Diffusion and its applications.

Diffusion and its applications. Concentration gradient diffusion In materials technology concentration gradients are used in order to change the surface composition. Self-diffusion: diffusion of isotopes. Important diffusion applications: Steel hardening via diffusion of N and C. Impurity diffusion in semiconductors (e.g. fabrication of the base and emitter region, resistors etc). Si oxidation Powder metallurgy which relies on mass transport for powder bonding or sintering. W. G. Pfann in 195 patented the application of diffusion for the introduction of impurities in semiconductors with the purpose of varying their conductivity. 1

Diffusion systems: 1. Diffusion at high T from a gas source (infinite source).. Diffusion from an oxide which is rich in the impurity (limited source). 3. Diffusion and annealing from an implanted layer (limited source). Diffusion theories Fick s theory describes the phenomenon using appropriate diffusion coefficients. Applies when the impurity concentrations are low D constant. Atomistic theory: Interaction of impurities with defects. Boltzmann-Matano analysis: is used when the diffusion coefficient depends on the impurity concentration.

Mathematical description of diffusion J = D dc 1 st Fick s law dx where J is the flux (atoms/cm s), C is the concentration (atoms/cm 3 ) & D is the diffusion coefficient (cm /s). The concentration gradient and the direction of diffusion. The quantity dc/dt 0 when J x J x+dx Mass conservation: the time dependent variation of the impurity concentration must be equal to the local reduction of the diffusion flux, i.e. C(x, t) t C(x, t) = D x x When the impurity concentration is low D= constant: C(x, t) t C(x, t) = D x nd Fick s law. It describes diffusion under non-steady-state conditions and under the assumption that D is constant. 3

Steady-state diffusion C (x, t) t = 0, i.e. linear variation of the concentration and J (mass flow) is constant at every cross section of the system. The most common solutions for D=constant are: Constant surface concentration (limitless source). Constant total impurity concentration (S) (limited source or predeposition condition). Constant surface concentration Initial condition: C(x,0)=0 (for t=0) Boundary conditions: C(0,t)=C s & C(,t)=0 C(x, t) = Cs erfc x Dt where C s is the constant surface concentration (atoms/cm 3 ) and erfc is the complementary error function which is defined by equation: erfc x Dt ( x 4Dt ) = 1 exp( z )dz π 0 For small values of the integral (όρισμα): erfc( x 4Dt ) 1 x 4Dt 4

Time evolution of the normalized impurity distribution described by an erfc. Predepostion diffusion- Constant total amount of dopant Initial condition: C(x,0)=0 Boundary conditions : C (x, t)dx = S και C(x, )=0 0 The distribution is Gaussian: C(x, t) = S πdt x exp 4Dt The surface concentration C s (x=0) is: C s = C(0, t) = S πdt Time evolution of the normalized impurity distribution described by a Gaussian. 5

Differences between erfc and Gaussian in logarithmic and linear scales x = Dt Correlates the 3 basic parameters x, t & Τ. The temperature dependence is introduced via the relation: D = Do exp( ΔED RT). 6

The microscopic theory of diffusion The diffusion coefficient. The atom movement between adjustment cross sections is possible when: 1. A suitable empty site exists, e.g. a vacancy.. The diffusing atom has sufficient energy to overcome the energy barrier ΔΕ D imposed by its neighboring atoms. When the atom occupies a neighboring vacancy, the vacancy moves in the opposite direction and occupies the initial atom s site. C 1 C C(x) C 1 S 1 α Atomic jumps of equal probability across the atomic planes 1 & C α x Diffusion occurs against the concentration gradient. Assumptions: 1. The atoms on the cross sections 1 & perform atomic jumps with frequency ν.. The atom jumps or or have equal probability the atoms tend to cross surface S only during 1/ of the number of the jumps. 7

The diffusing atom must have sufficient energy to overcome the energy barrier ΔΕ D. α is the distance between adjacent atom planes, n i (cm - ) & c i (cm -3 ) are the surface & bulk concentrations, respectively. n i =αc i. Atomic kinetic model -Ατομικό κινητικό πρότυπο. The diffusion current J is: 1 Replace n i =αc i J = αν( ) The concentration gradient J 1 c να x = (1) 1 1 J = n1ν n ν c 1 c c x is: c c c 1 = α x the minus sign indicates that diffusion occurs against the concentration gradient. Via a comparison of (1) with Fick s law c J = D x D 1 να = (-dimensions) & D 1 να 6 = (3-dimensions) 8

The temperature dependence of the diffusion coefficient: E It is experimentally demonstrated that: D = D exp D o where D o is kt a constant, E D is the activation energy for diffusion (diffusion introduces strain in the lattice =>energy barrier). Diffusion mechanisms Diffusion with mutual exchange of lattice-atoms Diffusion via hopping of interstitial atoms. Vacancy assisted diffusion. If the atom vibrates with frequency ν ο around its equilibrium position (i.e. makes ν ο attempts/sec to overcome the energy barrier), the probability that it makes a successful jump is given by the Boltzmann factor : exp(-e m /kt). Therefore, it changes position with a frequency: E ν = ν ο exp m kt The atom can move only if there is an adjustment empty lattice position in its 1 st nearest neighbor shell (coordination number Z). The probability that there exists and empty lattice position is exp(-e v /kt), where Ε v is the formation energy of a vacancy. the probability that an atom migrates from one lattice position to a neighboring one is: E ν = Z ν ο exp m kt E exp v kt E = Z ν ο exp D kt, where E D =E m +E v. 9

Therefore: 1 = ν α E D Z ο exp D 6 kt E = D exp D o kt, where D Zν α o 1 = ο. 6 Arrhenius plot (logd-1/t) D o & E D D(T) για Fe:Ni. D varies by 16 orders of magnitude for ΔΤ = 0-1000 o C. 10

Results based on experimental observations: 1. In solids: ΔΕ D = 0. - ev per atom.. In several materials ΔΕ D Τ m 3. In liquids D has a weak Τ-dependence (10-4 -10-6 cm s -1 ) 4. The atoms diffuse fast on surfaces, grain boundaries & dislocations Extrinsic diffusion When the material is doped and the carrier concentration > the intrinsic carrier concentration, the value of D depends on n. Representative values of n i at 1000 o C: n i (Si)= 5x10 18 cm -3 & n i (GaAs)= 5x10 17 cm -3. D depends on n because when the carrier concentration increases both the position of E F and the vacancy concentration change. The vacancy concentration C V as a function of T is E E C = C exp F i V i kt where C i & E i are the vacancy concentration and Fermi level for the intrinsic material. When the diffusion is vacancy assisted, the value of D will depend on their concentration. For small values of n, E F & E i coincide. However, when the semiconductor becomes extrinsic E F moves in the gap and thus 11

exp ( E E kt) F of D increase. i >1. both the vacancy concentration and the value Variation of D versus the carrier concentration. The extrinsic diffusion is described by the equation C t F = x = C D instead of x x C t = D x C where D=const. The variation of D versus the concentration C is given by γ C D = D S where D S & C S are the values on the surface. In this C S case the diffusion equations are solved arithmetically. Normalized impurity distributions in extrinsic diffusion under the assumption that the concentration on the surface is constant. 1

When γ>0 the value of D decreases with decreasing C the distributions look like step functions (abrupt junction) abrupt junctions can be fabricated via diffusion in a substrate with the opposite kind of dopant. 13

Diffusion systems. Open vertical or horizontal quartz reactors operating at 900-100 ο C. Sources : solid, liquid or gaseous (better control). The T-gradients cause thermal stress introduction of defects (e.g. dislocations, wafer deformation etc) precise T control is necessary. Typical T-gradient: 3-10 ο C/min while at the center of the diffusion system (length 10-100 cm) the T is controlled within ±0.5 o C. Schematic of a diffusion system with gas sources. 14

Diffusion applications. Metal hardening 1. Carburization of steel is a surface hardening process which relies on diffusion. Takes place at 900 ο C in C-containing atmosphere, e.g. CH 4 -CO-H. The thickness of the C-rich surface layer increases with a rate t.. Hardening via diffusion of N and/or N & C. Oxidation. Proceeds with oxygen or metal diffusion via the existing native oxide Native oxidation is undesirable since it is not precisely controlled. SiO Gate dielectric & mask against impurity diffusion in μ-electronics Oxidation saturates dangling bonds at the Si surface reduces the density of surface states in the gap and the density of immobile charge. The native SiO is 15-40Å thick & ragged useless 15

The Si oxidation takes place at 900-100 ο C in dry or wet Ο : Dry oxidation: Si+O SiO (density.5gr/`cm 3 ) Wet oxidation: Si+H O SiO +H (density.15gr/cm 3 ) Oxidation mechanism 1. The oxygen is transferred to the wafer surface via a stagnant layer (remember CVD). Oxygen (Ο or Ο) diffuses via the native SiO to the Si-SiO interface consumption of Si from the substrate and growth of SiO. When the thickness of the consumed Si is x the thickness of the grown SiO is.7x. The thickness of the grown oxide is x 1/ D N k t = 1 + o 1 k Dn Where k is the reaction constant, n is the number of oxygen molecules in the unit volume of SiO (n=.x10 cm -3 for dry O & 4.4x10 cm -3 for H O). Limiting cases 1. short t (x 40Å in dry Ο & x 1000Å in Η Ο) limiting step is the oxidation rate (diffusion is fast while the reaction constant k is small) : N k x = o t x = n B A t. For longer t growth is limited by diffusion 1/ 1/ NoD x = t x = n B 1/ 1/ t 16

Β/Α : is the linear growth rate constant & Β : is the parabolic rate growth constant. Si oxidation rate in the T-range 700-1300 ο C. For short t the oxidation rate depends linearly on t (reaction rate limited). For longer t the growth is parabolic (diffusion limited) The activation energy for oxidation (E ox ) is calculated from Arrhenius plots lnx-1000/t. Ε ox in the linear region is -.05 ev/molecule ( 1.83eV/molecule is needed to break Si-Si bonds). In the parabolic region Ε ox for dry oxidation is 1.3eV/molecule ( 1.18eV/molecule is needed for oxygen diffusion in SiO ). In the parabolic region Ε ox for wet oxidation: E ox =0.78eV/molecule and E d =0.79eV/molecule for H O diffusion in SiO ). 17

Wet oxidation is 3 times faster than dry due to the larger solubility of Η Ο in SiO, which counter balances the smaller diffusion coefficient. Modifications of thermal oxidation: rapid thermal oxidation, introduction of halogens (F, Cl etc) 18

Redistribution of impurities during oxidation. Κατά την οξείδωση του Si οι προσμείξεις που βρίσκονται κοντά στην επιφάνεια του υποστρώματος ανακατανέμονται λόγω : (1) της βαθμίδας συγκέντρωσης μεταξύ υποστρώματος και αναπτυσσόμενου οξειδίου και () της διαφορετικής συγκέντρωσης ισορροπίας της πρόσμειξης ανάμεσα στο Si και το SiO. Το φαινόμενο είναι ανάλογο της αποβολής πρόσμειξης από κρύσταλλο που αναπτύσσεται από το τήγμα. Ορίζεται ο συντελεστής k που περιγράφει την διαφορά της συγκέντρωσης ισορροπίας στο Si και το SiO : συγκέντρωση ισορροπίας της πρόσμειξης στο Si k =. συγκέντρωση ισορροπίας της πρόσμειξης στο SiO Ένας άλλος μηχανισμός που συμβάλλει στην ανακατανομή της πρόσμειξης απαντάται όταν ο συντελεστής διάχυσης της πρόσμειξης στο οξείδιο είναι μεγάλος οπότε είναι δυνατόν τα άτομα της πρόσμειξης να φτάσουν στην επιφάνεια του οξειδίου και να διαφύγουν στην αέριο φάση. Τέλος ανακατανομή της πρόσμειξης συμβαίνει λόγω του μεγαλύτερου όγκου του αναπτυσσόμενου SiO σε σύγκριση με τον όγκο του Si που καταναλώνεται (V SiO V Si ή d Si =0.44d SiO ). 19

Σχήμα ΙΙΙ.1: Ανακατανομή των προσμείξεων κατά την οξείδωση του Si. Στις περιπτώσεις (α) & (β) : k<1 το Si αποβάλλει πρόσμειξη στο SiO. Ειδικότερα στο (β) η πρόσμειξη διαχέεται γρήγορα μέσω του SiO ο βαθμός μείωσης της συγκέντρωσης της πρόσμειξης στο Si είναι μεγάλος. Στις περιπτώσεις ( c) & (d): k>1 το οξείδιο αποβάλλει πρόσμειξη στο Si. Ειδικότερα, όταν η πρόσμειξη διαχέεται βραδέως μέσα στο SiO, όπως στο ( c), εμφανίζεται συσσώρευση της πρόσμειξης στην επιφάνεια του Si. Αντίθετα όταν η πρόσμειξη διαχέεται γρήγορα μέσα στο SiO, συμβαίνει απώλεια στην αέριο φάση και εκκένωση της επιφάνειας του Si από πρόσμειξη. 0

Η επίδραση του κρυσταλλογραφικού προσανατολισμού στην ταχύτητα οξείδωσης. Παραβολική περιοχή: η ταχύτητα οξείδωσης είναι ανεξάρτητη του κρυσταλλογραφικού προσανατολισμού του Si (η ανάπτυξη περιορίζεται από την διάχυση των οξειδωτικών ριζών μέσα από το αναπτυσσόμενο οξείδιο). Στην γραμμική περιοχή η ταχύτητα οξείδωσης εξαρτάται από την ταχύτητα ενσωμάτωσης των ατόμων του Si στο πλέγμα του SiO εξαρτάται από την επιφανειακή πυκνότητα του Si και τον προσανατολισμό του υποστρώματος. 1 η προσέγγιση: Η ταχύτητα οξείδωσης του Si (111) >> του Si (100) Si Si κατά ( B A) (111) : ( B A) (100) 1.16:1. Ακριβέστερη προσέγγιση: 3D πλέγμα του Si και το σχετικό μέγεθος των ατόμων του Si και του οξυγόνου η πειραματική τιμή ( B A) Si (111) : ( B A) Si (100) 1.68:1. 1