.
2. – Experimental conditions
Experiments on a-Si » H deposition were carried out in a stainless-steel reactor with flat horizontal electrodes made also of stainless steel. The main part of the reactor is a rectangular diode cell, inside which the pressure 0.1–5 Torr is supported by a vacuum pump. This diode cell contains two parallel electrodes: the top electrodes, which the wafer is located on, is stationary and heated by the resistive heater, the bottom electrode is moved in the vertical direction, whereby the distance h between electrodes is changed, and just on the bottom electrode the r.f.-potential is supplied. The gas mixture moves in the r.f.-glow-discharge plasma through 74 apertures in a top of the bottom electrode (area of each aperture 4 0.005 cm2) in the direction normal to the
wafer surface.
Optimization of the electronic properties of a-Si » H films is usually made in an experimental way [1, 2, 6-10]. For the chosen geometrical dimensions of the reactor and the nature of r.f.-glow discharge the process parameters such as temperature of the wafer surface Ts(K), total pressure p (Torr) in the reactor, r.f.-power of glow discharge
P (W), gas mixture flow Q ( cm3Omin ) and distance h (cm) between the electrodes are
changed. Thus a correlation between variable technological parameters and optimum electronic properties of a-Si » H films is searched for.
The theoretical calculations of a-Si » H deposition rate are of great importance to understand the physical sense of this correlation.
3. – Statement of the problem
To solve the a-Si » H deposition rate problem it is necessary to offer a reasonably idealized hydrodynamic model of gas mixture current at the extreme point [11-18] in a diode reactor cell. The evaluation for standard conditions of the process (Q 4 50 cm3Omin , p 4 1 Torr, Ts4 500 K) gives a value of the Reynolds number (Re) of the
order 1021–1022, therefore the gas mixture flow is considered laminar. Thus we
neglect the flowing streams of the gas flow, that is justified in the first approach. As the dimension of the electrode is considerably greater than the distance between the electrodes, the kind of the gas flow should not have a strong influence on the boundary effects on the periphery of the substrate. As the gas mixture is strongly diluted by hydrogen, it is possible to neglect the dependence of the flow on the ratio of reagents and the degree of their conversion. The process is considered isothermal. Besides, we admit incompressibility of the gas mixture, as the flow has a small value of Max number. We admit that the physico-chemical properties of a gas mixture in a plasma are constant (i.e. hypothetic isothermal silane plasma at low pressure, with similar properties, is considered).
In our calculations a cylindrical system of coordinates r 4 (r, z) is used. The substrate of radius Rs(cm) is placed at a distance h from the r.f.-glow discharge
electrode. We admit that the gas mixture is entered in the reactor space (plasma) with similar axial rate vz0(cm Os):
vr(r , 0 ) 40 , vz(r , 0 ) 4vz0, 0 GrGRs.
(1)
On the surface of the substrate in conditions of strong dilution of the reagents by hydrogen, we have
vr(r , h) 40 , vz(r , h) 40 , 0 GrGRs.
(2)
On the axis of symmetry (r 40) the conditions of symmetry are given by ¯vz
¯r 4 0 .
(3)
The gas mixture is considered similar to a Newton liquid with constant properties. The equations of hydrodynamics describing distributions of rates v(r , t) are written in the following form:
˜v 40 , ¯v ¯t 1 v (v ˜) 4 2
g
1 rGh
˜p 1n˜2v . (4)Here rGis the density of a gas mixture (gOcm3), n is the kinematic viscosity (cm2Os ).
The Reynolds number is defined as
Re 4 vz0h
n .
(5)
The research of the system (4) was carried out in [13-18] on the basis of paper [11], where it was supposed that the form of the flow, bending around the substrate, does not strongly change due to the edge effects. It enables (at the small values of Re G4) to have the analytical solution of the system of the main hydrodynamic equations (4) with boundary conditions (1)-(3), i.e. to find components of rate v of the gas mixture flow, used in further calculations:
vz(z) 4vz0
g
1 2 3 z2 h2 1 2 z3 h3h
, (6) vr(r , z) 4 3 vz0rz(h 2z) h3 . (7)4. – One-dimensional model of the deposition rate
To solve the problem of deposition rate we use the one-dimensional Neumann-Zshauer model [11] of the a-Si » H deposition, in which the availability of combined effects of convection and diffusion of chemical reagents to the substrate surface is admitted. The availability of surface reactions is implied in the first-order reaction mechanism. Thus, with the above-mentioned distributions of rates (6) and (7), the balance of
reagents in the diode reactor cell is defined by the ordinary diffusive-convective equation
˜(vci) 4˜(Di˜ci) 1F .
(8)
Here F is the function of source of the i-reagent, v is the flow rate of the gas mixture with components vzand vr, ciis the concentration of the i-reagent, and Diits molecular
diffusion coefficient. Further, everywhere we shall consider current processes on the key reagent SiH4, therefore, we shall omit the index i with reference to reagents. The
function of source F, included in (8), can be written as
F 42 (kv1c 1kv2c) 42 (kv11 kv2) c 42 kvic ,
(9)
where kviare the constants of rate of the following reactions:
SiH41 e2K SiH21 2 H ,
(10)
SiH41 e2K SiH31 H .
(11)
These constants are equal to
kvi4 anebasivdb .
(12)
Here aneb is the average value of the electron density in the plasma, sithe cross-section
of SiH2 and SiH3 radical generation by electron impact of SiH4molecules, vd the drift
velocity of electrons in plasma. For simplicity of the solution we have neglected volumetric reactions of SiH2 and SiH3radicals in plasma. The values asivdb represent
the average value of the si and vd product and are determined by the standard
expression of the gas-kinetic theory [1, 5, 8, 9]:
asivdb 4
em Q si(e) f (e) de 4o
2 e m em Q si(e) ke f (e) de , (13)where em is the threshold energy of radical generation (eV); f (e) is the function of
electron energy distribution; e and m are electron charge and mass. We can see that in formula (9) according to the definition kvis equal to
kv4 (kv11 kv2) 4 as1vdbaneb 1 as2vdbaneb .
Equation (8) is supplemented by the boundary conditions
c(r , 0 ) 4c0, for z 40 .
(14)
On the surface of the substrate two types of boundary conditions are used. In a mode of process limited by mass transport the concentration on the surface is equal to its equilibrium concentration ceq, i.e.
c(r , h) 4ceq, for z 4h .
(15)
Taking into account the surface process proceeding on the first-order reaction, the second type of boundary conditions is connected with the constant ks:
Fc4 2 D
dc
dz
N
z 4h4 ks
(
c(h) 2ceq)
.Similarly to eq. (8) the equations of SiH2 and SiH3 radical material balance must be
written as
˜(vcR) 4˜(DR˜cR) 2 anebasRvdb c 40 .
(17)
Here R f SiH2, SiH3. Boundary conditions near the input of a diode reactor cell can be
written as
cR( 01, r) 4c 0
R, for z 401.
(18)
The symbol 01 means that on the electrode, strictly speaking, cR( 0 , r) 40, and
near the boundary area of infinitely small dimensions, cR( 01, r) 4c 0
R. The boundary
conditions on the surface are still determined by expression (16).
Thus, in a general case, the problem of finding radical and monosilane concentrations in the diode reactor cell is reduced to the solution of three equations of the material balance (8) and (17). Thus we shall note that for small Reynolds numbers (up to about Re A 4) it is possible to use the analytical solutions (6), (7), included in (8) and (17), and for large Reynolds numbers for the rate components vr(r , z) and vz(r , z)
the nonlinear solution of Navier-Stocks equations [11] is used.
5. – Analytical solution of the deposition rate problem
We shall show how to simplify the problem of deposition rate using the experi-mental data [8, 9]. Under process standard conditions (p 40.05–5 Torr) and silane concentration xc4 0.1 moles P part the density of SiH4 molecules in a gas mixture is
about 1017–1018 cm23, while according to the last experimental data [8, 9] the density of
SiH3radicals is about 1011–1012 cm23, the density of SiH2radicals less than 1010 cm23,
and the density of SiH radicals is about 1010 cm23. In other words, in practice we have
a small degree of SiH4molecule conversion in plasma, and consequently it is possible to
consider c(r , z) 4c0. It relieves us of the necessity to solve eq. (8). Therefore, we
should concentrate our attention on the solution of eqs. (17) for SiH2and SiH3radicals.
For an analytical study we shall simplify the problem supposing that the a-Si » H deposition rate is basically determined by the SiH3radical flow [8, 9]. Then there is one
equation of kind (17) for SiH3radicals.
We shall normalize SiH3 radical concentration in eq. (16) dividing cR by DcR4
c0
R2 cReq. Then (16) may be written as
Sh 4FR h DcRDR 4 2dc * (z * ) dz *
N
z * 414 Da [c * ( 1 ) 2 (c eq R )* ] , (19)where z * 4zOh, c *R4 cRODcRand
Da 4 ksh
DR
(20)
is the nondimensional Damköller number. In its turn
Sh 4 FRh DcRDR
(21)
It is easy to show that in the one-dimensional case eq. (17) in nondimensional form can be written as d2c * R dx *2Pe v *z 2 dc *R dz * 2 Pe G 0 R4 0 , (22) where v *z 4 1 2 3(z * )21 2(z * )3, Pe 4 vh DR (23)
is the Peclet criterion and
G0
R4
kvc0h
vz0(cR02 cReq)
(24)
is the nondimensional SiH3radical generation rate.
In the general case the problem of deposition rate described by eq. (22) is reduced to its integration and finding c *R(z * ). Thus, the a-Si » H deposition rate Vd(h) (mkm Oh)
proceeding from (21) is determined by the expression [9]
(25) Vd(h) 43.6Q107 m( a-Si : H) MRQ r( a-Si : H) D0 R(cR02 cReq) h
g
Ts T0h
2 Sh 4 4 4 .62 Q 108 pp0 760 RT2 0 D0 R h Sh (x 0 R2 xReq) .Here m( a-Si»H ) is the molecular weight of a-Si » H, MRis the molecular weight of SiH3
radicals, r( a-Si»H ) is the density of a-Si » H, xRis the SiH3radical mole concentration,
p is the pressure in the reactor (Torr), p04 1 .013 Q 106dyn Ocm2, R is the universal gas
constant, T04 273 K.
6. – Analysis of process modes
The solution of eq. (22) on Neumann-Zshauer [11] can be written as
c *R(z * ) 4cR*02 erf
u
0 .9o
5 Pe 8v
1 erfy
o
5 Pe 8 (z * 20.9)z
erfu
0 .9o
5 Pe 8v
1 erfu
0 .1o
5 Pe 8v
, (26)where erf (x) is a function of errors. Hence
dc * (z * ) dz *
N
z * 414 Sh 4o
5 Pe 8 erfu
0 .9o
5 Pe 8v
1 erfu
0 .1o
5 Pe 8v
. (27)Thus, in the process mode limited by mass transfer, the a-Si » H deposition rate is given by the expression
(28) Vd(h) 4 4 4 .62 Q 108 pp0Ts 760 RT2 0 DR0 h x 0 Rh( Si , h)
o
5 Pe 2y
erfu
0 .9o
5 Pe 8v
1 erfu
0 .1o
5 Pe 8v
z
21 .Here h( Si , h) is a thermodynamic parameter, equal to
h( Si , h) 412 x eq R x0 R (29)
and determining the part of SiH3radicals reacting on the surface. Similarly, in case of
process mode limited by a surface reaction of the first order, according to [11], the deposition rate will be written as follows:
Vd(h) 44.62Q108h( Si , h) pp0Ts 760 RT02 D0 R h x 0 RDa
y
cR*01 Da Q g( 1 , Pe) 1 1DaQg(1, Pe) 2 c *eqz
, (30) where g(z * , Pe) 4o
2 p 5 kPe{
erfy
o
5 Pe 8 (z * 20.9)z
1 erfu
0 .9o
5 Pe 8v
}
. (31)We shall consider asymptotic cases of formulas (28) and (30). For small Pe numbers the mass transfer in a diode reactor cell is completely determined by radical diffusion and for large Pe numbers it is determined only by a thin boundary layer.
Thus, at Pe c 1 formula (28) will be
Vd(h) 44.62Q108h( Si , h) pp0Ts 760 RT2 0
o
5 8 pkPe D0 R h x 0 R (32) and at Pe b 1 Vd(h) 44.62Q108h( Si , h) pp0Ts 760 RT2 0 DR0 h x 0 R. (33)In its turn formula (30) for the process mode limited by diffusion (Da c 1 , Pe b 1 , so Sh 41) is transformed into formula (33). And for a deposition process determined by the surface reaction (Da b 1 , Sh 4Da ),
Vd(h) 44.62Q108h( Si , h) pp0Ts 760 RT02 D0 R h Da Q x 0 R4 4 .62 Q 108h( Si , h) ks pp0Ts 760 RT02 x0 R. (34)
determined by the well-known expression of gas kinetic theory: ks4 dvR 4 4 bk 1 2bkO2 k8 k0TsOpMR 4 , (35)
where d is the effective coefficient (probability) of SiH3 radical adhesion, vR is the
average thermal speed of radicals, bk the ordinary coefficient of adhesion, k0 the
Boltzmann constant.
Close analysis of experimental data [2, 6, 7] permits to conclude that for usual process conditions in the range p G1 Torr the deposition process is executed in a kinetic area and described by formula (34), and in the range p F1 Torr the diffusion mode is realized, where the deposition process is determined by radical diffusion through a thin boundary layer and is described by formula (32).
7. – Definition of x0 R
We shall begin now to define the mole part of radicals xR0 in the plasma mixture,
proceeding from the equation of the material balance [8]
dxR dt 4 anebasvdb xc2 xR tR 2 xR tD 1 FR. (36)
Here xc is the silane mole part in the gas mixture, FR the term adequate to the
processes of radical generation and recombination, such as, for example, H 1SiH4K
SiH31 H2, SiH31 SiH3K Si2H6, etc. (see [2]), tRis the time of radical presence in the
plasma, 1 tR 4 vz0 p0 p Ts Tin 1 h 4 Qin 60 Q S Q h Ts Tin p0 p , (37)
where S 47430.00540.37 cm2 is the total area of the apertures in the r.f.-electrode
for the input of gas mixture, Tinthe temperature of the gas mixture at the input of the
reactor, tD the time of radical diffusion.
Neglecting the term FR in the right part of eq. (36) and equating the right part of
this equation to zero, we have the expression for the definition of the mole part of radicals x0
Rin the plasma steady-state conditions:
x0 R4 anebasvdb 1 OtR1 1 OtD xc. (38)
8. – Main formulas for the calculations
So, for the kinetic region of the process at p G1 Torr the a-Si » H deposition rate is given by the following expression:
Vdk(h) 44.62Q108h( Si , h) ks pp0Ts 760 RT02 anebasvdb 1 OtR1 1 OtD xc. (39)
In the diffusion region at p D1 Torr the deposition rate is defined by the formula Vdd(h) 44.62Q108h( Si , h) pp0Ts 760 RT2 0 DR0 h
o
5 8 pkPe anebasvdb 1 OtR1 1 OtD xc. (40)For standard conditions of the a-Si » H deposition process [10] p 41 Torr,
Q 450 cm3O min, Ts4 500 K, P 4 40 W, xc4 0.1 moles P part, Tin4 300 K, h 4 5 .2 cm,
h( Si, h)B1, anebB108cm23, asvdbB5Q1029cm3Os, tDB5.2Q1023s, DR04400Op cm2Os [5].
In this case the calculation of ks under formula (35) at bk4 0 .1 [8, 9] gives ksB
1 .434Q103cm Os. The calculation of the a-Si » H deposition rate under formula (40) gives
Vdd(h) 40.432 mkmOh, while the calculation of deposition rate for kinetic region gives
Vdk(h) 41.171 mkmOh. The value of deposition rate for the general case of a transient
process from the general formula (25) is Vd(h) 40.316 mkmOh. The experimental
values of deposition rate under these conditons are 0.3–0.6 mkm Oh [5, 7, 10], i.e. the value of Vdk(h) is much greater than these experimental values of deposition rate. In
their turn the calculated values Vd(h) and Vdd(h) are close to the experimental ones
Vd(h), i.e. they are within the experimental interval of deposition rate. Thus, the
process of a-Si » H deposition seems to be in the diffusion or transient region. In general, this conclusion is confirmed by the present experimental data [5-7, 10]: at p E 1 Torr the process is in the kinetic region complicated by the mass transfer phenomena and at p D1 Torr the process is clearly in the diffusion region.
8. – Conclusion
Thus, the presented theoretical model of a-Si » H deposition in a r.f.-glow discharge in SiH4-H2 and SiH4-He gas mixtures permits to analyze in an evident way the
dependence of the deposition rate on the main technological parameters of the process and on the geometrical dimensions of the diode reactor cell and to have numerical evaluations of the deposition rate, that are of significant interest for the experts in the area of material science.
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