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Charge Accumulation in a Conductor

### Spin and Charge Transport

#### 3) charge accumulation due to a difference between work functions of metals of MTJ electrodes. It does not depends either on the allied voltage or the current. Fig.1 A Capacitor. Charge is accumulated on the boundary of the dielectric and it does not influence the transport properties of the metal

When the charge is accumulated inside materials, the magnetic and electrical properties of material may be changed. The best example is a semiconductor. In the case when the charge is accumulated inside the semiconductor, its conductivity significantly changes and the semiconductor may change from conductor into dielectric and vice versa.

A tunnel junction consists of two metal electrodes and dielectric between them. Because of a capacitance of the tunnel junction, a charge is accumulated in tunnel junction when the voltage is applied between electrodes. As it is shown in Fig. 1, this charge is accumulated on boundary of the dielectric. The polarization of the dielectric may induce the spin-orbit interaction, which may cause the voltage -induced the magnetization reversal (See here)

When current flows through the tunnel junction, it induces a charge accumulation due to a change of the conductivity. , which is inside the metal. Therefore, it may influence the transport properties. The charge accumulation is within thickness in the metal, which equals about the mean-free path in the metal.

## 1. Capacitor.

##### Classical model. Assumption of a point-like electrns in a metal.   Fig.2 Electrical potential, electric field and charge distribution in a capacitor Fig.3 Number of electrons accumulated in a capacitor under applied voltage of 1 V, dielectric is MgO Fig.4 The density of accumulated electrons under applied voltage of 1 V. The mean-free path in the metal is 1 nm.

A capacitor consists of a dielectric inserted between two metals (Fig.1). In case when voltage is applied between metals, the charge is accumulated on the surface of dielectric.The charge accumulation is of opposite signs at two interfaces.

The charge accumulated at electrodes of a capacitor is proportional to the applied voltage and it can be calculated as where V is the applied voltage and C is the capacitance, q is the charge of an electron, N_accum is the number of the accumulated electrons

The capacitance of a parallel-plate capacitor can be calculated as where A is area of the plates and d is the distance between plates and epsilon is permittivity of the dielectric. Result from Fig. 3:

1) The charge accumulation sharply increases when the thickness of dielectric becomes thinner.

2) To compare: The charge accamulated at surface state at GaAs/metal contact is about 1E12-1E14 1/cm2. The charge accamulated at surface state at H-terminated Si/metal contact is about 1E8-1E10 1/cm2. The charge accumulated at sides of a capacitor (See Fig. 3) is comparable to the charge accumulated at a semiconductor-metal contact.

The classical model

In the classical model the electrons are assumed to be point-like particles. The charge accumulation in metal at contact with dielectric is assumed to be described by the delta function.

The distribution of electrical potential, electrical field and charge density along the capacitor are where epsilon and d is permittivity and thickness of the dielectric, , C is the capacitance and delta is the delta-function.

The presented model.

In the presented model the finite size of an electron in a metal is not ignored.

The size of an electron approximatly equals to the mean-free path.

Figure 4 shows the density of accumulated electron in the metal in the vicinity of the MgO tunnel barrier.

To compare: For example, the largest possible doping concentation in GaAs is 4-6 E19 1/cm3 and in Si 1-2 E20 1/cm3. It is comparible with a charge accumulation in a metal near contact of the capacitor (Fig.4)

### The ruduction of the capacitance due to a finite size of electrons in the metal

Screening, the capacitance reduction

The capacitance of the charge accumulated region in the metal is At frequancy f the resistance of the charge accumulated region in the metal

Tab

The electrical potential induced by a charge accumulation can be calculated the Gauss law. The Gauss law reads where fi is the electrical potential, q is the electron charge. In the case when the potential varies only the x-axis and the charge accumulation constant, in the case of the constant charge accumulation Eq. (3.3) is solved as Therefore the electrical potential in the charge accumulation region can be expressed as Since there is a potential drop in each metal, the effective voltage, which is applied to the dielectric, is reduced and it is calculated as Since equal amounts of charge are accumulated at each side of the capacitor, the Eq. (5.1) is simplified to or where Q is the charge accumulated at the capacitor

The effective and intrinstic capacitances can be defined as where the intristic capacitance of the capacitor, is the capacitance when the size of charge accumulation region in the metal can be ignored.

Using definitions Eq. (5.4), Eq. (5.3) can be written as the effective capacitance can be calculated as in the case of paralel-plate capacitor, the inntristic capacitance can be calculated as The capacitance due the finite size of the charge accumulation region in the metal can be calculated as The effective capacitance can be calculated asthe capacitance of two capacitors in series ## 2. Charge accumulation due to the different work functions of the metals

Fig. 11. Charge accumulation due the different work functions of the metals

### F The number of accumulated electrons in each metal  The mean free path and prmittiviy is the same for both metals and it equals to 9.8. Only work functions are different.

At contact of two metals the charge can be accumulated due to a difference of the work functions of the metals at sides of the contact.

The work functions is energy difference between vacuum level and the Fermi energy in the bulk of the metal. When two metals with different work functions are contacted, the electrons form the metals with smaller work functions (shallower Fermi level) flows into the metals with larger work function (deeper Fermi level). The accumulated charge makes the Fermi level the same in both metals. This type of the charge accumulation does not depend on a voltage applied to the contact or a current flowing through the contact.

The difference between work functions of metal 1 and 2 equals can be calculated as

The the density charge accumulation in the metals 1 and 2 in the vicinity of their contact where V_DWorkFun is difference of the work functions of metal 1 and 2, epsilon is permittivity and lambda_mean is the mean-free path.

The amount of the charge, which is accumulated at each side of the contact, is the same, but the charge accumulation is of the opposite signs. It is calculated as Conclusions from Figure 11:

1) The charge accumulation sharply increases when the mean-free path decreases. Particulary the decrease is substantial when the mean-free path is less than 2 nm

2) The amount of the charge accumulation is comparible with the charge accumulation, which forms in a Schottky barrier.

To compare: For example, the largest possible doping concentation in GaAs is 4-6 E19 1/cm3 and in Si 1-2 E20 1/cm3. It is comparible with a charge accumulation in a metal (Fig. 11 left)

To compare: The charge accamulated at surface state at GaAs/metal contact is about 1E12-1E14 1/cm2. The charge accamulated at surface state at H-terminated Si/metal contact is about 1E8-1E10 1/cm2. The charge accumulated at a metal-metal contact (See Fig. 11 right) is comparable to the charge accumulated at a semiconductor-metal contact.

Calculation of the charge accumulation due to the difference of the work functions of the metals. Eq. (3.7). Click to expand

In the case when the charge accumulation is constant within the charge accumulation region, the charge accumulation can be expressed as where lambda_mean is the mean-free path in the metal 1 and 2. n_accum,1 is number of accumulated electrons in metal 1 and 2

The negative charge accumulation is equals to the positive charge accumulation in the metal 1, because the charge accumulation is only due to the move of the electrons from metal 1 to the metal 2. This condition is expressed as The density of charge accumulation can be calculated from the Gauss law. The Gauss law reads where fi is the electrical potential, q is the electron charge. In the case when the potential varies only the x-axis, Eq. (3.3) can be solved as where A and B are constants. From Eqs. (3.4) and (3.2), the electrical potential fi in the vicinity of the contact can be expressed as The potential difference due to the charge accumulation should be equal to the difference between the work functions of metals 1 and 2. From Eq.(3.5) the difference between work functions of metal 1 and 2 equals. In the case when the permittivity is the same for both metals using Eq.(3.2) we obtain In the case when the mean free path is the same for both metals Eq. (3.7) is simplified to From the density of the accumulated electrons can be found as where it is assumed epsilon=9.8

The total number of the electron accumulated at the contact interface can be found from integrating Eq. (3.9) or where it is assumed epsilon=9.8

Fig. 12. Charge accumulation due the different work functions of the metals

as function of a ratio of mean-free paths in metals 1 and 2

## Accumulated charge at both sides of a metal-metal contact

At one side of contact it is positive. At another side of contact it is negative.

### The number of accumulated electrons in each metal   pemittiviy is the same for both metals. The mean-free path in the metal 1 is 5 nm.

Conclusions from Fig.12

1) The amount of the accumulated charge increases at both sides of the contact even if only in one metal the mean-free path decreases.

## The limitation of the classical model

#### In the descriptions of the charge accumulation in a metal within theclassical model hits its limits for charge accumulation at a contact. The clasical model assumes the electron as a point like particle and within the clasical model the destribution of charge accumulation in a metal is described as the delta-function.

However, this incorrect assumtion leads to clearly incorrect prediction.

The reduction of size electron to a point means From Fig. 11, approximation of the classical model Eq. (3.12) leads to the infinite density of charge accumulation (Fig.11 left) and it leads to the infinite charge accumulation. The infinite charge accumulation has no physical meaning.

The limitation of the classical model:

Ignoring the finite size of electron leads to the infinite charge accumulation at an metal-metal interface, which has no physical meaning.

## 3. Charge accumulation due to a gradient of conductivity

#### When a current flows through a contact between two metals of different conductivities, a charge accumulated in the vicinity of the contact. The charge accumulation is linearly proportional to the current. The charge accumulated in a region, which thickness approximately equals to the mean-free path in the metal.

The cases when the the charge accumulation occurs when a current flows through:

1) a contact between two metals;

2) a tunnel junction;

3) a Schottky contact and pn-junction

The charge accumulation induced by a current can be calculated from Gauss and Ohm laws: where q is the charge of an electron, n_accum is the number of the accumulated electrons, epsilon is the permittivity of the metal. where sigma is the conductivity of the metal.

Substitution of Eq. (2.2) into Eq.(2.1) gives when along the axis the current and permittivity are constants, but there is a gradient of the conductivity, the charge accumulation can be calculated as ### The classical model vs the presented model

In the classical model of the electron transport, the electron is assumed to be a point-like particle without sizes. This assumption allows for the charge accumulation to be a point like and be described by the Dirac delta function. Also, it allows for conductivity to have a step and be described by the Heaviside step function.

In the classical model the distribution of the conductivity in the vicinity of the contact is assumed to be step-like. From one side of the contact it is equal to the bulk conductivity of one metal. From another side of the contact, it equals to the conductivity of another metal.

In the present model the finite size of an electron is not ignored. The electron size approximately equals to the mean-free path. Therefore, the size of charge accumulation region can not be smaller than the the mean-free path.

The conductivity does not have a step at the contact. It continuously changes from one metal to another. From the bulk conductivity of the first metal the conductivity decreases in the vicinity of the contact, it has a minimum at the contact interface and it increases toward the bulk conductivity of the second metal.

## Contact between two metals

There are 3 sources of the charge accumulation at the contact between metals:

1) Charge accumulation due to the different work functions of the metals;

2) Charge accumulation due to the different bulk conductivities of the metals;

3) Charge accumulation due to the contact resistance

. Fig.2 Electrical potential, electric field and charge distribution along contact between two metals

Let us consider a contact between two metals, which have equal work function, but different conductivity. The distribution of electrical potential, electrical field and charge density along the contact are It should be notice the charge accumulation occurs only when there is a current through the contact.

3. Charge accumulation due to the contact resistance

## Tunnel Junction

As it is shown in Fig.4, in the case of tunnel junction, there are two different charge accumulations. One is in the metal and another one is in the dielectric. The charge accumulation in the dielectric occurs due to the capacitance of the junction. This accumulation does not affect the transport properties of the metals. The charge accumulation in the metal occurs due to the tunneling current. This accumulation might significantly affect transport properties of the metals. The similar charge accumulation in the metal can occur due to a difference of work functions of metals. It also does affect the transport properties of the metals. In the case when metals have equal work function, but different conductivity and the tunneling current through contact is J, the distribution of electrical potential, electrical field and charge density along the tunnel junction are Fig.4 Charge accumulated in a tunnel junction. Charge accumulated in dielectric (yellow) due to junction capacitance. Charge accumulated in metals (green) due to (1) the screening of charge in the dielectric; (2) due the difference of the work functions of metal electrodes and (3) the current flow where are the accumulated charge in metal 1 and metal 2, respectively.

It should be noticed that the charge accumulation in the metals is induced by the tunneling current J.