V38 thesis - FAU · 2015. 2. 23. · Title: Microsoft Word - V38_thesis Author: smalzer Created Date: 2/19/2015 5:52:05 PM

Optical and electrical properties of micro-structured semiconductors

An experiment for Master students

Version 1.0 (February 2015)

Updated, modified and translated by Stefan Malzer

from the thesis of

Till Häusler

Friedrich-Alexander-Universität Erlangen-Nürnberg

(1998)

Content

0 Introduction ....................................................................................... 3

0.1 Motivation ................................................................................................... 3 0.2 Goal of the experiment ............................................................................... 3 0.3 The components ........................................................................................ 4 0.4 Structure of the thesis ................................................................................ 5

1 Fundamentals of the experiment ..................................................... 6

1.1 Black body radiation – Planck’s law ........................................................... 6 1.2 Diffraction grating ....................................................................................... 6 1.3 Imaging with lenses .................................................................................... 7 1.4 Microscope / loupe ..................................................................................... 9 1.5 Absorption ................................................................................................ 11 1.6 Reflection at interfaces ............................................................................. 12 1.7 Fabry-Perot oscillations ............................................................................ 12 1.8 Anti-reflection coating ............................................................................... 13

2 Band model of solids ...................................................................... 15

2.1 Preliminary remark ................................................................................... 15 2.2 Bands and band-splitting in solids ............................................................ 15 2.3 Electronic bandstructure and distribution function .................................... 16 2.4 Density of states and effective mass approximation ................................ 18 2.5 Doping in semiconductors ........................................................................ 21 2.6 Electron transitions between bands and absorption ................................. 23 2.7 Excitons .................................................................................................... 26 2.8 The pn-junction ........................................................................................ 27 2.9 Photodiodes and photocurrent ................................................................. 30 2.10 Light emitting diode (LED) ........................................................................ 31 2.11 Franz-Keldysh effect ................................................................................ 31 2.12 Quantum-confined Stark effect ................................................................. 34

3 The experimental setup .................................................................. 39

3.1 Sketch of the setup .................................................................................. 39 3.2 The individual components ....................................................................... 40

3.2.1 Light source .......................................................................................... 40 3.2.2 Monochromator ..................................................................................... 40 3.2.3 Optical fiber ........................................................................................... 42 3.2.4 CCD-camera ......................................................................................... 45 3.2.5 Photodetector and power meter ............................................................ 45 3.2.6 Picoamp meter with integrated voltage source ..................................... 45 3.2.7 Control software .................................................................................... 45 3.2.8 Samples ................................................................................................ 46

Appendix ............................................................................................... 49

A. Material parameters of (Al)GaAs .............................................................. 49 B. Fundamental physical constants .............................................................. 49

Bibliography ......................................................................................... 50

0 Introduction

0.1 Motivation

The last decades were affected by a rapid development of semiconductor electronics optoelectronics, data transmitting systems and data storage. The internet, smart phones and tablets are few of the latest outcomes of steadily improvement, downsizing and increasing the complexity in designing and fabricating semiconductor devices. Laser scanners, laser printers and fiber-optical sensing tools nowadays appear to be naturally as well. At the interface between optics and electronics we need devices that are able to translate signals fast and efficiently in both directions. In particular, we need a light source (laser) that can be modulated either directly by a voltage or indirect by using an electro-optical modulator, and a light detector that converts the light signal into an electrical signal. The requirements for an efficient interaction are best met by compound semiconductors which exhibit a direct bandgap. The composition of such (originally binary) materials can be extended to ternary and quaternary material systems whose properties can be tuned within a wide range. Moreover, the invention of artificial, periodic structures of different semiconductor materials, called superlattices, has opened a new “playground” for creating new and arbitrary electronic and optical properties. In this thesis, some of the physical basics are summarized to understand the electro-optical properties of semiconductor materials. As an experimental example, the effect of an electric field on a “standard” semiconducting device (a p-i-n diode) and on a superlattice device (a multiple quantum well structure) is investigated. In terms of quantum mechanics, this experiment is a test bed for quantum mechanical perturbation theory: Starting from the simple quantum well problem we study the electric field (Stark) effect by absorption of light (which is a time dependent perturbation, stationary approximated by Fermi’s “Golden rule”). Overall, we observe these effects in a system with a periodic lattice potential. Though this experiment is only scheduled as a two-day experiment, it contains more than enough substance for extensive investigation and interpretation.

0.2 Goal of the experiment

The experiment “Optical and electrical properties of micro-structured semiconductors“ deals with the fundamentals of solid state and semiconductor physics. The following goals are pursued:

I. Experimental goals: a) Development of the ability to build a setup for electro-optical experiments with

the provided components. This includes the measurement of the absorption spectrum by spectral transmission and photocurrent experiments for different semiconductor structures.

0. Introduction 4

b) Independent acquisition of up to date experimental methods, and performing, evaluating and discussing the results.

c) Acquiring fundamental knowledge of computer controlled experiments (LabView) and analysis tools.

II. Theoretical goals: a) Developing a deep knowledge of a common model for the bandstructure of

solids particularly with regard to semiconductors.

b) Capability of explaining the experimental findings in an own model, and finding the limits of the model.

At the end you should have an established knowledge on bandstructure and on the absorption behavior of semiconductors which can be extended easily by further studies. For more interested students there is the chance to get in touch with more details of the experiment. Those are kindly asked to get in touch with the tutor, also if they plan to work on their own proposal for experiments (including additional equipment!).

0.3 The components

The first part of the experiment is planned for building a transmission setup. For applying an electric field and / or measuring the photocurrent, electrical contacts are needed. For such experiments you need to get used to the contacting by probe needles. This typically takes some time for exercising (and should be made on a “test” sample). By a modular designed setup is quite easy to switch from one measurement to another. So you can directly compare photocurrent measurements with transmission experiments with the same experimental parameters. Fig. 0.1 shows schematically some of the different experimental situations for measurement.

Fig. 0.1 Different experimental methods for absorption measurements in semiconductors.

Light Light Light

LightLight

DetectorDetector

Detector

current

Current ++

+

--

-

a) Transmission c) Transmission with applied voltage

b) Photocurrent measurement

d) Photocurrent with applied voltage

e) Electroluminescence

0. Introduction 5

0.4 Structure of the thesis

This thesis is split in an experimental and a theoretical part. The experimental part covered the development and installation of the experimental setup including purchasing parts and designing work order for the machine shop as well as the programming of the control software and establishing the experiment. Superior to that an educational concept was elaborated for communicating mandatory educational objectives as well as leaving space for interesting additional experiments. With the present written part the following objectives are intended to be met: A complete documentation of all components in the setup and establishing the fundamental physical models and concepts should allow for the preparation to focus on this thesis (with only a few excursions into the given literature where the reader needs to bridge a gap). The thesis is organized in three chapters. Chapter 1 treats the physical foundations in short form. The individual reference for the experiment is given for each component. Chapter 2 follows with elaborating elementary concepts in solid state physics, in particular with the band model. The order is closely related to the experimental procedure. Starting with the basics we go more and more into the relevant details for the experiment. As more special topics (which are not necessarily found in standard text books) the electro-optic effects, namely the Franz-Keldysh effect and the quantum-confined Stark effect, are described for a deeper understanding of the electro-absorption of micro-structured semiconductors. Due to the wealth of subject matter of chapter 2 it can be split between the 2 days for the experiment. Chapter 3 introduces the setup with the individual components and samples. For a deeper understanding the reader is encouraged to deepen his knowledge by studying the textbooks cited in the bibliography.

1 Fundamentals of the experiment

This chapter gives a survey on the physical basics which are necessary to understand the components and techniques used in the experiment.

1.1 Black body radiation – Planck’s law

The light source used in the experiment is a combination of a tungsten lamp and a monochromator. The lamp can be assumed to be a black body. The emission spectrum of this black body will be measured in the experiment and compared to the theoretical spectrum of a black body of the same temperature. The spectral emission P(,T) of a black body can be calculated by Planck’s law (Eq. 1.1). In Fig. 1.1 it can be seen that the maximum of the spectral radiation density is shifting to shorter wavelengths with increasing temperature.

d

kT

hc

hcdTP

1exp

18),( 5

2

(1.1)

From the maximum of the measured spectrum and Wien’s law (Gl. 1.2) the temperature of the lamp (or, to be more precise, the temperature of the tungsten wire) can be determined. This temperature is used as the input for the theoretical spectrum.

max

2102898,0

Km

T

(1.2)

1.2 Diffraction grating

The monochromator is a kind of “light filter” for the spectrum of the tungsten lamp. In principle, there are different options for choosing a certain “wavelength window” from a continuous spectrum: Dispersion in a prism, multiple reflections at thin layers (Fabry-Perot interferometer), or diffraction from a grating. The monochromator in this experiment uses a diffraction grating for the light dispersion. The monochromator itself can be seen as an imaging instrument, imaging the entrance slit onto the exit slit. Together with the dispersive element, only a narrow spectral range is imaged on the exit. The principle of diffraction is shortly summed up. A more detailed description of the monochromator is given in section 3.2.2.

Fig. 1.1 Planck’s radiation spectra for different temperatures (source: Wikipedia)


For a common theory of diffraction from gratings (mostly discussed for transmission gratings) you are referred to standard textbooks in optics. The grating used in monochromators is typically a blazed reflection grating. The reflective surface consists of equidistant grooves (distance d) which are in the order of the wavelength of the light.

Fig. 1.2 Diffraction at a blazed reflection grating In Fig. 1.2 the situation for a blazed grating (with a blaze angle blaze) is shown. An incident wave under an angle with respect to the grating normal and the diffracted wave (with angle with respect to the grating normal) exhibit a path difference of = d (sin + sin 1.3) We expect constructive interference if the path difference is a multiple of the wavelength = z z = 0, 1, 2, 3 ... For the maxima we get z d(sin + sin ) (1.4) In the monochromator, we use the first order diffraction (z = 1) and turn the grating for the selection of desired wavelength on the exit slit.

1.3 Imaging with lenses

It is a fundamental issue in this experiment to align the components for an imaging of the light source and the sample. Therefore the fundamental imaging equations and important optical instruments are summed up. The imaging of an object by a lens is shown in Fig. 1.3. We recall that parallel optical beams are refracted and cross the opposite focus point of a (thin) convex lens, while central rays are not refracted.


Fig. 1.3 Image construction at a convex lens. The focal length is f, the height of the object (AP) and (real) image (BP1) are y1 and y2, the object distance is a, and the image distance is b. The triangles AF1P and OF1C as well as AOP and P1BO are similar. With the intercept theorem this yields:

fy

fay 21

(1.5a)

and

ba

yy

2

1 (1.5b)

Elimination of 2

1

y

y from (1.5a) and (1.5b) gives the lens equation

fba

111 (1.6)

The magnification is given by the ratio between image size and object size.

f

b

a

f

a

b

y

y ´

´1

2 (1.7)

Fig. 1.4 shows the situation for different distances (a > f) between object and lens where a real, reversed image on the opposite side is formed. For large distances (∞ > a > 2f) the image size is reduced, and f < b < 2f. For a = 2 f, the image has the same distance (b = 2f) and size on the opposite side (This is called the “4f – setup” as this is the minimum total length for an imaging setup). For 2f > a > f, the situation is contrary to the first case and we get a magnified image in the range 2f < b < ∞. What happens when the object is moved in (or even further) the focus f of the lens? Obviously, the image is moving to infinity on the opposite side – and is appearing again from the object side as the object is crossing the focal point. For this case we get a virtual (enlarged) image on the same side (see Fig. 1.4).

Fig. 1.4 Imaging with a convex lens.

ff

OF2

F 1

2 f

12

23

3

a b

a´

b´f

f

A B

C

O

P

F2F1

P1

y2

y1


F2

P´

P

AA´ F1

The reciprocal value of the focal length is the refraction power. Its unit is [1/m] and called “diopter”. A lens with a focal length of f = 0.25 m has a refraction power of 4 diopters. Fig. 1.5 Image construction of a virtual image with the object placed within the focal length of a convex lens.

In the experiment, the sample with an active size of 1 mm2 has to be imaged on a detector area of ~1 cm2 with a microscope objective (f ~ 10 mm). Think about magnification and distances between object, lens and detector!

1.4 Microscope / loupe

This subsection on optical instruments is not mandatory for the experiment but is written to give a broader background. The size of an object G in the eye of an observer is dependent on the distance between the eye and G. With this distance the angular diameter is determined. It is common to view an object with unity magnification in the distance of 25 cm in front of the eye. This is called the clear visual range s0. The related angular diameter is 0. For larger distances, G appears to be smaller, for smaller distances G appears to be larger. The distance cannot be made arbitrarily small as the accommodation1 of the eye is limited (typically, the near point of vision is around 10 cm). For an enhancement of an object one needs optical instruments, e.g. a loupe, a microscope, or telescope. As microscope objectives are used in the experiment for imaging the light source and the sample, we will briefly address the function of a loupe and a microscope. The magnification of such an instrument is defined by

Vinstrument = 0ε

εinstrument withoutdistancecm25indiameter angular

instrumentwithdiameter angular (1.8)

The loupe: A loupe is a convex lens of a small focal length. It allows looking at an object in the focal plane of the lens with a totally relaxed eye (focused to infinity). The image is virtually (e.g., cannot be displayed on a screen). The geometric image con-struction is sketched in Fig. 1.6. Fig. 1.6 Imaging with a loupe.

1 Accommodation means the change of the refraction power of the eye lens by the ciliary muscle.

ff

A

B

image at 8

eye

s

eye0

0


The magnification Vloupe is given by geometrical considerations and Eq. (1.8):

f

AB and

00 s

AB :

fs

εεv 0

0loupe (1.9)

If the object is shifted towards the lens, the virtual image appears at finite distance. Commonly, the virtual image is appearing in the clear visual range s0. The magnification in this case is

1f

sV 0loupe (1.10)

As a typical example, with a convex lens of 10 mm focal length a magnification of 25 is achieved. The microscope A microscope allows much larger magnifications than a loupe. An objective at a little larger distance than its focal length f1 from the object yields a magnified image at a certain distance (the real intermediate image). The magnification is (cf. Eq. (1.7)):

11 f

t

f

´b

The real intermediate image at the distance t (the mechanical tube length) is observed with the ocular as a loupe. This gives an additional magnification (Eq. (1.9)) of

fsV 0

loupe .

The total magnification by the microscope Vmicroscope is the product of from the objective lens and the ocular Vloupe

2

0

1microscope f

sftV (1.11)

However, it is not enough to increase the magnification (to arbitrarily large values). The resolution is also important and is given by

sinαnλg

, (1.12)

With the wavelength of the used light and the numerical aperture, sin of the microscope. The index n is from the medium in front of the objective (n = 1 in air). The angle 2 is the angular range for the edge rays incident on the objective lens. At least the first diffraction order of the incident light from the object has to be in the range of incidence, which defines g n sin =1 Fig 1.7 Microscope. This is the condition for resolving two neighboring lines according to Abbe’s theorem. It gets better for short wavelengths of the used light and using immersion liquids with n > 1.

t

f2

f1

ocular

intermediateimage plane

objective

object


1.5 Absorption

The samples in this experiment will be investigated by transmission experiments. We can deduce information on the material properties by recording the transmitted part of the light but also try to measure directly the absorbed part of the light by measuring a photo-induced current. For the interpretation of both methods it is important to understand the absorption mechanism. Nearly any material absorbs light partially. The energy of the light (photons) is transformed into other forms of energy (mostly heat). Materials of low absorption in the visible range (350 nm to 700 nm) are called “transparent”; materials with high absorption are called “opaque”. To determine the attenuation of the incident light power P0 in a solid, we first look at the absorption of monochromatic light of wavelength in a homogeneous, isotropic material. On the surface we lose part of the power by reflection, RP, with the reflection coefficient R (0<R<1). The remaining part, P-RP= P(1- R), propagates into the medium (see Fig. 1.8). We denote the direction of propagation as the x-direction and expect the absorption to be proportional to the power for any small direction dx. P(x) is decreasing with increasing x by dP(x). The power at x+dx is therefore

dxdx

xdPxPdxxP

)()()( (1.13)

The decrease of power is given by xP

dxxdP

and

dx

xdPαxP , (1.14)

Reconverting and integrating Eq. (1.14) from x0 = 0 to x, results in

xRPxP )1(ln)(ln 0 or x

0 e)1()( RPxP (1.15) Fig. 1.8 Absorption

This relation is called Lambert-Beer’s law. The absorption constant (or coefficient) has the unit [1/m] and is a function of the wavelength and depending on the material. However, it is not depending on x (therefore “constant”). A thickness of the absorbing material of 1/reduces the power by a factor 1/e. This power is reduced

by the reflection at the backside and transmitted. The transmitted power T can then be measured. The attenuation of the power P(x) is plotted in Fig 1.9. Fig. 1.9 Attenuation of optical power through a medium of thickness d.

P0x

reflected part

absorbingmaterial

incident light power

P(1-R)0

0PR

x =00

P

P(x)

loss by frontside reflection

loss by backside reflection

transmitted power

0 x(thickness of the absorbing material)

0

d


Solving Eq. 1.15 for the absorption through a material with thickness d we get

20 )1(

ln1

RP

T

d . (1.16)

As is function of the wavelength we refine Eq. (1.16) to

20 ))(1(

)(ln

1)(

RP

T

d (1.17)

We will discuss the reflection R( in the following section.

1.6 Reflection at interfaces

The transmission through (and reflection at) dielectric interfaces as a function of angle of incidence and polarization is described by Fresnel’s formulae. Here, we are only interested in vertical incidence and for this very special case the ratio of reflected and incident amplitude (Er and Ei) of the electromagnetic wave from a medium M1 with refraction index n1 into a medium M2 with index n2 is given by

21

21

i

r

nnnn

EE

(1.18)

We are measuring intensities and therefore have to use the squared value as the reflection coefficient

R = 2

21

212

e

2r

nnnn

EE

, (1.19)

For an air/medium interface with nair~nvac = 1, R reduces to 2

1

1

HL

HL

n

nR (1.20)

The refraction index is given by the ratio of wavelength in vacuum to wavelength in medium, nmed = n(vac/med. Typically, the dispersion (dn(/d) is much smaller than the change of absorption d)/d (at least for semiconductors around the bandgap) and the reflection coefficient R( can be assumed to be constant. In Eq. (1.17) reduces the wavelength dependence to the transmission signal

20 )1(

)(ln

1)(

RP

T

d

(1.21)

1.7 Fabry-Perot oscillations

If the attenuation through a (thin) medium is weak we can expect multiple reflections at both interfaces (see Fig. 1.10), which give rise to interference. All the reflected and transmitted beam parts have to be summed up by considering the right phases and amplitudes. The result is given by Airy’s formula for the transmitted intensity

12

200 141

kdR

RRkdRkd rt sin

)(),(),( (1.22)


with the wavenumber k = plate =n2vac. Fig. 1.11 shows a plot of Eq. (1.22). For kd=N, N=1,2,3, … the total intensity is transmitted through the platelet without any losses.

Fig.1.11 Transmitted intensity It through a coplanar platelet of thickness d as a function of k.

With k = 2n2Vac and kd = N we get

dn

Nd

nN

VacVac 2

2

2

12

(1.23)

The wavelength of the Nth and (N+1)th maxima is taken for the difference 1/N- 1/N,

)(22

1

2

111

12

1

12

!

1

1

1VacN

VacN

VacN

VacN

VacN

VacN

VacN

VacN

VacN

VacN n

ddndn

NN

(1.24)

With this formula, the thickness of the platelet can be determined from the extrema of the Fabry-Perot oscillations (FPO). Keep in mind that the refraction index can be considered to be constant only for a narrow range of wavelengths.

1.8 Anti-reflection coating

Multiple reflections can be useful for the determination of the thickness of a platelet. However, for investigating the wavelength dependent absorption, FPOs should be suppressed. This can be achieved for perpendicular incidence by a special anti-reflective coating (ARC) of at least one (or odd multiples of) /4-layer of the index n2

(see Fig. 1.12). The reflected beams from the interface M1/M2 and from M2/M3 then interfere destructively. Ideally, both beams should have the same amplitude and this requires for the index n2 of the ARC material (“index-matching”)

312 nnn (1.23)

/4

n 1

n 2

n 3

Phase change

Fig. 1.12 Reflection at a /4-layer.

Thickness d

n n > nn 1 2 1 1

I 0

T 0 R 0

R 1

R 3

R 2 T 1

T 2

T 3

Fig. 1.10 Multiple reflections at a coplanar platelet.


In most cases one interface is towards air so that n1 = 1 and n2 3n (1.24)

For the thickness of the ARC-layer M2 we get

ARC

reflARC n

kd

4

)12(

(1.25)

With the order k = 0,1,2,3, … of the ARC layer. This is strictly true only for the considered wavelength. Practically, however, the range where the reflection is reasonably suppressed is relatively broad. As an example we consider an ARC-layer for the semiconductor GaAs (nGaAs = 3,59 @ = 900 nm). We find from Eq. (1.24) for the ARC-layer index nARC = 1,894 and for the minimum thickness (Eq. (1.25), k = 0), dARC = (900/41,894) nm = 119 nm. If a broad range anti-reflection range is desired (e.g., for eye glasses) this can be achieved by multi-layer coatings.

15 Band model of solids


2.1 Preliminary remark

The bandstructure of a solid describes the energy states of an electron in an ordered crystal. Absorption and emission processes are described as optically induced transitions between such energy states. A simple model for calculating the energy states is the free electron gas in a periodic crystal (like in a metal) while the “tight-binding” approach seems to be better suited for the covalent binding in most semiconductors. It is beyond the scope of this introduction to discuss all these models. At the same time it is expected that the reader is familiar with elementary quantum mechanics and solid state physics. Therefore we will restrict ourselves to introduce the most important terms and concepts. It is therefore strongly recommended that the reader studies the respective chapters in textbooks (e.g., Ibach/Lüth, “Solid State Physics”).

2.2 Bands and band-splitting in solids

An electron in its atomic potential is only allowed to have distinct energy levels. These are the single electron states of a spherically symmetric potential with 1s; 2s, 2p; 3s, 3p, 3d; 4s, 4p, 4d, 4f; ... states with the numbers being the principal quantum numbers n and the letters s, p, d, f correspond to the orbital angular momentum quantum numbers (l = 0, 1, 2, 3...(n-1)) of the electron. In addition, there is the magnetic quantum number m, which can take (2l+1) values. The Pauli-principle states that each energy level can be occupied by at most two electrons with differing spin. Successive filling of the electron states results in the periodic table of elements. If we approach in a “Gedanken” experiment several atoms, there will be two consequences: First, we have to consider the Coulomb potential of all the other atoms at the place of the observed atom. This is simply an addition of all the other Coulomb potentials and leads to a lowering of the total potential (and a drop of the Eigenstates). Second, the electron wave functions of the neighboring atoms extending far enough will cause a splitting of their energy states. This splitting becomes larger the closer the atoms get. For each additional atom we get a new splitting state, which finally ends up in a quasi-continuum of states (for N ~1023 cm-3!) which is called a band. If between such bands a range of energy remains without any energy states, this is called a bandgap and the respective solid is typically a semiconductor. The wave function of an electron can be described by a linear combination of atomic orbitals (LCAO) and a factor for the periodicity of the crystal lattice which manifests in the wave function as well (the Bloch-wave function). The highest occupied band is called the valence band and the lowest unoccupied band is called the conduction band. Fig 2.1 shows a very simple sketch for the approach of neighboring atoms with the according state splitting. This is done for the most common case of having s- and p-orbitals occupied.


Fig. 2.1 Schematic drawing of the band splitting of a semiconductor as a function of the

interatomic distance (taken from Ibach/Lüth. p.142).

2.3 Electronic bandstructure and distribution function

Electrons are fermions, i.e. they possess a half-integer spin, and obey Pauli’s principle. Therefore, in the lowest energy state of a multi electron system, the electrons fill the available states up to a certain energy level. In metals, for instance, this energy is in the range of a few eV, which is much larger than the additional thermal energy electrons might gain at room temperature (kT @ 300 K ~ 25 meV). In general, one can deduce the partition function from thermodynamical principles for distributing undistinguishable particles into single particle states. This is the Fermi (-Dirac) distribution function

1

1

kT

ETEf

exp),( , (2.1)

giving the probability of finding an electron at the energy E and a system temperature T. The chemical potential µ is determined by minimizing the free energy of the system with a constant number of particles. At zero temperature, the chemical potential µ is identical to the Fermi energy (EF

0). At finite temperature, the probability for occupying states with higher energy increases (see Fig. 2.2). It is obvious that the particles occupying states above EF leave empty states below. Therefore it is not the Fermi-energy that is fixed but the number of electrons. The Fermi level balances the probability of creating an electron at a higher energy state (E-EF > 0) and the probability of creating a “hole” in the sea of electrons at lower energy (E-EF < 0).

Fig. 2.2 Fermi-distribution function.

f(E)10

E

E0

FT=0K0

T > 1

T0

T > 2

T1


For the representation of the energy states and wave functions in a solid the description in the reciprocal (or k-) space is preferred. The k-space is constructed from the base vectors of the lattice in the real space (cf. Fig. 2.3), a1, a2 and a3 by

)aa(a

aa2g

321

321

and cyclic. (2.2)

Fig. 2.3 A two-dimensional lattice in real space and the respective reciprocal lattice. The vector g1 is perpendicular on the plane spanned by a1 and a2. Bear in mind that the real space is drawn in in the dimension m and the reciprocal space (k-space) with dimension m-1.

Within the reciprocal lattice we can define a „Brillouin zone“ which is the analog to a Wigner-Seitz cell in real space. It is defined by the smallest polyhedron centered in the origin and spanned by the perpendicular bisectors of reciprocal lattice vectors as sketched in Fig. 2.4.

a) b)

Fig 2.4 a) Construction of the 1st Brillouin zone in a rhomboid lattice. b) Brillouin zone of a cubic face-centered lattice. The points of high symmetry are labelled L, etc.

The zone boundaries are special, as for every wave with a k-vector from the origin to these boundaries a Bragg reflected wave arises. Bragg reflection only occurs for discrete energies ( kcpcE ) and it is a good idea to plot an E(k)-diagram

along the high symmetry directions. As the k-space is the Fourier-transformed real space, it obeys the same symmetry, i.e. the periodicity in k. therefore it is sufficient to investigate the first Brillouin-Zone. Fig. 2.5 shows the bandstructure of GaAs as a typical representative of direct semiconductors. The drawing follows the direction of high symmetry, i.e. X-direction for the (100)-direction and equivalent (from the center towards the 6 sides of a cube), or L-direction towards the 8 corners of a cube. The drawing in Fig. 2.5 is made from the L-point to the -point to the X-point (cf. Fig. 2.4b), then going to the equivalent points U and K, and back to .

a2

a1

g2

g1


Fig 2.5 Bandstructure of GaAs

The energy scale is set to zero at the maximum of the valence band. In semiconductors, this band is completely occupied and the next empty state for an electron would be at the bottom of the conduction band which is separated by the bandgap energy. In a filled band, no transport of carriers can occur as no empty states are available and the carriers cannot gain momentum. If a few electrons are in the conduction band (which then is still mostly empty), they find states of higher k-values and therefore can gain momentum (and energy) in an applied electric field. One way to bring electrons in the conduction band is by optical absorption where an electron from the valence band is lifted into the conduction band (satisfying energy and momentum conservation!). The empty electron state in the valence band is considered to be a “hole” (in the sea of electrons). We will see later that this picture is very strong and allows a reasonable simple and realistic description of electrical and optical properties. An important quantity for these properties is the number of energy states in the bands, or as a more adequate quantity, the density of states (DOS).

2.4 Density of states and effective mass approximation

If we zoom into the region of interest in Fig. 2.5a we see that the top of the valence band and the bottom of the conduction band can be well approximated by a parabolic E(k) function

.k2m︶k︵Eandk

2mE︶k︵E 2

*v

2

v2

*c

2

gapc

(2.3)

This is identical to the free electron dispersion E(k) = p2/2m with the classical momentum p replaced by the crystal momentum ħk and the free electron mass m is replaced by the “effective” mass mc* in the conduction and mv* in the valence band, respectively. Obviously, the effective mass is determined by the curvature of the E(k)-dispersion,

region of interest

bandgap

valence band edge

conduction band edge


2

2

2* kE1

m1

(2.4)

The effective mass is constant as long as the dispersion is strictly parabolic. The model can be extended so that an effective mass can be defined for any k-region. However, the effective mass is then k-dependent as shown in Fig. 2.6.

Fig. 2.6 Schematic drawing of the effective mass for a one-dimensional bandstructure E(k) for a) strong curvature and b) weak curvature.

The k-space is constructed from states of all atoms of the solid, N. The volume of the first Brillouin zone (with the borders kmax = ±/a, and the lattice constant a) is

3

k a2πV

(2.5a)

and the real space volume of the solid is simply N times the volume of the unit cell

3a NaV (2.5b)

where we used cubic symmetry for simplicity. We normalize the density of states, dN/dk, with the total volume of the crystal to get the expression for the density of states in k-space (which is not only true for electrons but for any quantized “particles”!)

3

ak3

3

2π1

VVN

dkNd

(2.6a)


Each state can be occupied by two electrons of different spin. So, the spin degree of freedom provides a factor of two for the number of states. (This argument also holds for other “particles”, i.e. for photons where we have two orthogonal polarization states.)

3

33 dk

2π2Nd (2.6b)

From this point it is straightforward to calculate the DOS in energy space. We now have to insert the respective energy-momentum relation, which is in our case

dEmkdkorm

kdkdEandE2mkor

2mkE 2

*

*

221

2

*

*

22

(2.7)

and reduce the 3-dimensional integral dk3 to a 1-dimensional integration over a sphere, 4k2dk,

dEEπm2 dEmE2m

2π4π2EdN 3

23*

2

*21

2

*

3

2

(2.8)

The 3-dimensional density of states in an energy interval E+dE is

Eπm2

dEEdNED 3

23*

dim.3

(2.9)

with a square root dependence of energy. The calculation of the DOS for 2- and 1-dimensional systems can be performed as an exercise. The resulting densities of states for 3, 2, 1, and 0 dimensions are summarized in Fig. 2.7.

E

Fig. 2.7 Density of states for 3-, 2-, 1-, and 0-dimensional electronic systems. The probability of finding a state occupied is given by the Fermi distribution of Eq. (2.1). The absolute carrier concentration in the conduction and valence band is given by the product of f(E,T) and the DOS, D(E). This is shown in Fig. 2.8 for finite temperature and equal DOS for conduction and valence band (a) and for the (realistic) case of a higher DOS in the valence band (b). The number of electrons

D(E)

E

D(E)

E

3-dim.D(E)

E

2-dim.

E2

D(E)

EE 2

D(E)

EE 2

0-dim.1-dim.


must be equal to the number of holes, so the Fermi level in Fig. 2.8b is shifted towards the conduction band

Abb. 2.8 Fermi-distribution f(E), DOS D(E), electron and hole concentration for a) equal DOS in conduction and valence band b) different DOS (higher DOS in valence band).

The electron concentration in the conduction band is given by

LE

c dET ︶f ︵E,︵E ︶Dn (2.10a)

and the hole concentration in the valence band by

VE

v dET ︶]f ︵E,[1︵E ︶Dp (2.10b)

For finite temperature, the free carriers (n = p = ni, “intrinsic carrier density”) are taking part in transport. The number of intrinsic carriers ni at room temperature can

be estimated with the approximation f(E,T) ~

TkEEexp

B

F for (E-EF >> kBT) and

an “effective band edge density of states” Nc and Nv (accumulating all “thermally relevant” states - see Fig. 2.8 - at the band edge, which allows us to substitute E with Ec and Ev, respectively).

TkE

expNNTkEEexpN

TkEEexpNnpn

B

gapvc

B

vFv

B

Fcc

2i

(2.11)

In GaAs this gives an intrinsic carrier density at 300 K (kBT = 25 meV) of ~106 cm-3. This carrier density is way too small to allow for reasonable currents. In practice, sufficient (and not temperature dependent!) carrier densities are created by doping.

2.5 Doping in semiconductors

The samples used in the experiment are composed as a p-i-n diode, built by a p-doped, an undoped (“intrinsic”), and an n-doped layer. We will briefly explain “doping” at the example of n-doping which leads to free electrons in a semiconductor (while p-doping is responsible for free holes).


If we replace the atom on a regular lattice site by an atom with a higher valence (e.g., P instead of Si in a silicon semiconductor, see Fig. 2.9a) we can pretend to still have a silicon atom plus a proton (fixed at the same place) and an electron. Simply speaking, this is a hydrogen atom in the background of a semiconductor (Si) with a dielectric constant sc. This modifies the binding energy and the Bohr radius of the electron of such a donor. The hydrogen binding energy EH = 13.6 eV is reduced by

H2sc

*e

D EεmE (2.12a)

and the Bohr radius (aH = 0.5 Å) is increased by

H*e

scD a

mεa (2.12b)

If we put in typical numbers for the effective electron mass and dielectric constant, we find the Bohr radius in the ground state to be a few nm to a few tens of nm and binding energies of a few meV up to a few tens of meV. This energy is in the range or lower than the thermal energy at room temperature and practically all the donors are ionized and their electrons fill the conduction band. It is straightforward to locate the energy level of a donor, ED right below the conduction band (according to the zero energy level for the quasi-hydrogen atom). The energy levels are sketched in Fig. 2.10 for the donor and acceptor level in a semiconductor. In addition, you are requested to draw the Fermi level in Fig. 2.10 a) and b). It is an analogue consideration for p-doping with acceptors in the valence band (like B in Si, see Fig. 2.9).

Fig 2.9 Schematic drawing for n-doping with phosphor and p-doping with boron. a) b)

Fig. 2.10 Band diagram with a) an acceptor level and b) donor level. It is a unique strength of semiconductors that the conductance of the material can be accurately tuned by doping. The GaAs semiconductor in the experiment is made of

p-doped semiconductorEL

EV

AE

En-dopedsemiconductor

EL

EV

D

E

E

x x


group III and group V elements but it behaves equivalent if we use Si as a donor and beryllium (Be) as an acceptor, both occupying Ga sites.

2.6 Electron transitions between bands and absorption

We will now focus on electronic transitions between energy states in semiconductors. If a photon with an energy ≥ Egap is absorbed it will lift an electron in an empty conduction band state and leave an empty state in the valence band. This can be described as an electron-hole generation while the photon is annihilated. The reverse process is recombination, where an electron-hole pair is annihilated and the photon created. The respective transitions between valence and conduction band are interband transitions. If the electron is changing its state (energy and momentum) within the same band, these processes are called intraband transitions. All the possible transitions can be categorized into 4 classes (see Fig. 2.11) 1. Interband transitions a) Direct transitions between valence band and conduction band (absorption and

recombination). b) Indirect transitions between valence band and conduction band with phonon

contribution and/or excitons (bound state of an electron-hole pair). For the transition energy, the phonon energy or the Coulomb energy of the excitons reduce the transition energy with respect to the free particle case.

2. Electronic transitions via traps a) Donor-valence band transitions b) Donor-Acceptor transitions c) Conduction band–acceptor transitions d) Bound exciton transitions e) Phonon cascades and multi phonon transitions

Fig. 2.11 Electronic transitions in semiconductors.


3. Intraband transitions 4. Auger-processes

The interband transition energy is directly transferred to a free carrier. This is a non-radiative process which gets stronger with increasing carrier density.

Primarily, the bandstructure of the semiconductor has a great influence on which processes dominate the transition. A fundamental issue is given by the alignment of the absolute extrema of valence and conduction band of a semiconductor. While the maximum of the valence band is always at the -point, the situation in the conduction band can be very different. Silicon as the most famous representative of an indirect semiconductor exhibits the conduction band minimum at the X-point. Direct semiconductors are present in III-V- and II-VI compound semiconductors (with GaAs as a famous representative). However, also within the compound semiconductors, there are candidates with an indirect bandgap, e.g. GaP, see Fig. 2.12.

Fig. 2.12 Bandstructure of GaAs and GaP with direct and indirect transitions at k = 0.

For all transitions, energy and momentum (and angular momentum!) conservation has to be satisfied. In photon induced transitions, the momentum of the photon is by orders of magnitude (question to the reader: to which number?) smaller than the momentum of the electron so that the momentum in the valence band and the conduction band has to be the same (i.e. k remains unchanged). In k-space this is represented by a vertical transition between valence and conduction band. If the transition requires an additional momentum transfer (e.g., to reach the X-valley in the conduction band of an indirect semiconductor), this can only be delivered by a phonon. The phonon carries a large momentum but has a low energy (compared to the photon energy). In this case we have involved a third particle in the process which reduces the transition probability by orders of magnitude. This is one of the crucial reasons why indirect semiconductors cannot emit light or - to be more accurate - are extremely inefficient in such a process. In the experiment the absorption of a semiconductor material is measured as a function of the wavelength of the incident light. This probes the energy dependent


transition probability and provides information on the bandstructure of the material. Ideally, the photon is absorbed only for energies larger than the bandgap, > Egap. For lower energies, the material is transparent. This should give a clear signature for the bandgap. The photon induced transition of an electron from the valence band into the conduction band in a direct semiconductor can be described by 1st order perturbation theory, known as „Fermi’s golden rule“,

)(´ˆ)( ´´,,,

vkck

kkvcrcv EEvkpeck

cnm

ekP

2

00

2

(2.13)

In this equation, e is the polarization vector, p̂ the momentum operator, m0 the free electron mass, the energy of the incident photon, and Ec,k und Ev,k´ are the eigenvalues of the electron and hole states in the conduction and valence band with the respective momenta k and k. For a simple consideration, we assume the valence band to be completely filled and the conduction band to be empty so that the distribution function can be neglected. The refractive index, nr, can be assumed to be constant over the measured wavelength range. The energy conservation is fulfilled by the -function in Eq. (2.13). The wave functions ck in the conduction band and vk’ in the valence band are Bloch functions (complying with the lattice periodicity) with a plain wave envelope. From the effective mass approximation (Eq. 2.3) for the DOS we get the Eigenvalues in valence and conduction band

cck m

kE

2

22 and

vvk m

kE

2

22´ . (2.14)

For a bulk semiconductor (without applied electric field and in single particle approximation) the absorption coefficient )( is derived

)()(/

ggcv

r

EEPcnm

e

2

32

23

00

2

34

2 (2.15)

P2 is the momentum matrix element at k = 0 and can be assumed to be constant for small k. The effective interband mass CV is defined by

LeLV mm

111

. (2.16)

The calculated single particle absorption, a more realistic model including excitonic interaction and a measured spectrum is shown for GaAs in Fig. 2.13. The single particle absorption represents the energy square root dependence of the combined DOS in 3 dimensions, gE .

The difference to the measured spectrum is considerably. The onset of the absorption is shifted by ~4 meV and the measured absorption strength is by a factor of two higher with a steeper onset. When the excitonic interaction is taken into account for the calculation, the accordance becomes much better


We will now briefly explain what an exciton is.

2.7 Excitons

The excitation of an electron from the valence into the conduction band leaves a hole in the valence band. Obviously, the creation of these two particles happens at the same

time at the same place and as charged particles they are subject to the Coulomb interaction, resulting in a bound electron hole state called excitons. Very similar to the dopants in semiconductors, excitons can be considered as a hydrogen-like system. The only difference to the dopant is the much lighter mass of the positively charged partner (typically ~ 5 to 10 times the effective mass of the electron). For our considerations, we therefore have to take the reduced effective electron-hole mass (µeh)

-1 = (me*)-1 + (mh)*

-1, which is not too much different (but a little smaller) than the effective electron mass (the reader should be able to explain why!). The resulting binding energy of an exciton is similar but smaller than the donor binding energy. In GaAs, the exciton binding energy is about 4 meV. This is much lower than the thermal energy at room temperature (~25 meV) and the reason why excitonic effects are much more pronounced at low temperatures, as shown in Fig. 2.14. In 2-dimensional structures, however, where a strong confinement of electrons and holes also supports a stronger excitons confinement, the binding energy is substantially higher. While theory predicts a factor of 4 higher binding energy, real structures exhibit a two times higher value. This is already enough to observe excitonic effects at room temperature.

Fig. 2.13 Calculated (dotted lines) and measured (full line) absorption spectra of bulk GaAs material at room temperature. (M. Kneissl, PhD thesis, “Physik mikrostrukturierter Halbleiter, Vol 1 , 1997)

Fig. 2.14 Absorption in GaAs at 21 K. the dashed line is the calculation without excitonic interaction.


2.8 The pn-junction

The samples for investigation are double hetero p-i-n diode (DH pin) structures. First, the function of a pn junction will be explained. Then, we add the i-layer and will finally address the hetero junction. In the pn-junction, we have a p-doped and an n-doped layer which are brought into contact as shown in Fig. 2.15. (Actually, this is a “Gedanken”-experiment. The interface of a pn-junction has to fit the crystal symmetry without defects, which can only be achieved directly by doping during epitaxy or by diffusion of dopants in an existing crystal layer). Fig. 2.15 Band scheme of a p- and n-semiconductor (a) completely decoupled; b) in thermal equilibrium. c) space charge (x) d) space charge potential V(x). a) Pn-junction in thermal equilibrium From Fig. 2.15a it can be seen that the Fermi energy in the n-semiconductor is at much higher energy (close to the conduction band) compared to the p-doped semiconductor. The band edges are aligned to the vacuum level (not shown). Bringing the semiconductor into contact means we allow electrons to flow into p-region (which lowers their energy) and holes into the n-region. Both carrier types are minority carriers in the respective doped layers and will recombine (i.e. the electrons in the p-layer recombine with holes which are present at a huge number and vice versa for holes in the n-layer). This process will proceed until a new equilibrium has settled with a common electro-chemical potential (given by the Fermi-energy). This new equilibrium is stabilized by ionized dopants (which are spatially fixed!) in the region close to the interface giving rise to a space charge. According to Maxwell’s equation charges are the source for an electric field, or with Poisson’s equation, the space charge is the origin for a potential (“space charge potential”). The calculation of this potential is a straightforward exercise and results in a form as shown in Fig. 2.15d.

x

c)

+

-

(x)

p field current

p diffusion current

x

d)V(x)

0

oo(- )Vp

Vn oo( )

p-semi-conductor

n-semi-conductor

Ec Ec

EVEV

FE

FE

DEAE

E

a)

E c

F

EA

E

E c

EV

FEDE

E

-eV(x)

--

++

x

p

E

Vp

n

n-VD

b)


In this dynamic equilibrium one would describe the majority carriers (i.e. electrons in the n-layer, holes in the p-layer) diffusing into the minority layer (p-layer for electrons and vice versa) against an energy barrier (diffusion potential, eVD) which becomes larger with increasing space charge. On the other hand, electrons at the p-layer experience an electric field which transports them back in the n-layer (field current). Therefore, the equilibrium is achieved when the diffusion current of electrons (and of holes) balances the field current of electrons (holes) without having applied an external voltage (thermal equilibrium). The resulting space charge is shown in Fig. 2.15c. It can be assumed to be constant (at a constant doping level) throughout its width. The resulting space charge potential V(x) is calculated by Poisson’s equation

02

2

sc

x

x

xV )()(

(2.17)

by the twofold integration over the space charge (x). The current densities for diffusion and field currents are given by

︶xpD

xn

︵Dej pndiff

(2.18a)

xpnfield E︶μpμe ︵nj (2.18b)

with the dynamic equilibrium condition at zero bias jdiff = jfield. We have used the usual abbreviations e for the elementary charge, n and p for the mobility of electrons and holes, n and p for the electron and hole density, Dn and Dp for the diffusion constants of electrons and holes. The latter are connected with the mobility

by e

kTD . The electric field

x

xVEx

)(is given by the derivative of the space

charge potential.

b) Pn-junction with bias Applying an external voltage U to the pn junction leads to a change of the diffusion potential VD. A positive voltage (forward bias) reduces the potential to V = VD – U while a negative voltage (reverse bias) increases the potential. In the latter case, the diffusion current will rapidly decrease with reverse bias as the diffusion current flows against an energy barrier (increasing with increasing reverse bias) which is expressed by a Boltzmann factor

T]/kUVeexp[jj BD0Diff (2.19)

A positive voltage decreases the diffusion potential and leads to an exponential increase of the current. The field current, however, is independent of the applied voltage. In particular, the field current balances the diffusion current a zero bias.

0 ︶︵Uj0 ︶︵Uj0 ︶︵Uj pn,field

pn,field

pn,diff (2.20)

(The indices n and p indicate that we have to consider both electrons and holes individually).From Eq. (2.19) and (2.20) we can calculate j0 and write for the current

kTeUexpjj pn,

fieldpn,

diff (2.21)


For the total current we have to subtract the field current (again, for electrons and holes) and find by adding the electron and hole field current to a total field current jfield

][exp)(

1

kT

eUjjjUj field

ptot

ntottot (2.22)

This is the IV-characteristics for a pn-junction shown in Fig. 2.16. The potential for the above discussion on forward bias, zero bias and reverse bias is illustrated in the band diagrams in Fig. 2.17.

Fig. 2.16 IV-characteristics of a pn-junction.

Fig. 2.17 Band diagrams for (left to right) forward bias (U > 0), zero bias (U = 0) and reverse bias (U < 0).

c) Pn-junction with intrinsic layer, the p-i-n diode We have seen that the interesting region of a pn-junction is the space charge region, where the photogenerated carriers are efficiently separated. The width of this region depends on the doping density (~N-1/2) and is typically a few tens of nm for doping densities in the range of 1018 cm-3. On the other hand, the absorption length 1/ is in the range of µm in semiconductors. So a good solution is to bring an undoped (“intrinsic”) layer between the n- and p-doped regions. From the above

considerations we get the (linear) field increase (or decrease) only for the doped regions, while in the undoped region (space charge ≡ 0) the field stays constant. The potential in the i-region will therefore show a linear behavior. The strength of the electric field can now easily be tuned by applying a bias voltage. Fig. 2.18 p-i-n structure at zero bias. Space charge is only located at the bending between n- and i-region (positively charged donors) and the i- and p-region (negatively charged acceptors).

x

E(x)p n

LE

EV

x

E(x)p n

LE

EV

x

E(x)p n

LE

EV

U>0V U<0VU=0V

p n p n

U

reverse bias forward bias- -++

j tot


d) Hetero p-i-n junction2 For the p-i-n structure shown in c) we would have absorption also in the p-and n-region of the diode. At least, when we want to investigate the field effect on the absorption, this structure is not optimal suited. Instead, it would be advantageous to have the absorption restricted on the intrinsic region only. This can be achieved by using doped layers with a higher bandgap which would be transparent for the wavelength range of interest. Fortunately, this can be achieved by replacing part of the Ga-content in GaAs by aluminum, yielding an AlxGa1-xAs semiconductor alloy

(with 0<x<1 for the Al-content). As a gift of nature, AlAs (x = 1!) has nearly the same lattice constant as GaAs and these semiconductors can grow perfectly on each other. Fig. 2.19 DH-p-i-n structure at zero bias. The n- and p-region is made of higher bandgap semiconductor (AlxGa1-xAs) as well as part of the i-layer. The central i-layer is the lower bandgap material (GaAs). If the low bandgap material would extent further to the doped layers, the edges would cross the Fermi-level and free carriers would occupy the triangular potential well. This would result in band bending of the i-layer and a constant field region in the i-layer would not be possible.

2.9 Photodiodes and photocurrent

A pn-junction is well suited for the detection of light with an energy larger than the bandgap. The mechanism is shown in Fig. 2.20.If the photon (with sufficient energy > Egap) is absorbed within the junction region, it generates an electron hole pair which is separated in the electric field. This current is called the photocurrent and this has the same direction as the field current of the junction. As the field current is not depending on the applied field we also expect the photocurrent to be independent of the applied voltage. As an open circuit device the photocurrent will increase the majority carriers in the doped layers and screen the space charge, giving rise to a “photovoltage” as shown in Fig. 2.20. Having a short circuit (or reverse bias) would allow for recording the photocurrent.

Fig. 2.20 Electron-hole generation by absorption of photons in the junction region.

The photocurrent adds as a constant current (in reverse direction) to the total current (cf. Eq. 2.22). Photogenerated carriers in the doped region will not contribute to the

2 Nobel prize in 2000 for Herbert Kroemer and Zhores I. Alferov "for developing semiconductor heterostructures used in high-speed- and optoelectronics".

p-region n-region

cE

Ev

junction

+ +

- -

V R L

+ -

h h


photocurrent and recombine. (A more detailed inspection shows that photogenerated carriers can also diffuse into the junction region and therefore can contribute to the photocurrent. In this case one has to calculate the characteristic diffusion length of the carriers). The samples in the experiment are especially designed to guarantee that only the junction region is used for photocurrent generation. The average photocurrent is determined by the generation rate G <jphoto> = eG (2.23a) with the elementary charge, e. At an optical power of P we have )/(0 P photons

per second arriving on the semiconductor. Only the fraction P is absorbed in the junction and contributes to the photocurrent. With ext, the external quantum efficiency, the photocurrent is

chλPeηω/Peηj 0

ext

0ext

photo

(2.23b)

The current under illumination jill is now the sum of the photocurrent and the diode current from Eq. (2.22), which is called the “dark current”, jdark for obvious reasons.

photodarkill jjj (2.24)

Fig. 2.21 IV-characteristics of a photodiode: Dark current jdark and added photocurrent <jPhoto> yields the total current.

2.10 Light emitting diode (LED)

If a forward bias is applied to the pn- (or DH-p-i-n)-junction, electrons and holes are injected in the space charge region and will recombine. In direct semiconductors, the recombination energy is converted into a photon. Consider Fig 2.19 with an applied forward bias in the range of U = eEgap. This will lead to a strong diffusion current of electrons and holes in the i-layer, which now behaves as a box for keeping electrons and holes confined so that they can efficiently recombine. It is also a welcome exercise to evaluate the spectral form of this (radiative) recombination.

2.11 Franz-Keldysh effect

The absorption in a semiconductor at the presence of an electric field (electroabsorption) is now going to be discussed. In particular, the question, if and how the absorption of light is influenced by the presence of an electric field is addressed. The underlying Franz-Keldysh effect is named after Walter Franz und Leonid V. Keldysh, who independently investigated the electric effect in semiconductors theoretically in 1958. The effect of an electric field in a bulk semiconductor is shown in Fig. 2. 22 (left part). It can be regarded as a central part of

U

jDunkel

jDiode

jPhoto< >


a p-i-n structure from Fig. 2.19. An electric field in a semiconductor breaks the translation symmetry of the semiconductor in one direction. An infinitely extending Bloch function experiences a strong perturbation when approaching the band edge. At this particular point it can only extent into the bandgap with an exponential decay. Classically, this situation corresponds to a reflection of the electron at the band edge, with a corresponding higher probability to find the electron at the band edge. The resulting wave function (perturbation of Bloch functions by an electric field) is an Airy function which is a periodic function with the discussed modification at the band edge. The situation is identical in the valence and conduction band. Looking at energies lower than the bandgap, the exponential tail of conduction and valence band wave functions is responsible for a finite, exponentially decreasing absorption towards lower energies. For higher energies we expect an oscillatory behavior as the wave functions interfere constructively and destructively with increasing energy. This electro-absorptive effect superposes the absorption at zero field and is expressed in respective absorption changes as shown in Fig. 2.22 (right part).

Fig. 2.22 Schematic drawing of the (real space) band profile and wave functions in an electric field (left). Absorption coefficient of GaAs determined for different electric field strengths by photocurrent measurements. (M. Kneissl, PhD thesis, “Physik mikrostrukturierter Halbleiter, Vol 1, 1997)

Measuring the photocurrent is a direct tool for determining the absorbed photons. The transmitted photons can be used as well for the spectral absorption experiment. In the experiment, both methods spectral electro-transmission and spectral photocurrent measurements are applied for comparison. a) Determination of from photocurrent measurements The absorption of photons is proportional to the photogenerated carriers and can therefore be measured by the photocurrent. A field induced change in the absorption can be detected as a change in the photocurrent. The relation between incident light power Popt and generated photocurrent jPhoto is given by Eq. (2.23b)

extoptextoptpnPhoto ηe

hcλP

ηeω

PU,j

.


The external efficiency ext accounts for reflection and transmission losses. An internal efficiency int describes the conversion of photons into electron-hole pairs carrying the current. We can write

int)1()1( dext eR (2.25) When a good anti reflection coating is used (R = 0) and the conversion is efficient, int = 1, the photocurrent is

︶e︵1ehc

PU,j αdopt

pnphoto

(2.26)

and

opt

pkotopn Pe

hcj

dU

11 ln),( (2.27)

By measuring the photocurrent spectra for different voltages and calculating i(,Ui), one can determine ij from the difference i(,Ui)-j(,Uj). From the above made assumptions for R and int it can be expected that the results might be flawed with some errors. Therefore it might be more accurate to determine from transmission experiments. b) Determination of from electro-transmission experiments The interesting question is how the absorption changes with the applied voltage for different wavelengths compared to the case at zero bias. Therefore, it is better to derive the change of absorption U/U0instead of U. This can be done directly from electro transmission experiments. For a sample thickness d, the transmitted power is

deRPT 20 )1( (2.28)

Both reflections at the front side and at backside of the sample have been considered. For the transmitted power measured at different voltages Upn

1 and Upn2

one gets for the transmission ratio

)exp()exp()1(

)exp()1(

12

0

22

0

1

2 ddRP

dRP

T

T

(2.29)

where we have neglected field dependent changes in reflectivity. The absorption change as a function of voltage is

1

21T

T

dln (2.30)

By this method, we get the field induced absorption change without knowledge of the reflection of the sample and absolute power of the light source. As these are the more accurate experiments, one can correct from photocurrent measurements.


2.12 Quantum-confined Stark effect

The second sample that is investigated in the experiment is a multiple quantum well (MQW) structure. Such structures are made of semiconductors with different bandgaps. For an undisturbed epitaxial growth the used materials need to have a very similar lattice constant. Fig 2.23 shows a “map” of semiconductors where the energy gap is plotted versus the lattice constant. As a further prerequisite, a suitable substrate is needed for the epitaxial growth. All these requirements are perfectly met for the GaAs/AlAs semiconductor system. GaAs is also available as a substrate material (as “wafers” up to 8 inch in diameter) and the bandgap can be continuously tuned between the GaAs bandgap and the AlAs bandgap by substituting Ga by Al in the compound.

Fig. 2.23 Plot of bandgap Eg of the most important binary and elementary semiconductors vs. lattice constant.

For a periodic structure of low and high bandgap material (as for the special case of GaAs / AlxGa1-xAs, 0 < x < 1) the band profile is shown in Fig. 2.24.

Fig. 2.24 Band profile of a GaAs/AlGaAs- hetero structure. This is a one-dimensional quantum well structure in growth direction. This (periodic) potential perturbs the wave function of electrons in this direction which gives new solutions. These will now be briefly revised.

Wachstumsrichtung

E

E

E L

V

GaA

s

GaA

s

Ga

As

GaA

s

AlG

aAs

AlG

aAs

AlG

aA

s


a) The one-dimensional potential well with finite height The potential well for an electron is described by

axfürUaxfür0

xU0

Fig. 2.25 Potential well in the conduction band of a AlGaAs / GaAs / AlGaAs layer system.

We separate the x-direction from the 3-dimensional equation and write for the time-independent Schrödinger equation

axforEψxψ

2m 2

2

0

2

and

axforψ︶U︵Exψ

2m 02

2

0

2

(2.31)

This equation has the solution for 0 < W < U0

axforαxcosBαxsinAψ ︵x ︶

and axfor βxexpDβx ︶exp ︵Cψ ︵x ︶ (2.32)

with the abbreviations

200

20 E ︶/︵U2mβ,E/2mα (2.33)

From the normalization of the wave functions it follows D = 0 for x > a and C = 0 for x < a. The continuity of the wave function and its derivative x) and d(x)/dx at the walls of the well at x = a and x = -a yields

)exp(sincos

)exp(cossin

)exp(sincos

)exp(cossin

aDaBaA

aDaBaA

aCaBaA

aCaBaA

(2.34)

With the straight consequences:

)exp()(sin2

)exp()(cos2

)exp()(cos2

)exp()(sin2

aDCaB

aDCaB

aDCaA

aDCaA

(2.35)

For DCand0A , DCand0B it follows

a-a 0

U(x)

U0

GaAs

AlGaAsAlGaAs

x


acot and

atan . (2.36) Both equations cannot be satisfied at the same time as it would yield an imaginary from tan2a = -1 and therefore to negative energies E we can either set

βαatanαandDC0,A or

βαacotαandDC0,B (2.37) and relocate to

aaa tan or

aaa cot . (2.38) With Eq. (2.33) we find

2200

200

220

222 /2/)(2/2)()( aUmEUmaEmaaa (2.39)

As a graphic solution of the characteristic equation is given by the intersection of the graphs of Eqs. (2.38) and (2.39) as shown in Fig. 2.26. Obviously, there exists only a finite number of intersections and accordingly the solutions for wave functions with the required boundary conditions exist only for discrete energy values. These are bound states in potential well and also shown in Fig. 2.26 Alternatively, the reader can solve the Schrödinger equation (2.13) by direct integration with a computer to find realisitic numbers for the allowed energy levels. This would be a particularly interesting exercise to compare these values with the values of the infinite well in the next subsection.

Fig. 2.26 Graphical solution of the characteristic equation with energy levels and corresponding wave functions.

b) The one-dimension potential well of infinite height A much easier problem is the infinitely high potential well. There is a straightforward calculation of the allowed energy levels by the quantization condition that only half waves of the electron must fit the well (width b = 2a). The momentum k is related to the wavelength by /2k and the energy is E(k) = m2k2 / . Allowing for only

0 2

2

4

4

6

6

8 a

atan(a)

acot(a)

acot(a)

atan(a)

a

a-a 0

U=U

U=0

0

x

E

EEE

E

12

3

4

x

4

x

2

x

1

x

3


values that fit into the well (i.e. the amplitude has to be zero at the walls) requires b = n/2. If we put all together we find

22

22

2

22

n 8 n

424E n

mb

h

bm

(2.40)

The ground state energy E = h2/8mb2 increases with decreasing (effective) particle mass and quadratically with decreasing well width. Now we treat the wavefunctions (in the same terms as in subsection a). As U0 , the exponential decaying wave functions for x<-a and x>a vanish and we have = 0. The conditions now simplify to

0acosBasinA

0acosBasinA

(2.41)

0cos,0sin aBaA (2.42)

For a given or E, sin(a) or cos(a) cannot vanish together at the same time. Therefore, two classes of solutions exist. Symmetric solutions for

0αacosand0A (2.43) and antisymmetric solutions for

00 aandB sin (2.44) This requires a = n/2 with an integer n resulting in the energy equation (2.10) and for the wave functions

a

xnBx

ora

xnBx

22

212

sin)(

,)(cos)(

(2.45)

(Note that the total width of the well is b = 2a!). The MQW structure is embedded in the i-layer of a p-i-n-structure and therefore experiences an electric (built-in) field. This field can be tuned by an additional external voltage. The electric field effect is schematically shown in Fig. 2.27. The tilted potential gives rise to a lowering of the ground state energy. The excitonic binding energy is not affected in 1st order as the confinement of carriers is not much changed. This effect shifts the transition energy for the fundamental electron-hole absorption to lower energies with increasing field and is called the quantum confined Stark effect. This redshift of the transition wavelength is accompanied by a reduction of the transition strength as the center of mass of the electron and hole wavefunction is shifted apart, reducing their overlap. With increasing field the ionization of the excitons becomes increasingly faster giving rise to a broadening of the transition. In Fig. 2.28 typical spectra are shown for different applied voltages.


Fig. 2.28 Spectral absorption at different applied voltages (reverse bias) for a MQW-structure as a result of the quantum confined Stark effect.

The electro-absorption in this structure will be measured by photocurrent spectroscopy. For low applied voltages (or even small forward bias), the spectra appear to be much smaller in intensity as compared to higher revers voltages where the high energy part of the spectra shows a saturation. An interpretation of this behavior by the reader is strongly encouraged.

F=0 F>0

e0

hh0

E0(F=0) E(F>0) < E0

Ec

Ev

Fig. 2.27 Sketch of band edge and ground state envelope wave functions of a quantum well without and with electric field.

3 The experimental setup

3.1 Sketch of the setup

Fig 3.1 is a sketch of the setup with all necessary components for the experiment. All voltage dependent spectral measurements (transmission and photocurrent) can be performed with this setup. Fig. 3.2 shows an enlarged sketch of the sample probe station with the probes for electrical contacting of the devices.

Fig. 3.1 Schematic drawing of the setup.

Fig. 3.2 Scheme of the probe station with probes to contact the sample.

Enlarged image on CRT

lens

lens

lamp

Photo -detector

CCD-camera

Powermeter

entrance slit

exit slit

optical fiber

x

yz

XYZ-translation stage

Probe needles for devicecontacting

Connection from the probes to thepicoamp meter for applying a voltage and measuring the current

x

yz

Optical fiber with xyz- translation for illuminationof the sample

3. The experimental setup 40

3.2 The individual components

3.2.1 Light source

The light source is a (250 W) tungsten halogen lamp working at 10 A maximum current. It is fixed in a home-built housing in front of the entrance slit. At the bottom of the housing, a fan ensures permanent cooling so that a steady state temperature can be reached within a short time. An additional concave mirror helps increasing the incident light power. The emission spectrum of the lamp can be approximated by a black body of the same temperature as the tungsten filament. The respective color temperature has to be determined in the experiment.

3.2.2 Monochromator

The monochromator used in the experiment is a TRIAX 180 (now at HORIBA Scientific). It comprises different holographic optical gratings for wavelength ranges of 190 to 1200 nm and from 500 to 1500 nm, as well as a ruled grating for the range of 1000 to 2500 nm. All three gratings are mounted on a turret so that for any desired wavelength the right grating can be chosen. The monochromator has entrance and exit slits of a height of 15 mm and a variable slit width from 0 to 2 mm, tunable in steps of 2 m. The focal length of the monochromator is 190 mm. All settings and the tuning of the gratings and the slits can be made via the computer and the stepper motors in the monochromator. The monochromator is built according to the Cross Czerny Turner principle (see Fig. 3.3a).The light source is imaged on the entrance slit with a concave mirror. In the monochromator, the entrance slit is imaged 1:1 onto the exit slit with 2 parabolic mirrors. Between these two mirrors the optical path is parallel. This configuration ensures full illumination of the grating independent in which position it is turned. This is important as the resolution of the grating is dependent on the numbers of grooves n that contribute to diffraction,

nz

(3.1)

with diffraction order z. A typical grating with 1200 grooves/mm and a size of 5 cm x 5 cm provides 6104 grooves. In first order this means that wavelengths around 600 nm can be separated when they differ by

nmmm

nz010101

10610600 11

4

9

.

(3.2)

However, to achieve this resolution for the experiment, we have to make the exit slit reasonably small to get only this narrow wavelength region through it. (A grating delivers only an angular dispersion. To translate this into a wavelength range on the exit slit we need to include the focal length of the monochromator.) This obviously also reduces the optical power transmission of the monochromator. Therefore a compromise is often made between high intensity and high resolution. In our case the resolution at 100 m slit width is about 0.3 nm (at a dispersion of 3.5 nm/mm). We use the monochromator as a tunable light source and are interested in high


power instead of high resolution. For a typical step width of 2 nm a slit width of ~1 mm is appropriate.

Fig.3.3a Optical path in the monochromator for the zero diffraction order. In Fig. 3.2b the grating is tuned such that the red part of the spectrum diffracted in the first order is entering the exit slit. The blue part of the spectrum is diffracted under a smaller angle and does not reach the exit slit. The blue and the red part of the spectrum are further separated the larger the focal length of the monchromator is.

Fig. 3.3b Optical path in the monochromator for the first diffraction order.

entrance slit

exit slit

turnable grating

Zero diffraction order

entrance slit

exi slit

tunable grating

First diffraction order

0 diff. o.

mirror

mirrorlamp


In Fig. 3.3c the grating is in a position that the red part of the spectrum in the first diffraction order is diffracted into the same direction as the second diffraction order of the blue part of the spectrum. In this case, red and blue light arrive at the exit slit. The reader should think at this point what could be done to solve this problem.

Fig.3.3c Optical path for 1st and 2nd diffraction order.

3.2.3 Optical fiber3

For the sample illumination an optical fiber with a core diameter of 50 m is used. This is a flexible method for the experiments under consideration. In particular, it is not necessary to place all components (from the lamp to the detector) within one optical axis. A 3-D translation stage at the end of the fiber allows for aligning the illumination spot on the sample. Another advantage is the very low damping of the fiber. A standard single mode fiber experiences a loss of only 0.2 dB/km at =1.55 µm. After 15 km the optical power is reduced by 50% (3 dB). a) Principle of operation and structure

The fundamental principle in optical fibers is the total internal reflection of light between a high index medium with n1 and a lower index medium with n2. For an angle of incidence > k (critical angle for total internal reflection) given by sin k = n2 / n1 (3.3) the optical beam is completely reflected (no part is transmitted). 3Nobel Prize in 2009 for Charles K. Kao for “groundbreaking achievements concerning the transmission of light in fibers for optical communication"

entrance slit

exit slittunable grating

First and second diffraction order

o. d

iff. o

.

mirror

mirror lamp


For < k the beam is refracted according to Snell’s law

1

2

nn

sinsin

(3.4)

with an emergent angle of < 90° as shown in Fig. 3.4. This angle reaches 90° for an angle of incidence of k.

Fig. 3.4 Optical path at the interface of a slow medium with index n1 to a fast medium with index n2 (with respect to the speed of light in the medium). The angle of incidence is , the emergent angle is , and k is the critical angle.

An optical fiber is made of a light guiding core and the surrounding cladding. For safe handling there are more layers around, a buffer made of acrylate polymer surrounded by strength fiber polymers (like Kevlar) and a plastic jacket with a color code (i.e. orange for a multimode fiber).

Fig. 3.5 Internal of an optical fiber. a) cross section b) longitudinal section. The core index is higher than the cladding index.

If the index of these fiber sections is constant and has a step like change at the interface it is called a “step-index”- or “SI”-fiber. Fig. 3.6 shows the beam path of such a (multimode) SI-fiber. The maximum angle of incidence max for coupling light into a fiber is given by the numerical aperture NA of the fiber N A = n0 sin max, (3.5) It can be calculated straightforward from the core and cladding indices and is given for a SI-fiber (Fig. 3.6) with k, by

NA 22

21k1k1max0 nncosn)90sin(nsinn (3.6)

Cladding

Cladding

Core

a) b)

n1 n2

n1 n2

n1 n2

n1 n2

refraction

total internal refraction

k k

0n

1n

2n

2n

Fig. 3.6 Optical path in a multimode SI-fiber with the angle of incidence and the reflection angle at the interface between core and cladding; indices of refraction n1 > n2.>n0.


For the 50 m-fiber used in the experiment, the maximum acceptance angle is max 12°. b) Further remarks

The simple picture from geometric optics is a good approximation as long as the core diameter is large compared to the wavelength (> 10 µm). For smaller core diameters we have to solve electro-magnetic wave propagation in dielectric materials by Maxwell’s equations. This is especially the case in a single mode fiber (the geometrical analog would be a perpendicular incoming beam which goes straight through the fiber without any reflection at all – which is a dubious picture). Another issue is the fiber dispersion which is important for optical data transmission. A short optical pulse carries a finite wavelength range and “normal” dispersion allows the red part to propagate faster than the blue part (as the index of refraction increases with shorter wavelengths). This leads to a (temporal) broadening of the pulse and requires a minimum temporal distance to the next pulse to be well resolved. In data communication this limits the maximum bit rate. Optical fibers show a minimum in dispersion at 1300 nm together with a local absorption minimum of ~0.5 dB/km which established to another commonly used wavelength range for high bit rate data transmission. Another local absorption minimum allows using also a window around 850 nm. Recently, plastic optical fibers emerged to an economic solution for short range fiber communication (i.e. for LAN or in cars) in the visible (~650 nm). They have a much larger core diameter (and a high NA) but the damping is typically in the range of 1 dB/m. Bending of the fiber is also an important issue. We can understand this with Fig. 3.7 in the optical ray picture. For a small radius of curvature the angle can get smaller than the critical angle and part of the beam gets lost in the cladding. In most cases these losses can be avoided if the bending radius is kept larger than 10 cm. Fibers have nowadays very wide areas of applications, mostly in fiber optic communications and fiber optical sensing (strain, temperature, pressure, magnetic field strength, rotation, etc.). Special fibers can be fabricated which are doped with rare earth elements, the most famous representative being the Erbium doped fiber amplifier (EDFA). This is used to “intrinsically” amplify signals within the fiber. This has become a key technology for long haul data communication systems.

Last not least, photonic crystal fibers (PCF) should be mentioned, which appeared to be a new class of fibers (similar to artificial semiconductor structures). The periodic two-dimensional arrangement of the core (made of glass/air) can form a bandstructure (with a bandgap) for light. Fig. 3.8 Micrograph of a photonic crystal fiber (from P. Russell, Science 209, 358 (2003))

r

1

2

Fig. 3.7 Optical path in the core of a bended fiber. The radius of curvature is r, the angles of incidence are , with and is the critical angle.


3.2.4 CCD-camera4

Contacting the sample would be very challenging just looking by eye. The core size of the fiber is 50 µm and the optical window on the device about 100 µm. We therefore use a CCD-camera to have an enlarged image on a TV-screen. CCD-sensors (CCD = charge coupled devices) are an array of individual light detectors with a line shift register. The vertical CCDs are clocked with the line frequency of the sensor. During the integration time carriers are created by the absorption of incident light. These carriers are shifted in the read-out CCD for serial read-out and image transmission to the screen.

3.2.5 Photodetector and power meter

For the spectral measurement of the light power two different semiconductor detectors are used for different wavelength ranges. They work as photon detectors and emit a photocurrent proportional to the number of detected photons. They are essentially a very similar structure as the p-i-n diode we are investigating, however made from different semiconductor materials. Photons therefore need a minimum energy to be absorbed at all (long wavelength cut-off). In the experiment a Ge-detector is used for the spectral range of 900 nm to 1700 nm and a Si-detector for 450 nm to 1020 nm. The responsivity R of a photodetector is defined by the ratio of generated photocurrent (i.e. the number of electron- hole pairs) and incident optical power (i.e. the number of incident photons),

chλeη

ωeη

PjR extext

Opt

Photo

(3.7)

The external quantum efficiency ext as defined in Eq. (2.25) is a measure how much optical power is converted into electron-hole pairs. It contains in particular losses due to reflection and transmission. To determine the optical power from the photocurrent, the wavelength has to be known and sent to the power meter.

3.2.6 Picoamp meter with integrated voltage source

The picoamp meter is a very sensitive current meter for the measurement of currents down to a few fA (= 10-15 A). It has two integrated voltage sources to allow for a wide variety of IV-measurements. A special feature is a protection circuit preventing damage of the device under test due to high currents (“current limit”).

3.2.7 Control software

All measuring devices and the monochromator have IEEE-interfaces to the computer to allow for a PC-controlled measurement and read-out of dat from the individual

4Nobel Prize in 2009 for Willard S. Boyle and George E. Smith “for the invention of an imaging semiconductor circuit – the CCD sensor”.


components. The control program is based on LabView for an easy control of the individual measurements.

3.2.8 Samples

A thin layer of semiconductor material would in principle be enough to measure the absorption of this semiconductor. However, if we want to apply an electric field and also measure photocurrent, we need samples with electrical contacts. In particular, p-i-n structures are well suited for that purpose. In our specific case we use a DH-p-i-n structure composed by a top p-AlGaAs layer, an intrinsic GaAs layer and a bottom n-AlGaAs layer. As discussed in section 2.8d, the important layer is the GaAs layer as the bandgap of the AlGaAs-layers is much higher. Spectral transmission and photocurrent measurements with applied electric field are easily accomplished by applying a voltage at the p and n contacts. Physically, the contacts are provided by metallic coatings on the n- and the p- doped layers. In the center of the contacts, an optical window is kept free for optical illumination. The actual sample layout deviates a little from our above simple description. It is shown in Fig. 3.9 for the DH-p-i-n structure.

Fig. 3.9 Layer composition and band profile of the DH-p-i-n sample.

The AlAs layer on top of the substrate is used as an etch stop layer for selective wet etching. This is necessary as the substrate has to be removed for transmission measurements. The sample structure starts on top of the AlAs-layer with an n-AlGaAs-layer. The i-layer is made of GaAs embedded in AlGaAs–layers with a higher bandgap. These “cladding” layers should prevent the accumulation of free carriers at the hetero junction and guarantee a constant filed in the (GaAs) i-layer (see section 2.8d). On top of the structure is the p-AlGaAs-layer. The actual thickness of the doped layers might be different and should be determined via the total thickness measurement by FPOs (see section 1.7). For the second part of the experiment, the measurement of the quantum confined Stark effect, a similar p-i-n structure is used. The intrinsic layer, however, is replaced by a multiple quantum well structure composed of a 50 periods of 10 nm GaAs (the

GaAs Substrat

i - AlAs - Buffer d=100nm

i - GaAs d= 500 nm

n - Al Ga As d=1000 nm

n=1*10 cm

0,4 0,6

18 -3

p - Al Ga As d= 500 nm

p=1*10 cm

0,4 0,6

18 -3

i - Al Ga As d= 100 nm0,4 0,6

i - Al Ga As d= 100 nm0,4 0,6

p - GaAs d= 10 nm+ ~20 kV/cm

~100 kV/cm


quantum wells) and 10 nm of AlGaAs (the barrier). The total thickness of GaAs is therefore identical with the DH-p-i-n sample but the total thickness of the i-layer is larger. The sample composition and band profile of the p-iMQW-n diode are shown in Fig. 3.10

Fig. 3.10 Layer composition and band profile of the p-iMQW-n sample.

a) Sample growth The sample are grown by molecular beam epitaxy (MBE) and processed in house. MBE and MOVPE (metal organic vapor phase epitaxy) are the most accurate methods for the epitaxial growth of semiconductor materials as they allow growth control down to sub-monolayer accuracy. The typical growth speed in such reactors is 1-2 monolayers per second (~1 µm / hour) but it can be even much slower if this is needed for sub-atomically precise growth control.

b) Sample design and processing After growth, the sample has to be structured and furnished with contacts. This is done by photolithography, wet chemical etching, and evaporation of materials (metals / dielectrics). In our special case we need to fabricate contacts to the n- and p-layers and to define a structure size. Furthermore, we need to place contact pads to allow a wiring or the contacting by probe needles. At the same time any shortcuts are to be prevented by an insulating layer and, finally, an anti-reflection coating is supplied. Prior to photolithographic processing, a photomask is designed with all the necessary structures and special alignment marks for adjusting the individual layers on top of each other. The process starts by spinning a (UV)-photosensitive resist on the sample. After a short thermal treatment, the sample is brought in a mask aligner which allows a very accurate alignment of the sample with respect to the photomask

GaAs Substrat

i - AlAs - Buffer d=100nm

i - GaAs d= 10 nm

i - Al Ga As d= 10 nm

n - Al Ga As d=1000 nm

n=1*10 cm

0,4

0,5

0,6

0,5

18 -3

i - Al Ga As d= 400 nm0,4 0,6

i - GaAs d= 10 nm

i - Al Ga As d= 10 nm0,5 0,5

p - Al Ga As d= 500 nm

p=1*10 cm

0,4 0,6

18 -3

p - GaAs d= 10 nm+

50 G

aAs/

AlG

aAs-

Dop

pels

chic

hte

n


and a subsequent exposure by UV-light. The exposure transfers the structure of the mask into the resist, as the exposed areas can be solved and removed in a developer solution (there exist also resists for the opposite process where the unexposed areas are removed). Now as the mask structure is transferred into the resist the respective processing step can be performed. This can be an etching to a certain depth or the evaporation of a metal in a “lift-off” process. For the latter the sample is completely evaporated by a metallic layer. The sample is subsequently dipped in a solvent (acetone) for the photoresist. The metal evaporated directly on the sample surface adheres, while the metal on the resist is removed with the solvent. For the complete sample structure as shown in Fig. 3.11 there are at least 7 processing steps necessary (p-contact, etching into the n-layer, n-contact, etching into the substrate, insulating layer, contact pads, ARC).

a) b)

Fig. 3.11 a) Sketch of all the layers of the photomask as they should appear on the device and b) an SEM image of a single device of the finished sample.

c) Removal of the substrate Finally, the DH-p-i-n sample needs to get rid of the substrate for transmission measurements. This final step has to be processed from the sample backside. A transmission window is photolithographically placed right in the center of the contacts. As the thickness of the (GaAs) substrate is much larger than the epitaxial sample layers, an “etch stop” layer is necessary to prevent the sample from being etched aswell. This is provided by the AlAs-layer which exhibits an etch rate for the used etchant that is orders of magnitude lower than the etch rate for GaAs. A cross section of the final sample is shown in Fig. 3.12.

Fig. 3.12 Sample cross section after processing and hole etching into the substrate.

Kontaktpad

Kon

takt

pad

Kon

taktpad

Kontaktpad

p p

n

n

500 m

250 m

250 m

1 mm1

mm

ARC-Fenster

Appendix

A. Material parameters of (Al)GaAs Bandgap of GaAs at 300 K Egap = 1.423 eV gap = 873 nm Effective masses (-point): me

* = 0.067 m0 mlh

* = 0.094 m0

mhh* = 0.34 m0

Bandgap of AlxGa1-xAs at 300 K:

1x0.45for0.45 ︶︵x1.1471.24

0.45x0forx1.24E ︵x ︶E GaAs

gapAsaGAl

gapx-1x

The refractive index nr of AlxGa1-xAs for an Al content from x = 0 to 0.3 is shown in Fig. A1. B. Fundamental physical constants c = 299 792 458 m/s Speed of light in vacuum e = 1.60219 10-19 As Elementary charge h = 6.6262 10-34 Js Planck’s constant = h/21.05459 10-34J/K

= 6.58218 10-16 eVs k = 1.380662 10-23 J/K Boltzmann constant

= 8.617343 10-31 kg m0 = 9.10956 10-31 kg Rest mass of the electron

Re

frac

tive

inde

x

Fig A1 Refractive indices nr of GaAs and AlxGa1-xAs with different Al-content

Bibliography

Harald Ibach and Hans Lüth Solid State Physics, An Introduction to Principles of Materials Science, Springer, Berlin 2003 Claus F. Klingshirn Semiconductor Optics, 4th ed. Springer, Berlin 2012 Peter Yu and Manuel Cardona Fundamentals of Semiconductors: Physics and Materials Properties 4th ed. Springer, Berlin 2010 John Davies The Physics of Low-Dimensional Semiconductors – An Introduction Cambridge University Press, New York 1998 Vladimir V. Mitin, Viatcheslav A. Kochelap, and Michael A. Stroscio Quantum Heterostructures – Microelectronics and Optoelectronics Cambridge University Press, New York 1999

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