Martienssen W., Warlimont H. (Eds.). Handbook of Condensed Matter and Materials Data
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1022 Part 5 Special Structures
L
D
(nm)
10
3
1
Resistivity (Ωcm)
10
2
10
1
10
–1
10
–2
10
–3
GaAs–n
Ge–n
Ge–p
Si–n
GaAs–p
Si–p
Fig. 5.2-63 Debye length as a function of resistivity for
n-type and p-type Si, Ge, GaAs
5.2.5.2 Useful Formulas
and Numerical Values
The space charge density and surface potential can be
obtained by solving Poisson’s equation for the one-
dimensional problem at the surface [2.93, 94]. For
nondegenerate conditions, one obtains
Q
sc
=∓q(n
b
+ p
b
)L
D
F
s
(u
b
,v
s
) (2.4)
(minus sign if v
s
> 0), where
F(u
b
,v)=
√
2
cosh(u
b
+v)
cosh u
b
−v tanh u
b
−1
1/2
.
(2.5)
The function F
s
is plotted in Fig. 5.2-64 as a function of
k
B
Tv
s
for various values of u
b
> 0.
To obtain values for u
b
< 0, the following relation
can be used:
F(u
b
, −v) = F(−u
b
,v). (2.6)
The profile of the potential barrier v(z) can be obtained
from
z
L
D
=
v
v
s
dv
∓F
s
(u
b
,v)
,
(2.7)
which can be solved numerically.
Figure 5.2-61 shows v(z) as a function of z/L
D
for accumulation and depletion/inversion layers and
Space charge function F
s
–70
20
15
10
5
0
–5
–10
Surface band bending in k
B
T
u
b
= 30
–60 –50 –40 –30 –20 –10 0 10
u
b
= 20 u
b
= 10
u
b
= 0
u
b
= 0
n-type sample
Fig. 5.2-64 The space charge function F
s
(v
s
, u
b
) for
n-type semiconductors (u
b
> 0) plotted versus band
bending v
s
k
B
T . The function for p-type semiconduc-
tors (u
b
< 0) may be obtained from the relation:
F(u
b
, −v
s
) = F(−u
b
,v
s
) [2.47, p. 25]
various values of u
b
. The profiles are plotted for
|v
s
|=20. The behavior, however, has a universal char-
acter, since
z
L
D
−
z
−z
L
D
=
v
v
s
dv
∓F
s
+
v
v
dv
∓F
s
, (2.8)
so that curves can be scaled for different values of v
s
simply by translating the origin.
The charge Q
ss
in the surface states can be calculated
once their energy in the gap and their donor or acceptor
character is known. (A surface state is called a donor if
it is neutral when filled with electrons, and an acceptor
if it is neutral when empty.)
For a discrete distribution of surface states of densi-
ties N
d
i
and N
a
j
, where the superscripts “d” and “a” refer
to donors and acceptors, respectively,
Q
ss
=−q
j
N
a
j
(E
j
) f(E
j
−E
F
)
+q
i
N
d
i
(E
i
)[1− f(E
i
−E
F
)] , (2.9)
where f is the Fermi function. A similar expression
holds for a continuous distribution of states. If we define
a reduced surface state energy
u
j
≡
E
j
−E
is
k
B
T
,
(2.10)
Part 5 2.5
The Physics of Solid Surfaces 2.5 The Space Charge Layer at the Surface of a Semiconductor 1023
Charge density (10
–9
Ccm
–2
)
–15
–0.5
15
Band bending v
s
–10 –5 0 5 10
0.5
1.0
1.5
2.0
0
N
a
= 10
12
cm
–2
Q
sc
–Q
ss
N
a
= 10
10
cm
–2
u
j
=0
u
j
=0
u
j
=–2
u
j
=2
Fig. 5.2-65 Sketchof the graphicalsolution of the equation
Q
sc
+Q
ss
=0, for an intrinsic semiconductor (u
b
=0) with
a single acceptor surface level at reduced energy u
j
.Two
valuesof acceptor densities N
a
are shown. In case of u
b
=0,
the curves −Q
ss
should be translated rigidly towards the
new origin v
s
=−u
b
. When, on the other hand, Q
total
=0,
the Q
ss
curves should be translated vertically by Q
total
.The
actual band bending is at the intersection of the two sets of
curves
where E
is
is the intrinsic level at the surface, the Fermi
function can be written as
f(E
j
−E
F
) =
1
e
u
j
−v
s
−u
b
+1
.
(2.11)
It is then possible to solve (2.1) numerically to ob-
tain the band bending v
s
. A graphical solution (valid
in the absence of an external field) is sketched in
Fig. 5.2-65 for an intrinsic semiconductor (u
b
= 0) with
a single surface acceptor level and various values of
u
j
and N
a
j
. For extrinsic semiconductors (u
b
= 0) the
procedure is similar, provided the origin is translated
to v
s
=−u
b
and the appropriate function for Q
sc
is
chosen. The same procedure can also be used in the
presence of an external field by translating the origin of
the coordinates.
5.2.5.3 Surface Conductivity
The excess carrier densities, defined as
∆N =
∞
0
(n −n
b
)dz , (2.12)
excess electron density (per unit area) at the surface, and
∆P =
∞
0
( p − p
b
)dz , (2.13)
the excess hole density (per unit area) at the surface, can
also be obtained from the function F
s
:
•
for majority carriers in accumulation layers:
∆N = n
b
L
D
G
+
(u
b
,v
s
)(u
b
> 0,v
s
> 0),
∆P = p
b
L
D
G
+
(u
b
,v
s
)(u
b
< 0,v
s
< 0) ;
•
for majority carriers in depletion/inversion layers:
∆N = n
b
L
D
G
−
(u
b
,v
s
)(u
b
> 0,v
s
< 0),
∆P = p
b
L
D
G
−
(u
b
,v
s
)(u
b
< 0,v
s
> 0) ;
•
for minority carriers in accumulation layers:
∆N = p
b
L
D
g
−
(u
b
,v
s
)(u
b
< 0,v
s
< 0),
∆P = n
b
L
D
g
−
(u
b
,v
s
)(u
b
> 0,v
s
> 0) ;
•
for minority carriers in depletion/inversion layers:
∆N = p
b
L
D
g
+
(u
b
,v
s
)(u
b
< 0,v
s
> 0),
∆P = n
b
L
D
g
+
(u
b
,v
s
)(u
b
> 0,v
s
< 0).
The four functions G
+
, G
−
, g
+
, g
−
are defined as
G
+
=
|v
s
|
0
e
v
−1
F(|u
b
|,v)
dv,
G
−
=
|v
s
|
0
e
−v
−1
F(−|u
b
|,v)
dv,
(2.14)
g
+
= e
−2|u
b
|
|v
s
|
0
e
v
−1
F(−|u
b
|,v)
dv,
g
−
= e
−2|u
b
|
|v
s
|
0
e
−v
−1
F(|u
b
|,v)
dv.
(2.15)
Part 5 2.5
1024 Part 5 Special Structures
Inversion and depletion layers
10
2
5
2
10
5
2
±1
–2
–5
–10
10
⏐v
s
⏐
Accumulation layer
50 40 30 20 10 0
⏐u
b
⏐=20
Minority
carriers
Majority
carriers
⏐u
b
⏐=15 ⏐u
b
⏐=10 ⏐u
b
⏐=5 ⏐u
b
⏐=2 ⏐u
b
⏐=0
u
b
=0
⏐u
b
⏐ⱜ 2
G
+
+1g
+
+1
Majority carriers
G
–
–1
Minority
carriers
g
–
–1
⏐u
b
⏐=20
⏐u
b
⏐=15
⏐u
b
⏐=10
⏐u
b
⏐=5
⏐u
b
⏐=2
⏐u
b
⏐=0
⏐u
b
⏐=2
⏐u
b
⏐=0
Fig. 5.2-66 Dependence of the functions G
+
, G
−
, g
+
, g
−
on |v
s
| for various values of |u
b
|. The functions defined in
(2.14), (2.15) can be used to obtain the excess carrier densities for majority and minority carriers in accumulation and
depletion/inversion layers [2.93, p. 151]
Table 5.2-26 Field-effect mobilities of semiconductors. References to original papers are given in [2.47]
Semiconductor Face Reconstruction Field-effect mobility (cm
2
V
−1
s
−1
)
Si
(111) 2×1 −(1 ±0.5)
7×7(?)
1
−(0.01±0.003)
Ge
(111) 2×1 −(100 ±50)
c(2 × 8) −(200 ±50)
1
Surface prepared in situ by IBA
The superscripts of the functions indicate the signs of
the respective integrals. The above functions are plotted
in Fig. 5.2-66 as a function of |v
s
| and, in greater detail,
in Fig. 5.2-67 for an inversion layer (as in a MOSFET).
Tabulations are given in [2.94].
The surface excess conductivity is defined as
∆σ =q(µ
ns
∆N +µ
ps
∆P), (2.16)
where ∆N and ∆P are given above. The surface mo-
bilities µ
ns
and µ
ps
are in general different from the
mobilities in the bulk, because of excess scattering at
the surface.
The surface field-effect mobility can be defined as
µ
fe
(u
b
,v
s
) =
d∆σ
dQ
total
. (2.17)
The presence of surface states, which may capture
the carriers, decreases the surface field-effect mobil-
ity. Values of µ
fe
for some surfaces are presented in
Table 5.2-26.
The very low field effect mobilities reported in
Table 5.2-26, show that the Fermi level is pinned in
the gap because of the large concentration of surface
states. Pinning positions of the Fermi level with respect
to the top of the valence band at the surface are given in
Table 5.2-27 for various semiconductors.
In a MOSFET, the surface conductance of the inver-
sion layer channel between the source and the drain
(Fig. 5.2-68) is controlled by the gate potential. The
above formulas for the surface conductivity are appli-
cable to MOSFETs under equilibrium conditions (low
frequency and low voltage). Here the interface states
play the role of surface states at the free surface. The
density of interface states should be kept to a minimum
for optimal performance. Figure 5.2-69 shows the den-
Part 5 2.5
The Physics of Solid Surfaces 2.5 The Space Charge Layer at the Surface of a Semiconductor 1025
Crystal Face Reconstruction E
F
−E
VS
(eV)
Diamond (111) 2×1 1.5
Si
(100) 2×1 0.32 (1)
(111) 2×1 0.38 (2)
7×7 0.69 (3)
Ge
(111) 2×1 0.01 (2)
c(2 × 8) 0.12 (2)
GaAs
(100) c(2 × 8) 0.47 (7)
c(4 × 4) 0.47 (7)
4×6 0.49
(110) 1×1 Flat bands
GaSb (110) 1×1 Flat bands
GaP (110) 1×1 Flat bands
InAs (110) 1×1 Flat bands
Table 5.2-27 Pinning position of the Fermi level at semi-
conductor surfaces. Data are average values from various
authors. Errors are given as standard deviations. Refer-
ences to the original papers and individual errors are given
in [2.47]
sity of interface states in the gap of Si for two different
orientations.
Channel mobilities are generally lower than bulk
mobilities. An example is reported for n- and p-type car-
riers at the silicon–silicon oxide interface in Fig. 5.2-70.
A moreextended discussion of thespace charge layer
may be found in [2.47,48, 93,94].
Oxide
Source
Gate
n-channel
p-type
Metal
electrode
z
y
x
z
–V
BS
(Substrate bias)
V
G
V
D
Source
n
+
n
+
D
Drain
L
d
Fig. 5.2-68 Schematic drawing of a MOSFET. The
n-inversion layer is obtained with a positive bias of the
gate with respect to the source [2.48, p. 345]
g
+
+1
–2
10
3
10
7
–1 0 1 2 3 4 5 6 7 8 9
G
–
–1
5
3
2
10
2
7
5
3
2
10
7
5
3
2
±1
–2
–3
–5
–7
–10
⏐v
s
⏐–2⏐u
b
⏐
Inversion layer
⏐u
b
⏐= 0.5
⏐u
b
⏐= 2
⏐u
b
⏐= 5
⏐u
b
⏐= 10
⏐u
b
⏐= 0.5
⏐u
b
⏐= 2
⏐u
b
⏐= 5
⏐u
b
⏐= 10
⏐u
b
⏐= 20
⏐u
b
⏐= 50
⏐u
b
⏐=20
Fig. 5.2-67 More detailed plots of the functions G
−
, g
+
for in-
version layers. Notice the change of the abscissa to |v
s
|−2|u
b
|
[2.93, p. 155]
Part 5 2.5
1026 Part 5 Special Structures
10
3
10
13
⏐Q
s
/q⏐(cm
–2
)
10
12
10
11
10
10
10
2
µ
eff
(cm
2
/V s)
a)
10
3
10
13
⏐Q
s
/q⏐(cm
–2
)
10
12
10
11
10
10
10
2
µ
eff
(cm
2
/V s)
b)
Electrons
Holes
Fig. 5.2-70 Effective mobility of electrons and holes in MOSFET’s
inversion channel. Q
s
= Q
total
is the charge on the gate electrode.
It is seen that mobility decreases at high transverse electric fields.
For comparison, bulk mobilities are: µ
n
=1600, µ
p
=600 cm
2
/Vs
[2.96]
Interface trap density N
is
(cm
–2
eV
–1
)
–0.6
10
13
0.6
Energy (eV)E
v
E
c
–0.4 –0.2 0.2 0.40
10
12
10
11
10
10
10
9
Si
(111)
(100)
Fig. 5.2-69 Interface state density in thermally oxidized
silicon for two different orientations. Notice the low density
around mid-gap [2.95, after [2.48] p. 340]
5.2.6 Most Frequently Used Acronyms
Surface physics, like many other fields, is plagued by the use of acronyms. The acronyms used in this chapter, as
well as those most frequently found in the literature, are listed below:
AES Auger electron spectroscopy
AFM atomic force microscope/microscopy
APS appearance potential spectroscopy
ARUPS angle-resolved ultraviolet photoemission spectroscopy
ARXPS angle-resolved X-ray photoemission spectroscopy
ATR attenuated total reflection
BB back bond
BBZ bulk Brillouin zone
BEP beam-equivalent pressure
bl blocking
BZ Brillouin zone
CB conduction band
CBM conduction band minimum
CCV core–core valence Auger transition
chan channeling
CISS collision ion scattering spectroscopy
Part 5 2.6
The Physics of Solid Surfaces 2.6 Most Frequently Used Acronyms 1027
CITS current imaging tunneling spectroscopy
CMA cylindrical mirror analyzer
CVD chemical vapor deposition
CVV core valence–valence Auger transition
DAS dimer–adatom–stacking fault
DB dangling bond
2D BZ 2-dimensional Brillouin zone
DOS density of states
DR differential reflectivity
ECS electron capture spectroscopy
EDC energy distribution curve
EELS electron-energy loss spectroscopy
ELEED elastic low-energy electron diffraction
ESA electrostatic analyzer
ESD electron-stimulated desorption
EXAFS extended X-ray absorption fine structure
FEED field emission energy distribution
FEM field emission microscope/microscopy
FIM field ion microscope/microscopy
FLAPW full-potential linearized augmented plane wave
HAS helium atom scattering
HATOF helium atom time-of-flight spectroscopy
HEED high-energy electron diffraction
HEIS high-energy ion scattering/high-energy ion scattering spectroscopy
HREELS high-resolution electron energy loss spectroscopy
HR-LEED high-resolution LEED
HR-RHEED high-resolution RHEED
IB ion bombardment
IBA ion bombardment and annealing
ICISS impact ion scattering spectroscopy
IPE inverse photoemission
IPES inverse photoemission spectroscopy
IS image state
ISS ion scattering spectroscopy
JDOS joint density of states
KRIPES K-resolved inverse photoelectron spectroscopy
LAPW linearized augmented-plane-wave method
LDA local-density approximation
LDOS local density of states
LEED low-energy electron diffraction
LEED-I/V LEED intensity/voltage measurements
LEIS low-energy ion scattering/low-energy ion scattering spectroscopy
LEPD low-energy positron diffraction
MBE molecular-beam epitaxy
Part 5 2.6
1028 Part 5 Special Structures
MD molecular dynamics
MEED medium-energy electron diffraction
MEIS medium-energy ion scattering/medium-energy ion scattering spectroscopy
ML monolayer
MOKE magneto-optical Kerr effect
MR missing row
NICISS neutral impact collision ion scattering spectroscopy
PBBS projected bulk band structure
PDS photothermal displacement spectroscopy
PED photoelectron diffraction
PES photoemission spectroscopy, photoelectron spectroscopy
PLAP pulsed laser atom probe
2P-PES 2-photon photoemission spectroscopy
RAS reflectance anisotropy spectroscopy
REM reflection electron microscope/microscopy
RHEED reflection high-energy electron diffraction
RPA random-phase approximation
SAM scanning Auger microscope/microscopy
SARS scattering and recoiling ion spectroscopy
SBZ surface Brillouin zone
SCLS surface core level shift
SDR surface differential reflectivity
SEM scanning electron microscope
sh shadowing
SHG second-harmonic generation
SIMS secondary-ion mass spectroscopy
SPARPES spin-polarized angle-resolved photoemission spectroscopy
SPIPES spin-polarized inverse photoemission spectroscopy
SPLEED spin-polarized LEED
SPV surface photovoltage spectroscopy
SS surface state
STM scanning tunneling microscope/microscopy
STS scanning tunneling spectroscopy
SXRD surface X-ray diffraction
TEM transmission electron microscope/microscopy
TOF time of flight
TOM torsion oscillation magnetometry
TRS truncation rod scattering
UHV ultra-high vacuum
UPS ultraviolet photoemission spectroscopy
VB valence band
VBM valence band maximum
VLEED very low-energy electron diffraction
XPS X-ray photoemission spectroscopy
Part 5 2.6
The Physics of Solid Surfaces References 1029
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Part 5 2
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Part 5 2
1031
Mesoscopic an
5.3. Mesoscopic and Nanostructured Materials
This chapter addresses the properties of nanostruc-
tured materials considered as statistical ensembles
of nanostructures. Emphasis is put on size and
confinement effects, although enhancements in
surface and interface properties are mentioned.
After a survey and a summary of basic definitions
and concepts in the introductory Sect. 5.3.1, the
properties associated with electronic confinement
are addressed in Sect. 5.3.2. Electronic confine-
ment affects the spectral properties, i. e. light
absorption and luminescence, mainly through
quantum size effects, and the electrical conduc-
tion properties through the Coulomb blockade.
Both two-dimensional systems (quantum wells)
and zero-dimensional systems (quantum dots)
are reviewed. Particular attention is drawn to
semiconductor-doped matrices. The effects asso-
ciated with confinement of electromagnetic fields
are treated in Sect. 5.3.3. Numerical relationships
and data for plasmon excitations of various metal
nanoparticles can be found in this section. Mag-
netic nanostructures are addressed in Sect. 5.3.4.
The two main applications of nanostructured
magnetic materials, namely spin electronics, or
spintronics, and ultrahigh-density data storage
media, are treated. Finally, we list and briefly de-
scribe in Sect. 5.3.5 some generic techniques for
the preparation of nanostructured materials, or-
ganized into the following groups of methods:
molecular-beam epitaxy (MBE), metal-organic
chemical vapor deposition (MOCVD), nanolithogra-
phy, nanocrystal growth in matrices, and ex-situ
synthesis of clusters.
5.3.1 Introduction and Survey...................... 1031
5.3.1.1 Historical Review..................... 1031
5.3.1.2 Definitions ............................. 1032
5.3.1.3 Specific Properties ................... 1034
5.3.1.4 Organization of this Chapter ..... 1034
5.3.2 Electronic Structure and Spectroscopy... 1035
5.3.2.1 Electronic Quantum Size Effects. 1035
5.3.2.2 Breakdown of the Momentum
Conservation Rule ................... 1036
5.3.2.3 Excitons
in Quantum-Confined Systems.. 1036
5.3.2.4 Vibrational Modes
and Electron–Phonon Coupling. 1040
5.3.2.5 Electron Transport Phenomena . 1042
5.3.3 Electromagnetic Confinement .............. 1044
5.3.3.1 Nanoparticle-Doped Materials .. 1044
5.3.3.2 Periodic
Electromagnetic Lattices........... 1048
5.3.4 Magnetic Nanostructures..................... 1048
5.3.4.1 Spin Electronics....................... 1049
5.3.4.2 Ultrahigh-Density Storage
Media in Hard Disk Drives......... 1060
5.3.5 Preparation Techniques ...................... 1063
5.3.5.1 Molecular-Beam Epitaxy .......... 1063
5.3.5.2 Metal-Organic Chemical Vapor
Deposition (MOCVD) ................. 1064
5.3.5.3 Lithography............................ 1064
5.3.5.4 Nanocrystals in Matrices........... 1064
5.3.5.5 Ex Situ Synthesis of Clusters ...... 1065
References ..................................................1066
5.3.1 Introduction and Survey
5.3.1.1 Historical Review
Except in the life sciences, there are only a few examples
of materials that are naturally structured on scales of
the order of a few to a few hundred nanometers. One
can cite, however, the natural zeolites, which constitute
a group of hydrated crystalline aluminosilicates con-
taining regularly shaped pores with sizes from 1 nm
to several nanometers. This provides them the abil-
ity to reversibly adsorb and desorb specific molecules.
Part 5 3