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PermalinkBrazilian Journal of Physics Teaching
Print version ISSN 1806-1117
Rev. Bras. Fís education. vol.23 no.2 São Paulo June 2001
http://dx.doi.org/10.1590/S1806-11172001000200004
Quantum dots: artificial atoms and atomic transistors
Quantum dots: artificial atoms and atomic transistors
Adenilson J. Chiquito
Department of Physics, Federal University of São Carlos
Rodovia Washington Luiz, Km 235, CP 676,
13565-905, São Carlos, São Paulo
Department of Physics, Federal University of São Carlos
Rodovia Washington Luiz, Km 235, CP 676,
13565-905, São Carlos, São Paulo
Received on 12/03/2001. Accepted on 18/04/2001
Quantum dots can be studied as macroscopic atoms artificially constructed. In this paper, we discuss some basic concepts related to the confinement of electrons in quantum dots, as well as some methods for obtaining it. A simple argument is developed in order to show some peculiarities and applications of quantum dots.The quantum dots can be seen the giant man-made atoms. In this work, some basic concepts related to the electron confinement in quantum dots are discussed. The process of fabrication is dots Also described. In addition, a simple discussion on the peculiar features and aplications of the quantum dots is presented.
I Introduction
Since 1947, after the demonstration of the effect transistor by J. Bardeen and W. Brattain on a germanium crystal [1], semiconductor materials have been responsible for numerous advancements, including the development of new technologies or in the area of basic science research. Many of these advances have occurred since the emergence of hybrid structures by 1970, calls heterostructures [2]. With the development and improvement of manufacturing techniques such as molecular beam epitaxy or by MBE (molecular beam epitaxy), made possible the growth of individual monoatômicas layers one after another, producing artificial lattices and almost perfect interfaces. Using these techniques and technology of crystal growth developed in the 80s and 90s, many new structures were produced in which the effects of quantization are fundamental. An example of these are the so-called quantum dots, where the electrons are confined in all three spatial directions and due to this characteristic, we often refer to them as giants [3] atoms. The production and development of quantum dots are closely linked to the optimization of existing electronic devices, the manufacture of new electronic devices and miniaturization of these.
II heterostructures and energy spectrum
A heterostructure is basically the combination of different semiconductor materials. When two different semiconductor gaps A and B are joined atomically as provided by epitaxial growth, for example, causes a discontinuity in-band energy of the resulting structure [4] which behaves like a potential well for movement of carriers in the growth direction of the structure, confining them, as illustrated in Figure One . In this we have a schema diagram of the energy bands in the two semiconductors (A and B) with different gaps, isolated [ Fig 1 (a)] and a junction at equilibrium [ Figure 1 (b) ]. At the junction AB, particles (electrons) migrate from one to another material to which the equilibrium condition given by the equality between the chemical potentials of both sides of the joint, m A = m B is achieved with the bending of the bands in the interface region. Thus, the interface have
And with AC and E CB representing the profile of the conduction band for the material on each side of the interface.
When the size of a system is comparable to the wavelength of de Broglie ( l B ), the movement of carriers becomes quantized, resulting in changes in the energy spectrum and the dynamic properties of the carriers [5, 6]. If only one direction is comparable to L B , it is said that the system behaves dynamically as a two dimensional system (or quantum wells). Limiting movement into two and then into three directions, one-dimensional systems (or quantum wires) will be obtained and zero-dimensional (or quantum dots) [7, 8].
To realize this is instructive to compare the wavelength of de Broglie associated with an electron and characteristic dimensions of a potential well of a typical heterostructure. For example, in InAs / GaAs system, the value E = 0.9 eV is typical for the confinement potential of electrons (in the conduction band discontinuity). The length of De Broglie associated is given by
Imagine now that we have built a structure of type ABA. In this case, the shape of the conduction band of the heterostructure ABA is as depicted in Figure 1 (c) . It is easy to recognize that we have a finite quantum well as those presented in the courses of quantum mechanics (at the junction AB type described above, were working with a roughly triangular well). Even more interesting, is that the characteristics of the well can be controlled externally: the well depth V 0 (discontinuity of the conduction band, D and C ) depends only on the values of the energy gaps of semiconductors chosen. Obviously, the pit width can also be varied. Recalling that the properties of the electronic levels allowed in a quantum well depend primarily on the physical characteristics of width and depth as well, then we have a versatile laboratory of Quantum Mechanics. We can describe the behavior of an electron in this structure using the Schrödinger equation
The presence of a quantum well changes the low energy spectrum to electrons inside the well and now only certain discrete energy values are allowed. This means that for each allowed state, the electrons can move freely in the direction x - y , but not in the direction z . The state corresponding to a given energy in this structure is given by
Based on the above considerations, for a potential that confine electrons in all three spatial directions or simply develop a quantum dot that limits the movement of electrons method also in directions x and y of the ABA structure. This is the subject of the next section.
III Quantum Dots - how to build them?
As mentioned previously, the presence of a quantum dot and sets a potential three-dimensional electronic movement is restricted in the three spatial directions x , y and z . The word "point" suggests a very small amount in the space to be occupied; nevertheless a quantum dot has millions of real and electrons associated with atoms. Most of the electrons is strongly linked to the atoms that form the structure of the quantum dot and only a few hundreds of them are free.
One of the manufacturing methods uses a material composed of three grown sequentially (ABA), wherein the electronic motion is confined to the growth direction of the frame (frame direction z ) as shown in the previous section. To limit the movement of electrons in the plane x - y are built so-called "tables" or islands by means of lithographic techniques. In this process, a mask pattern is deposited on the semiconductor and then the system + semiconductor mask is subjected to a chemical solution that defines the structure of tables, forming boxes or drums. When the desired dimensions for the device are small (of the order of 100 nm or less) is used by electronic lithography [9] bundles. Typically, the tables have a lateral dimension ( L x , y , z of about 100 nm - 50 nm [ Fig 2 (a) ].
There is another technique training of the islands through a process in which the quantum dots are built for themselves. Self-assembled InAs quantum dots embedded in a GaAs matrix are currently one of the systems studied, but other types can also be manufactured, for example, points of Ge embedded in a matrix of Si
There are three possible modes of epitaxial growth when the materials used are network parameters (spacing between atoms in the crystal lattice) different: the Frank-van der Merve mode, Stranski-Krastanow mode, and the Volmer-Weber [10] mode. At first, the growth of a material on another (as in a structure ABA) is made layer by layer, whereas in the latter, the system only becomes stable if the formation of islands or nucleation. The second case is an intermediate process in which there is complete formation of some layers (covering the entire surface of the substrate) to a critical thickness before the formation of the quantum dots. The self-organization of atoms and In The creating three-dimensional InAs islands occurs at the Stranski-Krastanow [11] mode. The details of the growth process are quite complex and not fully understood yet, but we can describe it simplistically as follows: the InAs and GaAs materials have lattice parameters that differ by approximately 7%. When it "grows" InAs on GaAs tension at the interface between the two materials [12] appears. This creates a force ( driving force ) that acts to "marry" local networks of GaAs and InAs, yielding three-dimensional InAs islands as shown inFig 2 (b) . In this sample we have about 10 10 per cubic centimeter quantum dots with a range of 10 to 25 nm in diameter [13] base. For the construction of any device, complete the structure of Figure 2 (b) , the growth of GaAs on another layer of InAs islands, resulting in an ABA structure with A = B = InAs and GaAs (now B represents islands and three-dimensional layers do not complete). From the standpoint of the total energy of the system, the formation of islands can be linked to the search for equilibrium with minimizing the total energy.
In summary, the self-assembled quantum dots are the result of forces caused by the attempted growth of materials with different network parameters.
So here we speak only of quantum dots built on semiconductors, but it is also necessary to remember the so-called metallic quantum dots, where small islands of aluminum or gold are introduced in insulating films as Al 2 O 3or Si 3 N 4 [14].
III.1 Energy spectrum - lithographic quantum dots and self-organized
The potential most appropriate to describe quantum dots fabricated by lithography can be cylindrical or cubic. On the other hand, the potential of self-organized quantum dots can be treated as pyramidal or hemispherical [Figure 2 (b) ].
Modeling quantum dots as three dimensional boxes, for example, we can determine the electronic states through the Schrödinger equation [Eq (3)], where the potential is defined by a box side L . Since we are interested in qualitative features, we consider that the potential is infinite outside the box and zero inside. The first step to solve Eq (3) with this potential is to try a separation of variables by writing the wave function as
Substituting eq. (6) in the Schrödinger equation [Eq (3)] and dividing the result by Y ( x , y , z ), we obtain
Since the function X ( x ) depends solely on x , change in y or z will not change the first term of Equation (7).Similarly, this consideration holds for y and z . We then three-dimensional differential equations of the type
The solution of each equation is quite straightforward (see, e.g., Ref [15]). For the variable x , we obtain:
To handle the case of self-organizing systems, we have to change the shape of the potential to then write the Schrödinger equation. Looking at Figure 1 (b) , we see that the self-organized system consists of hemispherical islands and the potential can be written as (remember a two-dimensional harmonic oscillator)
. The part where z is similar to that described in the previous case and hence the total energy state of an electron in a self-organized quantum dot is described by
Again, we obtain quantization of energy in three directions. Note that E n z refers to the energy quantization in the direction z , as in Eq (5). Both in the case of the potential box as hemispheric, an important feature can be observed: as in atoms, there is a well defined separation between the energy levels, but now, the whole system is "manageable". In a natural atom, the spectrum of energies almost can not be changed, primarily because it is determined by the interaction between the electrons and the periodic network between atoms and electrons. A quantum dot, in turn, may have its spectrum characteristic energy completely changed only if we change the geometry and composition of the materials that constitute (see equations above). Note that it is formed a structure sublevels as in natural atoms: it is often called the ground state of the quantum dots as state type s (n x n y n z = 100) and the other states as type p , d , f ... etc..
The characteristic size of a quantum dot can vary widely (e.g. from 5 nm to 100 nm) allowing the number of electrons contained in it vary from zero to tens or hundreds. This feature allows the study of the behavior of quantum depending on the number of electrons seeking, for example, points to determine if a system of a single electron is fundamentally different from a system with many electrons. Also with respect to the size of the quantum dots, it is possible the exploration of inaccessible regimes in atomic systems. As an example, it is possible to achieve a magnetic field through a quantum dot with a flow rate of 1 T per square micrometer in a laboratory. Now, to achieve the same flow through an atom would require a field of approximately 100 million T!
In the next section we review some properties and applications of quantum dots.
IV Some properties and applications of quantum dots
You can get lots of information about natural atoms and their properties by measuring the energies necessary to assign or remove electrons, which is usually done by spectroscopic techniques [17]. In artificial atoms is also very important to know these energies, but in this case, the information can be obtained by measuring the current through the quantum dot. Let us describe some properties of quantum related points with the occupation of the electronic states and the conduction of electrical current.
An always interesting property to be considered is the density of electronic states. This function describes the number of possible states in the conduction (or valence) band per unit energy and volume that can be filled with electrons. Will not be made here a rigorous derivation of the density of states, as found in Ref [18], but we can express it in the form:
), the density of states for the three dimensional case (homogeneous material), two-dimensional (eg, ABA structure), dimensional (quantum wire) and zero-dimensional (quantum dot) can be written as :
The use of low-dimensional systems in manufacturing electronic devices and opto-electronics is directly linked to the peculiar characteristics shown for example by the density of states. The closer the distribution of electronic states in energy, the smaller the influence of temperature effects and thus well defined separation between energy levels in a quantum dot, allows great selectivity in terms of energy. Quantum dots are therefore very favorable to the development of new device structures.
As we are talking about the occupation of the confined states with electrons, it is also interesting to discuss the processes of loading and unloading of quantum dots. These processes depend primarily on two characteristic energies: the one related to the Coulomb interaction and the other connected to the spatial confinement of the electrons. The net energetic effect of an electron moving in or out of a quantum dot, as in InAs / GaAs structure can be estimated by calculating the capacitance (or self-capacitance) system. Whereas a quantum dot can be represented by a sphere of radius R and recalling that Q = CV, the self-capacitance of the system is given by
, 0 are the relative permittivities of the InAs and absolute vacuum, respectively [19]). Then, the electrostatic energy involved is 
IV.1 The quantum transistor (Single Electron Transistor, SET)
Already in 1911, with his famous oil drop experiment, Millikan observed the effects of individual loads [20]. In solids, the first observation of the effects of tunneling of individual loads was performed in 1951 by Gorter [21].In a recent study, published in this journal [1], was given a brief historical description of the development of the transistor and also discussed the working principle of this device. Now, let's consider a more advanced version of transistors based on properties provided by electronic systems, when electrons are controlled individually.
Now imagine a device as shown in Fig 3 (a) and suppose that the discrete level in the quantum dot is already occupied by one electron. In Figure 3 (b) we can see the energy diagram versus distance showing the potential which an electron in the device shown in Fig 3 (a) is subject. Note that the region of the quantum dot, represented by only a finite quantum well in Figure 3 actually presents three-dimensional confinement for electrons.
To be current in the structure ( Fig. 3 ) it is necessary that the electrons Contact 1 (or 2) exceed the potential barriers, ie, the passage of electrons through the barrier contact 1 V1 for the quantized energy levels (quantum dot ), and then passing the quantized level for the contact 2 through the barrier V2. The presence of a fully quantized state between contacts 1 and 2, defined quantum dot, makes the transport of loads in very interesting structure because only one electron (two considering spin) can occupy discrete state within the quantum dot, as is energy described by the equations (11) and (14) and the density of states in Eq (19)One . When it tries to load quantum dots with electrons2 the electron-electron interaction, breaking the original degeneracy of the energy levels within the quantum dot and appears an energy interval (gap) and this new situation, no electrons can tunneling to or from the quantum dot as shown in the left side of Figure 3 (b) .
To transfer an electron to the quantum dot the energy described by Eq (21) is required. Neglecting thermal effects, the only power source is the battery V B . If the battery voltage is less than
This phenomenon enables some interesting applications, such as establishing a standard of current [22] or sensitive ammeters (electrometers) [23]. How can we detect and control the input or output of an electron quantum dot is possible to define two logical, states connected and disconnected , respectively: this is the principle on which are based many researchers currently devoting himself to building a computer scale atomic (quantum computer) and high speed [24].
Actually there is another way to induce electron tunneling in the case shown in Fig 3 (b) is the chemical potential of the quantum dots is increased D V Coul . For this, we introduce another terminal (B, base) and another battery (V G ) that are placed for optimal viewing in Fig 4 (a) . Over this terminal, complete what you might call a transistor of a single electron. Imagine now that V B < D V Coul structure, keeping V G = 0. Based on previous discussion that there is no current flowing between E and C or between C, E and B. If V L ¹ 0, the result is that the energy shift of the quantum dots (or in other words, change the chemical potential of the quantum dots) causing a resonance between the levels of E and B and the point, allowing the load or the downloading of the quantum dot for a electron. If the voltage V B is kept constant and satisfying the condition V B < D V Coul , the current through the terminals CE submit maximum representing the passage of electrons, one by one, through the quantum dot. In Figure 4 (b) we have measured current in the circuit as a function of V G : This behavior is analogous to the behavior of a common transitor [1].
The first SET transistors were fabricated by Fulton and Dolan [25] Kuzmin and Likharev and [26] in 1987. A transistor for applications such as this are quite large, due to its absolutely precise and rapid operation, such as memories and logic gates. The alternative use of self-organized structures for this purpose, such as InAs / GaAs or SiGe / Si system is quite promising due to the quantum dot thus obtained are free from defects. Additionally, the devices can be designed so as to minimize the influence of temperature in the occupation of quantum dots as proposed in a recent study [13].
At the end of this section, a question may arise: given the quantum nature of electrons, how is the Heisenberg Uncertainty Principle, when it says that it was possible to control an electron? To understand this, we should remember that we consider electrons moving in an electric field in a material and the resulting electrical current is determined by the load transferred to the material (wire). Here the charge is not important in a particular volume, but the amount of charge flowing through the wire or charge transfer. This charge is proportional to the sum of the displacements of all the electrons with respect to the atoms of the crystal lattice of the wire. Since electrons can be moved across the wire for distances as small as we please, the sum mentioned above can be changed continuously and therefore the charge transfer is continuous. If a junction similar to that of Figure 4 (a) is introduced into the wire, we have two behaviors for electrons: one is continuous, in which electrons accumulate on one side of the junction, producing a total charge Q, to be current through the circuit, electrons must pass through junction. Quantum mechanics shows that the charge Q can only decrease or increase (depending on the direction of current flow) through a discrete behavior, exactly where an electron is transferred through the point (remember the density of states in a zero-dimensional system) . Thus, at any time we determine the position and velocity of the electron and therefore the Heisenberg principle was not violated.
V Conclusion
In this paper some properties of quantum dots have been described in connection with concepts of quantization and familiar to undergraduate courses in physics electrostatics. Was also addressed in a simple way the idea of a transistor based on these structures. As transistors, which caused a revolution decades ago, quantum dots are promising candidates for a new big step in science and technology.
References
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1 If the quantum dots are replaced by a quantum well, although there is the process of tunneling, the situation is different because the density of states and the dispersion energy in the x and y directions can accommodate more electrons.
2 One possibility for this is a process of electron tunneling Terminal E (emitter) to the quantum dot. Tunneling is a quantum process absolutely, having no classical analog (for a basic description of tunneling, see Ref [5]).
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