Your attempts do not work because:

• focused is a property driving the mat-focused class on mdInputContainer. You can use it to know whether your input is focused or not. You cannot use it to change the focus state.
• focus is a method on mdInput that lets you programmatically focus the input. You can call the myInput.focus() with myInput being something like ViewChild('myInput') for instance.

But the simplest way to achieve what you want is to use the standard autofocus attribute :

<md-input-container>
    <input mdInput type="text" name="a" placeholder="zzzz" autofocus>
</md-input-container>

Have you try with cdkFocusInitial which will specify the element that will receive focus upon initialization

documentation can be found here

<form class="example-form">
  <md-input-container class="example-full-width" style="width: 300px; padding: 5px; border-radius: 10px;">
    <input mdInput #emailInput="matInput" type="email" name="to" placeholder="Email">
    <md-error></md-error>
  </md-input-container>
</form>

Then in the controller:

@ViewChild('emailInput') searchInput: MatInput;
....
ngAfterViewInit() {
  this.emailInput.focus();
}

What this.emailInput.focus() does is actually "elementRef.nativeElement.focus()" https://github.com/angular/material2/blob/master/src/lib/input/input.ts#L287

so you could do that your self like:

this.emailInput.nativeElement.focus()