Unformatted text preview: 6 Continued 611 Analogous calculation for valence band ⇒
3
2 2πm kT E − EF E − EF C exp − C
n = 2 = N eff exp − C
h kT kT *
n
2 (6.8) 3
2 2πm kT V exp − EV − E F = N eff exp − EV − E F p = 2 h
kT kT *
p
2 (6.9) V
C
N eff , N eff : effective densities of state (∝ T3/2 !) C
V
n ⋅ p = N eff ⋅ N eff e − Eg
kT 3 Eg kT * * 2 − kT
(mn m p ) e
= 4
2 2πh 3 (6.10) Eg = EC  EV, similarities with law of mass action.
Intrinsic semiconductor n =p
C
V
ni = pi = N eff N eff ⋅ e − Eg
2 kT kT = 2 2πh 2 3/ 2 3
4 *
(mn m* ) e
p − Eg
2 kT (6.11)
Eg [eV] ni [cm3] Ge 0.67 2.4 x 1013 Si 1.1 1.5 x 1010 GaAs 1.43 5 x 107 Cu / 8.5 x 1022
612 T=300K (6.8), (6.9) ⇒
C
n = p = N eff ⋅ e EF
kT ⇒e e 2 EF
kT − EC
kT = V
= N eff ⋅ e V
N eff N C
eff e − EF
kT e EV
kT EV + EC
kT V
EV + EC kT N eff
ln C =
⇒ EF =
+
2
2
N eff mp
EL + EV 3
+ kT ln *
2
4
m
14 4 n
23
* = 0 for m * = m *
p
n *
m * ≠ mn ⇒
p weak Tdependence of EF 613 (6.12) 6.2 Doping
n i, p i: far too small for sufficient current densities,
strongly Tdependent. Unwanted doping by impurities ⇒ n > ni.
ni = 5 x 107 cm3 E.g., GaAs (300K) n ≈ 1016 cm3
for highpurity single crystals (1990)
Donnors: donate electrons to conduction band ⇒ n increases.
E.g., group V atoms in Si and Ge: P, As, Sb
Si: 3s2 3p2 → sp3 hybride Donor: s2 p3 → sp3 hybride +e Tetrahedral bonding preserved by donors and acceptors.
Acceptors: accept electrons from valence band in sp3 hybride
⇒ p increases.
E.g., group III atoms in Si, Ge: B, Al, Ga, In 614 Fig. 6.9 Qualitative positionof the ground state levels of donors and acceptors relative to the
minimum of the conduction band EC and the maximum of the valence band EV. The quantities
Ed and Ea are ionization energies of the donor and acceptor, respectively. Fig. 6.10 Schematic representation of the effect of a donor (a) and an acceptor (b) in a silicon
lattice. The valencefive phosphorus atom is incorporated in the lattice at the site of a silicon
atom. The fifth valence electron of the phosphorus atom is not required for bonding and is thus
only weakly bound. The binding energy can be estimated by treating the system as a hydrogen
atom embedded in a dielectric medium. The case of an acceptor (b) can be described similarly:
the valencethree boron accepts an electron from the silicon lattice. The hole that is thereby
created in the valence band orbits around the negatively charged impurity. The lattice constant
and the radius of the defect center are not drawn to scale. In reality the first Bohr radius of the
″impurity orbit″ is about ten times as large as the lattice parameter.
615 Estimate of the ionization energy Ed of a donor
H atom Donor Energy levels: ε0 → ε0ε me e 4
1
E n = ( −)
2( 4πε 0 h ) 2 n 2 ε describes screening Ground state (n = 1):
E1 = 13.6 eV
(kT ≈ 25 meV at 300K) Bohr radius:
h2
rB = ε 0
= 0.053 nm
πme 2 εSi = 11.7 *
me → mn ≈ 0.3 me effective mass of a Si
conduction electron
Ed = 13.6 eV ⋅ 0.3
≈ 30 meV
11.7 2 Ge:
ε Ge = 15.8
⇒ E d ≈ 6 meV
*
m n ≈ 0.12me
r = rBεHL ⇒ ″smearing″ of the
bound donor valence electron
over ≈ 103 atoms
⇒ continum approximation
for ε justified. Exited states (n > 1) between ground state and conduction
band edge:
 spacing decreases with increasing energy
 finally join conduction band continuum.
616 Hydrogen model ⇒ Ed, Ea independent of group V donors
and group III acceptors.
Yields only order of magnitude.
(Hole bound to acceptor traeted like electron bound to donor). Experimental ionization energies
Donors (Ed)
P [meV] Acceptors (Ea) As Sb [meV] B [meV] Ga In [meV] [meV] [meV] [meV] Si 45 49 39 Si Ge 12 12.7 9.6 Ge Donor: Al 45 57 65 16 10.4 10.2 10.8 11.2 neutral for kT << Ed, positive upon exitation. Acceptor: neutral for kT << Ea, negative upon exitation.
Electrically active impurities in Si and Ge single crystals
> 1012/cm3 (1994).
⇒ Ge (ni = 2.4 x 1023 cm3 at 300K) intrisicly obtainable
Si (ni = 1.5 x 1010 cm3 at 300K) never intrisic
Many electrically inactive impurities: cannot bo ionized. 617 Carrier densities in doped semiconductors
Fig. 6.11 Explanation of the
notation commonly used for
carrier
and
impurity
concentrations in n and ptype semiconductors: n and p
are the concentrations of
″free″ electrons and holes. the
total concentrations ND and
NA of donors and acceptors
consist of the density of
0
0
neutral, N D and N A , and
+
−
ionised, N D or N A , donors
and acceptors, respectively.
Electrons in the conduction
band (density n) and holes in
the valence band (density p)
originate
either
from
interband exitations or from
impurities. ″Mass action law″ also holds for doped semiconductors
if E  EF >> 2kT:
C
V
n ⋅ p = N eff ⋅ N eff e − Eg
kT = const.(T ) (6.10) Note: ndoping ⇒ n increases ⇒ EF increases ⇒ E − EF p ∝ exp V kT decreases such that np = const.
Neutrality condition:
−
+
n + NA = p + ND 618 (6.13) Typical doping levels 1013  1017 cm3 ⇒ interaction between
doping atoms neglible ⇒ Fermi statistics applies:
Occupation probability of donors by electrons:
0
ND
1
=
ND 1 E − EF exp D +1
2
kT (6.14) Factor 1/2: donor can onle carry one electron (↑or ↓) due to
Coulomb repulsion.
⇒ statistics modified.
Occupation probability of acceptor by holes:
−
NA
=
NA 1 E − EF
2 exp A kT (6.15) +1 Factor 2: acceptor always occupied by 1 or 2 electrons:
no empty state posible! (when seen as energy level) Fig. 6.12 An acceptor, e.g., B in Si or Ge, can either be
occupied by one electron accepted from the valence
band if the resulting hole is bound to the acceptor (top)
or by two electrons if the hole is ionized (bottom). 619 ntype semiconductor
n=N e
C
eff − EC − E F
kT also holds for doped semiconductor (6.16) 0
+
ND = ND + ND (6.17) 1 E + EF
N = N D 1 + exp D kT
2
0
D −1 (6.14) ≡ (6.18) +
n = ND + p (6.19) +
Simplification: N D >> ni for Si: ni = 1.5 x 1010 cm2 at 300K ⇒ already obeyed for low doping levels. ⇒ 1
+
0
n ≈ N D = N D − N D = N D 1 − 1 + 1 exp E D − E F kT 2 = ND 1 E − ED 1 + 2 exp F kT 620 (6.20) (6.16) = (6.20) ⇒ E − EF
C
n = N eff exp − C
kT EF
⇒ e kT e EF
kT 1 = ND E − ED 1 + 2 exp F kT (6.21) 2 + 1 E D E F 1 N D ECkTE D + e kT ⋅ e kT −
e
=0
C
2
2 N eff E D
1 kT
1
= − e (±)
e
16
4 2 ED
kT N
+ De
C
2 N eff EC + E D
kT E −E
ED 1 kT N D CkT D 1+ 8
e
=e
− 1
C 4
N eff ⇒ ED
1 N D kT E F ≈ E D + kT ln 1 + 8 C e − 1 N eff
4 (6.22) in (6.21) ⇒ n≈ 2ND
E D
N D kT
1+ 1+ 8 C e
N eff +
(6.22), (6.23): approximation for N D >> ni not valid at very high temperatures.
621 (6.22) E d
N D kT
1) ″Freezeout range″ 8 N C e >> 1 , i.e. low temperatures
eff E ⇒ −d
1
C
n≈
N D N eff ⋅ e kT
2 (6.24) Sufficiently large number of donors retains their valence
electrons.
Cf.: V
C
ni = N eff N eff e − Eg
kT (6.11) V
N eff , E g → N D , E d 1 ND 1
1 E F = E D + E d + kT ln
C
2 N
2
2
eff (6.25) C
N eff ∝ T 3 / 2
C
N eff < N D for T → 0 C
N eff >> N D at high temp. Fig. 6.13 schematic coure of the Fermi energy
EF(T)at low temperatures according to eq. 6.25. The
maximum arises from the T 3/2 dependence of the
C
effective density of states N eff in the conduction
band. 622 E d
N D kT
2) Saturation range 8 N C e << 1 ⇒ n ≈ ND = const.
eff  All atoms are ionized
 Exitation of electrons from valence band still neglible.
3) Intrinsic range ni >> ND (6.22), (6.23) no longer valid! Behaviour like intrinsic semiconductor Fig. 6.14
a Qualitative
temperature dependence of the
concentration n of electrons in
the conduction band of an ntype semiconductor for two
different donor concentrations
ND´> ND. The width of the
forbidden band is Eg and Ed is
the ionization energy of the
donors;
b
Qualitative
temperature dependence of the
Fermi energy EF(T) in the same
semiconductor. EC and EV are
the lower edge of the valence
band, respectively, ED is the
position of the donor levels and
Ei is the Fermi level of an
intrinsic semiconductor. 623 Fig. 6.15 The connection n of free electrons in ntype germanium, measured using the Hall
effect. For the samples (1) to (6), the donor concentration ND varies between 1018 and 1013 cm3
. The temperature dependence of the electron concentration in the intrinsic region is shown by
the dashed line. ndoped Si with Nphosphorus = 3 x 1014 cm3:
Saturation range 45  500K ⇒ constant carrier density for microelectronic applications. 624 Conductivity of semiconductors σ = e(nµn + pµp) (6.1) Metals: µ = µ(EF) Semiconductors: µn: average values for electrons in lower
conduction band.
µp: average values for holes in upper
valence band Estimate of µ for ndoped semiconductors:
µn = e
τ
*
mn here τ ≠ τ (EF) ! complicated average value 1/τ ∝ Σ <v>, cf. metals: v = v(EF) E  EF >> kT ⇒ Boltzmann distribution
∞ < v >∝ ∫ v ( E )e mv 2
−
2 kT ∞ dE
∝ 0 ∞ ∫e mv 2
−
2 kT v 2e
∫ 0 ∞ 0 625 dv
∝T 0 ∫ ve dE mv 2
−
2 kT mv 2
−
2 kT dv Σ ph ∝< s 2 ( q ) >∝ T
Scattering from accustic phonons:
(scattering from optical phonons: relaxation time approximation no longer holds) µph ∝ T 3/2 ⇒ (6.26) Scattering from charged defects (ionized donors and
acceptors):
Σionized defects ∝ <v>4
Rutherford scattering via Coulombs interaction µionised defects ∝ T 3/2 ⇒ (6.27) 1
1
1
⇒
=
+
Addition of collision probabilities
τ τ ph τ ionized defects
⇒ µ= 1
(1 / µ ph ) + (1 / µ ionized ⇒ lim µ = µ ionized
T →0 defects defects ) ∝ T 3/ 2 lim µ = µ ph ∝ T −3 / 2 T →∞ Fig. 6.16 schematic temperature dependence of
the mobility µ for a semiconductor in which
scattering from phonons and charged inpurities
occurs. 626 (6.28) Fig. 6.16 Experimentally determined temperature dependence of the mobility µ of free
electrons. For the samples (1) to (6), the donor concentration ND varies between 1018 and 1013
cm3. The samples are the same as those used for the measurements in Fig. 6.15. 627 Fig. 6.18 Experimentally measured conductivity σ of ntype germanium as a function of
temperature. For the samples (1) to (6), which were also used for the measurements in Figs.
6.15 and 6.17, the donor concentration ND varies between 1018 and 1013 cm3. T dependence of n(T) much stronger that of µ(T) ⇒
µ(T) masked exept for saturation range where n(T) ≈ const. 628 ...
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